Deformations of path algebras of quivers with relations
aa r X i v : . [ m a t h . QA ] D ec DEFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITHRELATIONS
SEVERIN BARMEIER AND ZHENGFANG WANG
Abstract.
Let A = k Q/I be the path algebra of any finite quiver Q moduloany two-sided ideal I of relations. We develop a method to give a concreteand complete description of the deformation theory of A via the combinatoricsof reduction systems. We introduce a new natural notion of equivalence forreduction systems and obtain a one-to-one correspondence between formal de-formations of A and formal deformations of any reduction system for A , upto equivalence. Moreover, we give criteria for the existence of algebraizationsof formal deformations and give a wide range of applications in algebra andgeometry. Contents
1. Introduction 12. Quivers, path algebras and quotients 43. Reduction systems and noncommutative Gr¨obner bases 54. Projective resolutions 135. Homotopy deformation retract 166. Deformation theory 207. Deformations of path algebras of quivers with relations 288. Application to deformation quantization 479. Algebraizations of formal deformations 5510. Applications to geometry 7411. Deformations of reconstruction algebras 92References 1001.
Introduction
In this article we develop a method to study the deformation theory of pathalgebras of quivers with relations. These algebras naturally appear in various guisesfor example in representation theory, commutative and noncommutative algebraicgeometry and mathematical physics. Prime examples would be
Mathematics Subject Classification.
Key words and phrases. path algebras of quivers, reduction system, deformation quantization,Gr¨obner–Shirshov bases, noncommutative algebraic geometry.Both authors were supported by the Max Planck Institute for Mathematics Bonn and theHausdorff Research Institute for Mathematics Bonn.The second author was also supported by a Humboldt Research Fellowship from the Alexandervon Humboldt Foundation and by a National Natural Science Foundation of China (NSFC) grantwith number 11871071. ( i ) commutative algebras such as the polynomial algebra k [ x , . . . , x n ]( ii ) graded associative algebras such as N -Koszul algebras or universal envelop-ing algebras of finite-dimensional Lie algebras( iii ) finite-dimensional algebras( iv ) endomorphism algebras of tilting bundles on algebraic varieties( v ) diagram algebras obtained from the structure sheaf of algebraic varietiesrestricted to a finite affine open cover( vi ) noncommutative resolutions of singularities.The deformation theory of associative algebras is a systematic study of variationsof the associative multiplication along a single or along multiple parameters. Froma representation-theoretic point of view deformations of A can be identified withdeformations of the module category Mod( A ) as Abelian category [87, 88]. In thecontext of mathematical physics deformations are one way to study quantizations.But also “classical” deformations in algebraic geometry fit into this framework: afamily of (classical) deformations of a (quasi)projective variety X corresponds to afamily of deformations of the diagram algebra O X | U !, which can be viewed as thepath algebra of a quiver with relations.Applied to the above list of examples, our methods can be used to study( i ) deformation quantizations of Poisson structures, in particular on affine n -space A n (see § ii ) Poincar´e–Birkhoff–Witt (PBW) deformations of graded associative algebras(see § iii ) associative deformations of finite-dimensional algebras (see § iv )–( v ) deformations of the Abelian category Qcoh( X ) of quasi-coherent sheaves ona variety X (see §§ vi ) deformations of singularities and of their noncommutative resolutions (see § A as a k -linear map A ⊗ k A A . It turns out that first-orderdeformations correspond to Hochschild 2-cocycles and two such first-order deforma-tions are equivalent precisely when they differ by a 2-coboundary so that first-orderdeformations up to equivalence are parametrized by the second Hochschild coho-mology HH ( A ) and obstructions to extending such deformations to higher orderlie in HH ( A ).A cornerstone of the theory is the Gerstenhaber bracket, which endows the(shifted) Hochschild cochain complex with the structure of a DG Lie algebra —which is precisely the structure used in the Maurer–Cartan formalism of deforma-tion theory via DG Lie and L ∞ algebras. This description is both powerful andelegant. However, the theory works with multilinear maps and with the “higherstructure” on the Hochschild complex Hom A e (Bar , A ), where Bar is the bar res-olution · · · A ⊗ k A ⊗ k A A ⊗ k A A EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 3 and in a way the size of the bar resolution is the main obstacle for obtaining aconcrete description of the deformations of A . Indeed, such a concrete descrip-tion is difficult to come by even in the cases of, say, finite-dimensional algebras orcommutative polynomial rings.When A can be written as the path algebra k Q/I of a quiver with relations,we study deformations of A via a so-called reduction system, a notion which wasintroduced by Bergman [22] to state and prove his Diamond Lemma (see § P , which can then be used in place of the bar resolution to study thedeformation theory of A . We give explicit comparison morphisms between Barand P (see §
5) so that the deformation theory of A can be studied using an L ∞ algebra structure on Hom A e ( P , A ), obtained from the DG Lie algebra structure onthe Hochschild complex Hom A e (Bar , A ) via homotopy transfer (see § ∞ algebra — which also turns out to naturally control deformations of thereduction system — gives an equivalent and surprisingly workable description of thedeformation theory of A . Indeed, the Maurer–Cartan equation for this L ∞ algebracan be checked via an often elementary combinatorial computation. Moreover, thesame combinatorial operation also describes the deformed multiplication and onecan often give a complete description of the deformations. We have given a numberof examples to illustrate this and hope the reader will find these examples bothinteresting and instructive.For the rest of this article we shall use the following notations, which we list herefor reference: k a field of characteristic 0 Q a finite quiver, possibly with loops and/or cycles k Q the associated path algebra I ⊂ k Q a two-sided ideal of relations A = k Q/I ⊗ = ⊗ k Q ¯ A = A/ ( k Q · A ) R = { ( s, f s ) } a reduction system satisfying ( ⋄ ) for IS = { s | ( s, f s ) ∈ R } S overlap ambiguities S k higher ambiguities( P , ∂ ) = ( A ⊗ k S ⊗ A, ∂ ) the projective resolution of A associated to R ( P , ∂ ) = (Hom A e ( P , A ) , Hom A e ( ∂ , A )) p ( Q, R ) the L ∞ algebra with underlying cochain complex ( P , ∂ ).1.1.
Quickstart guide.
Given A = k Q/I all formal deformations of A as associa-tive algebra can be constructed as follows. • Choose any reduction system R = { ( s, f s ) } satisfying ( ⋄ ) for I (see § • Candidates for deformations over a complete local Noetherian k -algebra( B, m ) with maximal ideal m are elements g ∈ Hom( k S, A ) b ⊗ m where S = { s | ( s, f s ) ∈ R } is often a finite set (see § EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 4 • Given g one can define a combinatorial star product ⋆ g (see § A if w ⋆ g ( w ′ ⋆ g w ′′ ) = ( w ⋆ g w ′ ) ⋆ g w ′′ where ww ′ w ′′ ∈ S is any overlapambiguity (see Def. 3.7).Algebraizations of formal deformations are discussed abstractly in § §§ × § e A n ( § A n ( § § k (1 ,
1) surface singularities and their resolutions( § Quivers, path algebras and quotients
Quivers. A quiver Q consists of a set Q of vertices and a set Q of arrowstogether with source and target maps s , t : Q Q assigning to each arrow x ∈ Q its source and target vertices s( x ) and t( x ), respectively. A quiver Q is called finite if Q and Q are finite sets. All quivers in this article are finite quivers.
To each vertex i ∈ Q we associate a path of length e i with s( e i ) = i = t( e i ). For n ≥ path of length n in Q is a sequence p = p p · · · p n of n arrowswith t( p i ) = s( p i +1 ) for 1 ≤ i < n and we sometimes denote the path length of anarbitrary path p by | p | . We denote the set of paths of length n by Q n and write Q ≥ N = S n ≥ N Q n for the set of paths of length ≥ N and also Q for the set Q ≥ of all paths. For p = p p · · · p n , we call s( p ) the source of p , denoted by s( p ), andcall t( p n ) the target of p , denoted by t( p ). The paths p k · · · p l for 1 ≤ k ≤ l ≤ n arecalled subpaths of p .An arrow x ∈ Q with s( x ) = t( x ) is called a loop and a path p of length ≥ p ) = t( p ) is called a cycle . A finite quiver Q is called acyclic if it contains noloops or cycles in which case Q n = ∅ for n ≫ path algebra k Q = L n ≥ k Q n has the set of all paths as a k -basis and theproduct pq of two paths p and q is defined to be their concatenation if t( p ) = s( q )and zero otherwise. The paths of length 0 are orthogonal idempotents (i.e. e i e j = 0for i = j and e i = e i ) satisfying e s( p ) p = p = pe t( p ) for each path p and their sum P i ∈ Q e i = 1 k Q is the identity element of the path algebra k Q .Two paths p, q are said to be parallel if t( p ) = t( q ) and s( p ) = s( q ) and for alinear combination of paths f = P k λ k q k ∈ k Q , we say that p is parallel to f if p isparallel to q k whenever λ k = 0.The subspace k Q ≃ k ×· · ·× k is a subalgebra of k Q and k Q is a k Q -bimodule.The path algebra k Q is isomorphic to the tensor algebra of k Q over k Q k Q ≃ T k Q ( k Q ) = k Q ⊕ M n ≥ ( k Q ) ⊗ k Q n . The path algebra k Q can be written as a matrix algebra k Q = ( k e i Qe j ) ij where e i Qe j is the set of paths from i to j and the multiplication is given by matrixmultiplication.Given a two-sided ideal I ⊂ k Q , the algebra A = k Q/I is called a path algebraof a quiver with relations . EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 5 Q ( i )( ii )( iii ) x ,..., x n ... I ( x j x i − x i x j ) ≤ i Three examples of quivers with relations.Three simple examples of quivers with relations are given in Fig. 1, where thequotient algebras k Q/I are the following( i ) the polynomial algebra on n variables k [ x , . . . , x n ]( ii ) the endomorphism algebra of the tilting bundle O ⊕ O (1) ⊕ O (2) on P ( iii ) a noncommutative resolution of the toric k (1 , 1) singularity.(Deformations of these particular algebras will be described in § § 10 and § Q is acyclic, then k Q is finite-dimensional, and so is k Q/I for any two-sided ideal I (e.g. Fig. 1 ( ii )). We generally allow the quiversto have loops and cycles, in which case quotients of their path algebras are finitelygenerated, but may also be infinite-dimensional over k (e.g. Fig. 1 ( i ) and ( iii )).3. Reduction systems and noncommutative Gr¨obner bases The notion of a reduction system introduced by Bergman [22] is central to ourdescription of deformations. It formalizes the choice of a k -basis for the path algebra A = k Q/I of a quiver Q with ideal of relations I and deformations of A may thenbe described in this basis. For example, given a path s in Q which appears in arelation for I , we have that s − f s ∈ I for some linear combination of paths f s ∈ k Q and s = f s in A . A reduction system is a collection of pairs ( s, f s ), where s willbe “reduced” to f s . (And deformations of A may then be given by reducing s to f s + g s .)The choice of a reduction system will give a k -vector space basis of A consistingof “irreducible” paths (see the Diamond Lemma 3.8). Following Chouhy–Solotar[40] the combinatorics of the reduction system can be used to construct a projective A -bimodule resolution of A (see § A (see § Reduction systems.Definition 3.1 (Bergman [22, § . A reduction system R for k Q is a set of pairs R = { ( s, f s ) | s ∈ S and f s ∈ k Q } EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 6 where • S is a subset of Q ≥ such that s is not a subpath of s ′ when s = s ′ ∈ S • for all s ∈ S , s and f s are parallel and s = f s • for each ( s, f s ) ∈ R , f s is irreducible, i.e. it is a linear combination ofirreducible paths.Here a path is irreducible if it does not contain elements in S as a subpath and wedenote by Irr S = Irr S ( Q ) = Q \ Q S Q the set of all irreducible paths.Given a two-sided ideal I of k Q , we say that a reduction system R satisfies thecondition ( ⋄ ) for I if( i ) I is equal to the two-sided ideal generated by the set { s − f s } ( s,f s ) ∈ R ( ii ) every path is reduction finite and reduction unique (see Definition 3.3 below).We call a reduction system R finite if R is a finite set. Remark . It follows from the definition that a reduction system R = { ( s, f s ) } is uniquely determined by the set S ⊂ Q ≥ together with f ∈ Hom k Q e0 ( k S, k Irr S ),where k Q e0 = k Q ⊗ k k Q op0 is the enveloping algebra of k Q .Let ( s, f s ) ∈ R and let q, r ∈ Q = S n ≥ Q n be two paths such that qsr = 0 in k Q . Following [40, § 2] a basic reduction r q,s,r : k Q k Q is defined as the k -linearmap uniquely determined by the following: for any path p ∈ Q r q,s,r ( p ) = ( q f s r if p = qsrp if p = qsr. A reduction r is defined as a composition r q ,s ,r ◦ r q ,s ,r ◦ · · · ◦ r q n ,s n ,r n of basicreductions for some n ≥ S and so one may obtain differentelements in k Q after performing different reductions. Definition 3.3. We say that a path p ∈ Q is • reduction finite if for any infinite sequence of reductions ( r i ) i ∈ N there exists n ∈ N such that for all n ≥ n , we have r n ◦ · · · ◦ r ( p ) = r n ◦ · · · ◦ r ( p ) • reduction unique if p is reduction finite and moreover for any two reductions r and r ′ such that r ( p ) and r ′ ( p ) are both irreducible, we have r ( p ) = r ′ ( p ).Note that an element a ∈ k Q is irreducible if and only if r ( a ) = a for all reductions r . The combinatorics of reductions is described in § Examples 3.4. ( i ) Monomial algebras. Let A = k Q/ ( S ) be a monomial al-gebra where ( S ) is the two-sided ideal generated by the set S ⊂ Q ≥ ofminimal relations, i.e. s is not a subpath of s ′ when s = s ′ ∈ S . Any pathin the ideal ( S ), i.e. any path containing an element of S as a subpath, isequal to 0 in A and R = { ( s, | s ∈ S } is a reduction system satisfyingthe condition ( ⋄ ) for the ideal ( S ). In this case, it is clear that the set ofirreducible paths forms a k -basis of A (even without invoking the DiamondLemma 3.8 below). EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 7 ( ii ) Polynomial algebras. Let Q be the quiver with one vertex and n loops x , . . . , x n and let I = ( x j x i − x i x j ) ≤ i An example of reductions in a reduction system for thepolynomial algebra.Reductions can be described systematically by defining maps g split ∗ : k Q k Q ⊗ k S ⊗ k Q ∗ ∈ { R , L , ∅ } by g split ( p ) = X s ∈ Sqsr = p p ⊗ s ⊗ r g split R2 ( p ) = q ⊗ s R ⊗ r g split L2 ( p ) = q ⊗ s L ⊗ r (3.5)where s R (resp. s L ) is the right-most (resp. left-most) subpath of p which lies in S as illustrated in Fig. 3. (In § ∗ by passing to the quotient A = k Q/I on the first and last factor of the tensor product, whence the notation f here, as well as “higher analogues” of this map denoted by split ∗ n , whence theindex here.)The map g split R2 records the right-most subpath lying in S and can be used todefine the right-most reduction , which is the k -linear mapred f : k Q k Q EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 8 ∈ k Qp = ∈ S ∈ S ∈ S g split R2 ( p ) = ∈ k Q ⊗ S ⊗ k Q ⊗ ⊗ ∈ k Q ⊗ S ⊗ k Q g split L2 ( p ) = ⊗ ⊗ Figure 3. An illustration of the maps g split L2 and g split R2 .given by red f ( p ) = ( qf s r if p is a path such that g split R2 ( p ) = q ⊗ s ⊗ rp if p is an irreducible path(3.6)and will be used to define a combinatorial star product in § R is reduction finite, then for every path p we have red Nf ( p ) = red N +1 f ( p ) for N ≫ Nf ( p ) is irreducible for N ≫ ( ∞ ) f : k Q k Irr S as red ( ∞ ) f ( p ) = red Nf ( p ) for N ≫ . (This definition can also be recovered from a more general construction by taking z = 0 in (7.17).)We will now recall Bergman’s Diamond Lemma , which shows that a reductionsystem R satisfying ( ⋄ ) for an ideal I ⊂ k Q will give rise to a k -basis of A = k Q/I . Definition 3.7 [22, § . Let R be a reduction system for k Q . A path pqr ∈ Q ≥ for p, q, r ∈ Q ≥ is an overlap ambiguity (or simply overlap ) of R if pq, qr ∈ S .We say that an overlap ambiguity pqr with pq = s and qr = s ′ is resolvable if f s r and pf s ′ are reduction finite and r ( f s r ) = r ′ ( pf s ′ ) for some reductions r , r ′ . Theorem 3.8 (Diamond Lemma) [22, Thm. 1.2] . Let R be a reduction system for k Q , let I = ( s − f s ) ( s,f s ) ∈ R ⊂ k Q be the corresponding two-sided ideal and denoteby A = k Q/I the quotient algebra. If R is reduction finite, then the following areequivalent: ( i ) all overlap ambiguities of R are resolvable ( ii ) R is reduction unique, i.e. R satisfies ( ⋄ ) for I ( iii ) the image of the set Irr S of irreducible paths under the projection π : k QA forms a k -basis of A . (We will give a deformation-theoretic interpretation of the Diamond Lemma in § Remark . If R satisfies ( ⋄ ) for the ideal I = ( s − f s ) ( s,f s ) ∈ R , then it follows fromthe Diamond Lemma that there is an k Q -bimodule isomorphism A ≃ k Irr S so thatwe may also identify Hom k Q e0 ( k S, k Irr S ) ≃ Hom k Q e0 ( k S, A ) (cf. Remark 3.2).Note also that in this case red ( ∞ ) f does actually not depend on the reductionschosen.The Diamond Lemma also implies that there exists a unique k -linear map(3.10) σ : A k Q EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 9 which is a section of π , i.e. πσ = id A , and moreover satisfies σπ ( p ) = p for all paths p ∈ Irr S . Note that σπ ( s ) = f s for any ( s, f s ) ∈ R .The following result shows that the deformation theory developed in later sectionsvia reduction systems can be applied to any algebra A = k Q/I and the choice of areduction system can be thought of the choice of a basis in which the deformationscan be described explicitly. Proposition 3.11 (Chouhy–Solotar [40, Prop. 2.7]) . If I ⊂ k Q is an ideal, thenthere exists a reduction system R satisfying the condition ( ⋄ ) for I .Remark . In fact, Chouhy–Solotar [40] give a version of Buchberger’s algorithm(used to compute commutative and noncommutative Gr¨obner bases) for construct-ing such a reduction system. They also point out that there may be other naturalchoices of reduction systems which cannot be obtained from this algorithm (see [40,Ex. 2.10.1] for an example). Definition 3.13. Let MC ⊂ Hom( k S, k Irr S ) denote the subset of elements whoseassociated reduction systems (cf. Remark 3.2) are reduction finite and reductionunique. Given any f, f ′ ∈ MC, let R and R ′ denote the corresponding reductionsystems. We say that R and R ′ (or f and f ′ ) are equivalent if there exists a k -linearautomorphism T ∈ GL( k Irr S ) satisfying T ( u ) = red ( ∞ ) f ′ ( T ( u ) · · · T ( u m )) u = u · · · u m (3.14)for all irreducible paths u ∈ Irr S such that for every s ∈ Sf s = T − (red ( ∞ ) f ′ ( T ( s ) · · · T ( s n ))) s = s · · · s n . (3.15)Let G ( f, f ′ ) denote the set of all k -linear automorphisms of k Irr S satisfying(3.14). Then we have a groupoid G MC with T ∈ G ( f, f ′ ) acting by theformula (3.15) and R and R ′ are equivalent precisely when G ( f, f ′ ) = ∅ . Remark . Note that T ∈ G ( f, f ′ ) is determined by its value on arrows andmay be viewed as a map in Hom( k Q , k Irr S ). Also note that red ( ∞ ) f ′ is precisely theproduct in the algebra k Q/ ( s − f ′ s ) so that T induces an algebra isomorphism k Q/ ( s − f s ) ≃ k Q/ ( s − f ′ s ). In other words, an equivalence of reduction systems induces anequivalence (isomorphism) of associative algebras. We will see the formal version ofthis in § Reduction systems from noncommutative Gr¨obner bases. In the theory ofnoncommutative Gr¨obner bases (also called Gr¨obner–Shirshov bases), an admissibleorder ≺ is a total order on Q = S n ≥ Q n such that( i ) every non-empty subset of Q has a minimal element, and( ii ) for any elements p, q, r ∈ Q the following conditions hold • if p ≺ q then pr ≺ qr whenever pr and qr are non-zero • if p ≺ q then rp ≺ rq whenever rp and rq are non-zero • and q (cid:22) pqr whenever pqr is non-zero. Later MC will be seen to be the set of Maurer–Cartan elements for the L ∞ algebra p ( Q, R )controlling the deformation theory of the reduction system R . EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 10 Fix an admissible order ≺ on Q . Let F = P p ∈ Q λ p p be a non-zero element ofthe path algebra k Q , where λ p ∈ k and almost all λ p are zero. Then we definetip ≺ ( F ) = tip( F ) := p if λ p = 0 and p ≻ q for all q with λ q = 0 . If X ⊂ k Q then let tip( X ) = (cid:8) tip( F ) | F ∈ X \{ } (cid:9) .Furthermore, an element F ∈ k Q is called uniform if it is a linear combinationof parallel paths. Definition 3.17. Let I be an ideal of k Q and let ≺ be an admissible order on Q .The (reduced) noncommutative Gr¨obner basis for I with respect to ≺ is the set G of uniform elements in I such that h tip( I ) i = h tip( G ) i and such that the coefficient λ tip( F ) = 1 for any F ∈ G and tip( F i ) is not a subpathof tip( F j ) when F i = F j ∈ G , where h X i denotes the two-sided ideal generated by X .For any fixed admissible order ≺ the (reduced) noncommutative Gr¨obner basisis unique (see e.g. [64, Def. 3.2]).Let { F j } j ∈ J be a noncommutative Gr¨obner basis for an ideal I ⊂ k Q (withrespect to an order ≺ ). Then one immediately obtains a reduction system(3.18) R = (cid:8)(cid:0) tip( F j ) , − F j + tip( F j ) (cid:1)(cid:9) j ∈ J satisfying ( ⋄ ) for I .So every noncommutative Gr¨obner basis gives a reduction system, but converselynot every reduction system is obtained from a Gr¨obner basis (cf. [40, Ex. 2.10.1]).In the commutative case, any ideal admits a finite commutative Gr¨obner basiswhich may be computed by Buchberger’s algorithm. For ideals in noncommuta-tive algebras a similar algorithm computes a noncommutative Gr¨obner basis andthis algorithm terminates if and only if the ideal admits a finite Gr¨obner basis.However, the property of admitting a finite Gr¨obner basis is undecidable (see e.g.[25, 95, 96]). That said, noncommutative Gr¨obner bases have been computed fora growing number of noncommutative algebras — see e.g. [24, 35] for examplesincluding Weyl algebras, Iwahori–Hecke algebras, quantum groups, Ore extensions,universal enveloping algebras of finite-dimensional Lie algebras, [114] for preprojec-tive algebras, and also [29] for noncommutative Gr¨obner bases in the more generalsetting of algebras over an operad. All of these can be used to describe the defor-mations of these algebras using the deformation theory via reduction systems asdeveloped in the present article.A basic example is the Gr¨obner basis for the polynomial algebra. Example 3.19. Let R = { ( x j x i , x i x j ) } ≤ i A 4-ambiguity of five overlapping elements in S .3.2. Higher ambiguities. Let Q be a finite quiver. Let S be any subset of Q ≥ such that s is not a subpath of s ′ when s = s ′ ∈ S . We now define n -ambiguitiesfor n ≥ Definition 3.20. Let p ∈ Q ≥ be a path. If p = qr for some paths q, r we call q a proper left subpath of p if p = q .Now let n ≥ 0. A path p ∈ Q is a (left) n -ambiguity if there exist u ∈ Q andirreducible paths u , . . . , u n +1 such that( i ) p = u · · · u n +1 ( ii ) for all i , u i u i +1 is reducible, and u i d is irreducible for any proper left subpath d of u i +1 .The notion of right n -ambiguity may be defined analogously (as p = v · · · v n +1 with v n +1 ∈ Q and v , . . . , v n irreducible paths satisfying analogous conditions),but these notions turn out to be equivalent.The set of n -ambiguities can be visualized as “overlaps” of n + 1 elements in S ,as illustrated in Fig. 4 where we have also illustrated the paths appearing in thedefinition of left or right n -ambiguity. (Note that for 0 < i ≤ n the overlap of u i and v i is exactly the overlap of the elements in S .)Now set S = Q S = Q S = S and let S n +2 ⊂ Q ≥ n for n ≥ n -ambiguities.When S ⊂ Q , then overlaps can have no “gaps” and can thus be visualized asfollows ∈ S ∈ S ∈ S ∈ S ∈ S so that in this case S n = (cid:8) x · · · x n ∈ Q n (cid:12)(cid:12) x i ∈ Q for all i and x i − x i ∈ S (cid:9) (cf. [40, Prop. 3.4]). Remark . The definition of n -ambiguities was first given in Anick [4] under thename of “( n + 1)-chain”. This indexing might create confusion in the context of thisarticle since in § a ⊗ w ⊗ b for w ∈ S n +2 will be ( n +2)-chainsin a projective A -bimodule resolution of A (see Theorem 4.1 and Proposition 4.2). EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 12 Although our numbering also does not agree with the degree of the resultingelement in the chain complex, we have that a 0-ambiguity is just an element in S (performing reductions is unambiguous) and a 1-ambiguity is an overlap ambiguity(see Definition 3.7).3.2.1. The combinatorics of reductions. We now study the combinatorics of reduc-tions. The maps defined here will be used to give a recursive formula for thedifferential in the projective resolution of Theorem 4.1 and later to give an explicitdescription of the deformations in § ∗ n , δ n and γ n for any n ≥ k Q A ⊗ k S n ⊗ A A ⊗ k S n − ⊗ A split ∗ n δ n γ n ∗ ∈ { ∅ , R , L } by split n ( p ) = X w ∈ S n qwr = p π ( q ) ⊗ w ⊗ π ( r ) split R n ( p ) = π ( q ) ⊗ w R ⊗ π ( r )split L n ( p ) = π ( q ) ⊗ w L ⊗ π ( r )(3.22)where w R (resp. w L ) is the right-most (resp. left-most) subpath of p which is anelement of S n .Define δ n as the A -bimodule morphism determined by δ n (1 ⊗ w ⊗ 1) = π ( w ) ⊗ − ⊗ π ( w ) if n = 1split R n − ( w ) − split L n − ( w ) if n > n − ( w ) if n ≥ γ n as the morphism of left A -modules determined by γ n (1 ⊗ w ⊗ π ( u )) = ( − n split n ( wu ) for any u ∈ Irr S .We will use the following identities for n ≥ γ n γ n − = 0 γ n δ n γ n = γ n where we set γ ( a ) = a ⊗ a ∈ A .The first identity in (3.23) can be seen to hold since for any w ∈ S n − and anyirreducible path u the path wu never contains elements in S n as a subpath. FromSk¨oldberg [117, Thm. 1] it is not difficult to show γ n = γ n γ n − δ n − + γ n δ n γ n sothat the second identity in (3.23) follows then directly from the first.We make repeated use of the map split : k Q A ⊗ k Q ⊗ A which for p = p · · · p m with p i ∈ Q is given bysplit ( p ) = 1 ⊗ p ⊗ π ( p · · · p m ) + P
The Hochschild cohomology of an associative algebra A can be calculated fromany projective A -bimodule resolution of A . We recall the definition of the standardand the normalized bar resolutions ( § § Bar resolutions. The bar resolution of an associative k -algebra A is given by · · · A ⊗ A ⊗ n ⊗ A · · · A ⊗ A ⊗ A A ⊗ A A ⊗ = ⊗ k .For A = k Q/I one can consider the ( k Q -relative ) normalized bar resolution Bar · · · A ⊗ ¯ A ⊗ n ⊗ A · · · A ⊗ ¯ A ⊗ A A ⊗ A A ⊗ = ⊗ k Q and ¯ A = A/ ( k Q · A ) is the quotient k Q -bimodule. We shall refer to Bar simply as “the bar resolution” (cf. Remark 5.14).The differential d n : Bar n Bar n − is given by d n ( a ⊗ ¯ a ...n ⊗ a n +1 ) = a a ⊗ ¯ a ...n ⊗ a n +1 + n − X i =1 ( − i a ⊗ ¯ a ...i − ⊗ a i a i +1 ⊗ ¯ a i +2 ...n ⊗ a n +1 + ( − n a ⊗ ¯ a ...n − ⊗ a n a n +1 , where for i ≤ j we have written ¯ a i...j to denote ¯ a i ⊗ · · · ⊗ ¯ a j ∈ ¯ A ⊗ ( j − i +1) . It is wellknown that Bar is a projective A -bimodule resolution of A (see e.g. [85, Ch. 1]).4.2. Projective resolutions from reduction systems. For A = k Q/I , a reduc-tion system R satisfying ( ⋄ ) for I gives rise to a much smaller resolution that uses theset of n -ambiguities (see § R = { ( s, f s ) } , replacing subpaths which lie in S = { s | ( s, f s ) ∈ R } by linear combinations of irreducible paths. (Under certain conditions this resolu-tion can sometimes be shown to be minimal, see [40, Thm. 8.1].) Theorem 4.1 (Chouhy–Solotar [40, § . Let A = k Q/I , let R be a reductionsystem satisfying ( ⋄ ) for I , let S m = Q m for m = 0 , and let S n +2 for n ≥ denote the set of n -ambiguities. ( i ) There is a projective A -bimodule resolution P of A · · · ∂ n +1 P n ∂ n P n − ∂ n − · · · ∂ P ∂ P ∂ A where P n = A ⊗ k S n ⊗ A and the augmentation map ∂ : A ⊗ A A isgiven by the multiplication of A . ( ii ) For each n ≥ , there is a homomorphism of left A -modules ρ n : P n − P n , where we set P − = A , such that for any n ≥ ∂ n ρ n + ρ n − ∂ n − = id P n − and ρ n +1 ( a ⊗ w ⊗ 1) = 0 for any w ∈ S n and a ∈ A . Here we set ∂ − = 0 and ρ − = 0 . The actual maps in the resolution P of Theorem 4.1 shall be useful later, so wegive explicit formulae for ∂ n and ρ n using the maps γ n and δ n defined in § EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 14 Proposition 4.2. Let A = k Q/I and let R be a reduction system satisfying ( ⋄ ) for I . Then the differential ∂ and the homotopy ρ in Theorem 4.1 can be definedfrom ∂ ( a ⊗ b ) = ab and ρ ( a ) = a ⊗ by the following recursive formulae for n ≥ ∂ n ( a ⊗ w ⊗ b ) = a (cid:0) (id − ρ n − ∂ n − ) δ n (cid:1) (1 ⊗ w ⊗ b a, b ∈ A, w ∈ S n ρ n = γ n + P i ≥ ( γ n δ n − γ n ∂ n ) i γ n . Note that the sum in the formula for ρ n is well defined since the terms appearingin the sum are performing reductions and all elements are by assumption reductionfinite. Proof. We first claim that the formula for ρ n can be rewritten as ρ n = γ n + P i ≥ γ n (id − ρ n − ∂ n − − ∂ n γ n ) i . Indeed, by γ n γ n − = 0 in (3.23) we get that γ n ρ n − = 0 and thus γ n (id − ρ n − ∂ n − − ∂ n γ n ) i = γ n (id − ∂ n γ n ) i = γ n ( δ n γ n − ∂ n γ n ) i where the second identity uses γ n δ n γ n = γ n in (3.23). This proves the claim.We now prove that ∂ n is a differential, i.e. that ∂ n ∂ n +1 = 0, and ρ n is a homotopy,i.e. ∂ n ρ n + ρ n − ∂ n − = id. (Note that this implies that ( P , ∂ ) is exact.) The proofproceeds by induction on n . For this, it is clear that ∂ ∂ = 0 and ∂ ρ + ρ − ∂ − =id, where we set ρ − = 0. Assume that this holds for all i ≤ n − 1. We need toprove it holds for n . For simplicity, we set C n := id − ∂ n γ n − ρ n − ∂ n − . so that ρ n = γ n + P i ≥ γ n C in . Note that ∂ n − C n = 0 since ∂ n − C n = ∂ n − − ∂ n − ρ n − ∂ n − − ∂ n − ∂ n γ n = ∂ n − − (id − ρ n − ∂ n − ) ∂ n − = 0where by the induction hypothesis we have ∂ n − ∂ n = 0 = ∂ n − ∂ n − .Let us prove that ( ∂ n ρ n + ρ n − ∂ n − )( y ) = y for any y ∈ P . Since C in ( y ) = C n · · · C n ( y ) = 0 for i ≫ 0, there exists m ≥ y ) such that C mn ( y ) = 0but C m − n ( y ) = 0. Note that( ∂ n ρ n + ρ n − ∂ n − )( C m − n ( y )) = C m − n ( y )since ρ n ( C m − n ( y )) = γ n ( C m − n ( y )) and thus0 = C mn ( y ) = (id − ∂ n γ n − ρ n − ∂ n − )( C m − n ( y )) = (id − ∂ n ρ n − ρ n − ∂ n − )( C m − n ( y )) . More generally, for any 0 ≤ i ≤ m we have(4.3) ( ∂ n ρ n + ρ n − ∂ n − )( C m − in ( y )) = C m − in ( y ) EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 15 which can be proved by induction on i as follows. Note that C n ( y ) = y and assumethat (4.3) holds for i − < m . Then it also holds for i since( ∂ n ρ n + ρ n − ∂ n − )( C m − in ( y ))= ( ∂ n γ n + P j ≥ ∂ n γ n C jn + ρ n − ∂ n − )( C m − in ( y ))= C m − in ( y ) − (id − ∂ n γ n − ρ n − ∂ n − )( C m − in ( y )) + P j ≥ ∂ n γ n C m − i + jn ( y )= C m − in ( y ) − C m − ( i − n ( y ) + ∂ n ρ n C m − ( i − n ( y )= C m − in ( y ) − ρ n − ∂ n − ( C m − ( i − n ( y ))= C m − in ( y )where the first and the third identities follow from ρ n = γ n + P i ≥ γ n C in , the fourthidentity from the induction hypothesis, and the fifth identity from ∂ n − C n = 0. Inparticular, we get that ( ∂ n ρ n + ρ n − ∂ n − )( y ) = y .Let us prove that ∂ n ∂ n +1 = 0. For this, we take y = ∂ n δ n +1 (1 ⊗ w ⊗ 1) into( ∂ n ρ n + ρ n − ∂ n − )( y ) = y and get ∂ n ρ n ∂ n δ n +1 (1 ⊗ w ⊗ 1) = ∂ n δ n +1 (1 ⊗ w ⊗ 1) for w ∈ S n +1 .This yields ∂ n ∂ n +1 (1 ⊗ w ⊗ 1) = ( ∂ n δ n +1 − ∂ n ρ n ∂ n δ n +1 )(1 ⊗ w ⊗ 1) = 0 . (cid:3) Lemma 4.4. ( i ) For n ≥ we have ρ n = γ n + ρ n ( δ n γ n − ∂ n γ n ) . ( ii ) For u ∈ Irr S an irreducible path and for x ∈ Q , we have the followinguseful identities ∂ ( a ⊗ x ⊗ b ) = aπ ( x ) ⊗ b − a ⊗ π ( x ) bρ ( a ⊗ π ( u )) = − a split ( u ) ρ ( a ⊗ x ⊗ π ( u )) = ( if xu ∈ Irr S a ⊗ s ⊗ if xu = s ∈ S .Proof. The first assertion follows since by definition ρ n = γ n + (cid:16) γ n + P i ≥ ( γ n δ n − γ n ∂ n ) i γ n (cid:17) ( δ n γ n − ∂ n γ n ) = γ n + ρ n ( δ n γ n − ∂ n γ n ) . Let us prove the second assertion. We have ∂ ( a ⊗ x ⊗ b ) = a ((id − ρ ∂ ) ◦ δ )(1 ⊗ x ⊗ b = a (id − ρ ∂ )( π ( x ) ⊗ − ⊗ π ( x )) b = aπ ( x ) ⊗ b − a ⊗ π ( x ) b. This also shows that ∂ = δ . Thus by ( i ) for n = 1 we have ρ ( a ⊗ π ( u )) = γ ( a ⊗ π ( u )) = − a split ( u ).If xu ∈ Irr S is irreducible, then γ ( a ⊗ x ⊗ π ( u )) = 0. It follows from ( i ) that ρ ( a ⊗ x ⊗ π ( u )) = 0.If xu = s ∈ S , then γ ( a ⊗ x ⊗ π ( u )) = a ⊗ s ⊗ ρ ( π ( s ) ⊗ 1) = 0 we have( δ − ∂ )( a ⊗ s ⊗ 1) = aρ ∂ δ (1 ⊗ s ⊗ 1) = aρ ( π ( s ) ⊗ − ⊗ π ( s )) = a split ( f s ) . EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 16 This yields that ρ ( a ⊗ x ⊗ π ( u )) = a ⊗ s ⊗ ρ ( a split ( f s )) = a ⊗ s ⊗ , since bythe first case we have ρ ( a split ( f s )) = 0. (cid:3) We give two simple and well-known examples of the resolution P in Theorem4.1 which may be obtained from the reduction systems of Examples 3.4. Examples 4.5. ( i ) Bardzell’s resolution of monomial algebras. Let A = k Q/ ( S )be a monomial algebra and let R = { ( s, | s ∈ S } be the reduction systemfor A . Then the resolution P coincides with Bardzell’s resolution [13, 14]and the homotopy ρ with the one given in Sk¨oldberg [117].( ii ) Koszul resolution of the polynomial ring. Let A = k [ x , . . . , x n ] and let R = (cid:8) ( x j x i , x i x j ) (cid:9) ≤ i 1) = k X j =1 ( − k − j (cid:0) x i j ⊗ x i k · · · x i j +1 b x i j x i j − · · · x i ⊗ − ⊗ x i k · · · x i j +1 b x i j x i j − · · · x i ⊗ x i j (cid:1) for any x i k x i k − · · · x i ∈ S k . Note that P coincides with the usual Koszulresolution of A via the map 1 ⊗ x i k · · · x i ⊗ ⊗ x i k ∧ · · · ∧ x i ⊗ Homotopy deformation retract In this section we construct a homotopy deformation retract between the projec-tive resolution ( P , ∂ ) given in Theorem 4.1 and the bar resolution (Bar , d ) (see § Notation 5.1. Let ̟ : A ¯ A = A/ ( k Q · A ) denote the natural projection andfor n ≥ ̟ n = id A ⊗ ¯ A ⊗ n ⊗ ̟ so that ̟ n : A ⊗ ¯ A ⊗ n ⊗ A A ⊗ ¯ A ⊗ n ⊗ ¯ A. (5.2)We also set ̟ − = id A .Given a path p ∈ k Q we denote ¯ p = ̟π ( p ), where π : k Q k Q/I = A is thenatural projection.Note that by the definition of d n , the maps ̟ n satisfy the following identities d n +1 ( ̟ n ( y ) ⊗ b ) = ̟ n − d n ( y ) ⊗ b + ( − n +1 yb for n ≥ . (5.3)5.1. Comparison maps and homotopy. We first define comparison maps( P , ∂ ) (Bar , d ) FG where F and G are morphisms of complexes of A -bimodules satisfying G F =id P . (In the monomial case similar comparison maps were given in Redondo–Rom´an [107].) EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 17 The complexes P and Bar are zero in negative degrees. However, it is convenientto set P − = Bar − = A and F − = G − = ̟ − = id A and let ∂ = d : A ⊗ A A denote the augmentations.We can then define A ⊗ k S n ⊗ A A ⊗ ¯ A ⊗ n ⊗ A F n G n n ≥ F n ( a ⊗ w ⊗ b ) = ( − n ̟ n − F n − ∂ n ( a ⊗ w ⊗ ⊗ b (5.4) G n ( a ⊗ y ⊗ b ) = ( ρ n G n − d n ( a ⊗ y ⊗ b (5.5)where ρ n : A ⊗ k S n − ⊗ A A ⊗ k S n ⊗ A are the morphisms of left A -modules inProposition 4.2. (Note that P = Bar = A ⊗ A and F = G = id A ⊗ A .) Remark . Since ̟ (1) = 0, the definitions of F n and ̟ n immediately give that ̟ n F n ( a ⊗ w ⊗ 1) = 0 for n ≥ . (5.7) Lemma 5.8. F n and G n satisfy the following identities for n ≥ i ) d n F n = F n − ∂ n ( ii ) ∂ n G n = G n − d n ( iii ) G n − F n − = id P n − .Proof. Each part can be proved by induction on n using the identity (5.3), therecursive definitions of F n and G n given in (5.4) and (5.5), the fact that ρ is ahomotopy retract and the fact that d n − d n = 0 and ∂ n − ∂ n = 0. For each partthe case n = 0 is immediate from the definitions. A proof can also be found in [71,Lem. 2.4 & Lem. 2.5].We illustrate the proof of ( iii ), where indeed G F = id A ⊗ A by definition. For n ≥ 1, we then have G n F n ( a ⊗ w ⊗ b ) = ( − n ( ρ n G n − d n ( ̟ n − F n − ∂ n ( a ⊗ w ⊗ ⊗ b = ( − n ( ρ n G n − ( ̟ n − F n − ∂ n − ∂ n ( a ⊗ w ⊗ ⊗ b + ( ρ n G n − F n − ∂ n ( a ⊗ w ⊗ b = ( ρ n ∂ n ( a ⊗ w ⊗ b = a ⊗ w ⊗ b where the second identity follows from (5.3) and d n − F n − = F n − ∂ n − , the thirdidentity from ∂ n − ∂ n = 0 and the induction hypothesis G n − F n − = id, and thelast identity from id P n = ρ n ∂ n + ∂ n +1 ρ n +1 and ρ n +1 ( a ⊗ w ⊗ 1) = 0. (cid:3) We can now give a homotopy h by A -bimodule morphisms h n : Bar n Bar n +1 for n ≥ 0, defining h n ( a ⊗ ¯ a ...n ⊗ a n +1 ) = n X i =1 ( − i +1 ̟ i F i G i ( a ⊗ ¯ a ...i ⊗ ⊗ ¯ a i +1 ...n ⊗ a n +1 . (5.9) EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 18 Note that for n = 0 the sum is empty and h = 0. Alternatively, h n can be definedrecursively for n ≥ h n ( a ⊗ ¯ a ...n ⊗ a n +1 ) = ̟ n h n − ( a ⊗ ¯ a ...n − ⊗ a n ) ⊗ a n +1 + ( − n +1 ̟ n F n G n ( a ⊗ ¯ a ...n ⊗ ⊗ a n +1 . (5.10) Theorem 5.11. We have a homotopy deformation retract ( P , ∂ ) (Bar , d ) FG h (5.12) namely G F = id and F G − id = h d + d h . Moreover, the homotopy deforma-tion retract is special , i.e. it satisfies h F = 0 G h = 0 h h = 0 . (5.13) Proof. The first identity is proved in Lemma 5.8 ( iii ). The other identities can beproved similarly by induction, using the identities of Lemma 5.8 as well as identities(5.3)–(5.7) and (5.10) as well as ρ n ( a ⊗ w ⊗ 1) = 0 (cf. Theorem 4.1). (cid:3) Remark . For an augmented algebra A ǫ k Q (i.e. augmented over k Q ≃ k × · · · × k ), the maps ¯ A A given by ¯ A ≃ ker ǫ ⊂ A and ̟ : A ¯ A = A/ ( k Q · A ) induce comparison maps between the ( k Q -relative) normalized barresolution and the standard bar resolution. Moreover, it is straight forward toconstruct a homotopy deformation retract using a formula similar to (5.10), so thatBar in Theorem 5.11 can also be taken to be the standard bar resolution. Thehomotopy h is inspired by He–Li–Li [68, Def. 3.3].5.2. Maps in low degrees. In this section we give explicit expressions for themaps ∂ , F and G in low degrees, namely for the labelled maps in · · · P P P P · · · Bar Bar Bar Bar . ∂ ∂ ∂ d d d F F F G G ρ ρ These maps will be used in § A = k Q/I .Recall from Lemma 4.4 that the homotopy ρ is given in low degrees by ρ ( a ⊗ b ) = − a split ( σ ( b )) ρ ( a ⊗ x ⊗ b ) = a X i ≥ ( γ δ − γ ∂ ) i γ (1 ⊗ x ⊗ b ) x ∈ Q . (A formula for split was given in (3.24).) Lemma 5.15. In low degrees we have the following formulae for ∂ , F and G . ( i ) G ( a ⊗ ¯ u ⊗ b ) = a split ( u ) bG ( a ⊗ ¯ u ⊗ ¯ v ⊗ b ) = aρ (split ( u ) π ( v )) b ( ii ) ∂ ( a ⊗ x ⊗ b ) = aπ ( x ) ⊗ b − a ⊗ π ( x ) b∂ ( a ⊗ s ⊗ b ) = a split ( s − f s ) b∂ ( a ⊗ w ⊗ b ) = δ ( a ⊗ w ⊗ b ) + G ( a ⊗ ¯ u ⊗ ¯ f s ⊗ b ) − G ( a ⊗ ¯ f s ′ ⊗ ¯ v ⊗ b ) EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 19 ( iii ) F ( a ⊗ x ⊗ b ) = − a ⊗ ¯ x ⊗ bF ( a ⊗ s ⊗ b ) = a̟ split ( s − f s ) ⊗ bF ( a ⊗ w ⊗ b ) = − a̟ F (1 ⊗ s ′ ⊗ π ( v )) ⊗ b + a̟ F G (1 ⊗ ¯ f s ′ ⊗ ¯ v ⊗ − ⊗ ¯ u ⊗ ¯ f s ⊗ ⊗ b where a, b ∈ A , and in ( i ) u, v are arbitrary irreducible paths and in ( ii )–( iii ) x ∈ Q , s, s ′ ∈ S and w ∈ S with w = us = s ′ v for some (irreducible) paths u, v .Proof. The formulae for ∂ , ∂ , G and F , F follow from the definitions (cf. Lemma4.4). Let us compute G : G ( a ⊗ ¯ u ⊗ ¯ v ⊗ b ) = ( ρ G d ( a ⊗ ¯ u ⊗ ¯ v ⊗ b = aρ ( π ( u )split ( v ) − split ( σπ ( uv )) + split ( u ) π ( v )) b = aρ (split ( u ) π ( v )) b where the third identity follows from the identity ρ ( a split ( u )) = 0 for any irre-ducible path u (see Lemma 4.4 ( ii )).To compute ∂ , let w ∈ S . Then ∂ ( a ⊗ w ⊗ b ) = δ ( a ⊗ w ⊗ b ) − a ( ρ ∂ δ (1 ⊗ w ⊗ b = δ ( a ⊗ w ⊗ b ) − aρ ( π ( u )split ( s )) b + aρ ( π ( u )split ( f s )) b + aρ (split ( s ′ ) π ( v )) b − aρ (split ( f s ′ ) π ( v )) b = δ ( a ⊗ w ⊗ b ) − aπ ( u ) ⊗ s ⊗ b + aρ (split ( s ′ ) π ( v )) b − aG (1 ⊗ ¯ f s ′ ⊗ ¯ v ⊗ b = δ ( a ⊗ w ⊗ b ) + G ( a ⊗ ¯ u ⊗ ¯ f s ⊗ b ) − G ( a ⊗ ¯ f s ′ ⊗ ¯ v ⊗ b )where the third identity follows from the formula of G , Lemma 4.4 ( ii ) and (3.25);and the fourth identity follows from Lemma 4.4 ( ii ): aρ (split ( s ′ ) π ( v )) b = aπ ( u ) ⊗ s ⊗ b + aρ (split ( u ) π ( f s )) b. It remains to compute F . For w ∈ S we have F ( a ⊗ w ⊗ b ) = ̟ F ( π ( u ) ⊗ s ⊗ ⊗ b − ̟ F (1 ⊗ s ′ ⊗ π ( v )) ⊗ b + ̟ F G ( a ⊗ ¯ f s ′ ⊗ ¯ v ⊗ b ) − ̟ F G ( a ⊗ ¯ u ⊗ ¯ f s ⊗ b )= − ̟ F (1 ⊗ s ′ ⊗ π ( v )) ⊗ b + ̟ F G ( a ⊗ ¯ f s ′ ⊗ ¯ v ⊗ ⊗ b − ̟ F G ( a ⊗ ¯ u ⊗ ¯ f s ⊗ ⊗ b where the first identity follows from the formula of ∂ and the second identity followsfrom ̟ F ( π ( u ) ⊗ s ⊗ 1) = 0. (cid:3) The following lemma gives more explicit formulae for G . Lemma 5.16. Let u be an irreducible path. ( i ) For x ∈ Q such that xu = sv with s ∈ S we have G (1 ⊗ ¯ x ⊗ ¯ u ⊗ 1) = 1 ⊗ s ⊗ π ( v ) + G (1 ⊗ ¯ f s ⊗ ¯ v ⊗ . In particular, if xu = s , then G (1 ⊗ ¯ x ⊗ ¯ u ⊗ 1) = 1 ⊗ s ⊗ . EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 20 ( ii ) Let u, v be irreducible paths such that uv is also irreducible. Then G (1 ⊗ ¯ u ⊗ ¯ v ⊗ 1) = 0 . ( iii ) Let u = u · · · u n be an irreducible path with u i ∈ Q for i = 1 , . . . , n . Thenfor any (irreducible) path v we have G (1 ⊗ ¯ u ⊗ ¯ v ⊗ 1) = n X i =1 π ( u · · · u i − ) G (1 ⊗ ¯ u i ⊗ u i +1 · · · u n v ⊗ . Proof. Let us prove ( i ). It follows from Lemma 5.15 that G (1 ⊗ ¯ x ⊗ ¯ u ⊗ 1) = ρ (split ( x ) π ( u ))= γ (1 ⊗ x ⊗ π ( u )) + (cid:0) ρ ◦ ( δ − ∂ ) ◦ γ (cid:1) (1 ⊗ x ⊗ π ( u ))= 1 ⊗ s ⊗ π ( v ) + ρ (split ( f s ) π ( v ))= 1 ⊗ s ⊗ π ( v ) + G (1 ⊗ ¯ f s ⊗ ¯ v ⊗ i ).To prove ( ii ) note that γ (1 ⊗ ¯ u ⊗ π ( v )) = 0 since uv is assumed to be irreducible.From Lemma 5.15 again, we get that G (1 ⊗ ¯ u ⊗ ¯ v ⊗ 1) = ρ (split ( u ) π ( v )) = 0 . Let us prove ( iii ). We have G (1 ⊗ ¯ u ⊗ ¯ v ⊗ 1) = ρ (split ( u ) π ( v ))= n X i =1 ρ ( π ( u · · · u i − ) ⊗ ¯ u i ⊗ π ( u i +1 · · · u n v ))= n X i =1 π ( u · · · u i − ) ρ (split ( u i ) π ( u i +1 · · · u n v ))= n X i =1 π ( u · · · u i − ) G (1 ⊗ ¯ u i ⊗ u i +1 · · · u n v ⊗ (cid:3) Deformation theory We give a brief review of general deformation theory as used in this article. In § ∞ algebras as explained in § Deformations of associative algebras. An associative k -algebra A is a pair( V, µ ) where V is a k -vector space and µ ∈ Hom k ( V ⊗ k V, V ) a bilinear map satisfyingthe associativity condition(6.1) µ ( u ⊗ µ ( v ⊗ w )) = µ ( µ ( u ⊗ v ) ⊗ w ) . This condition of associativity can be expressed in the following form. Proposition 6.2. Let µ ∈ Hom k ( V ⊗ k V, V ) . Then µ is associative if and only if [ µ, µ ] = 0 . Here [ − , − ] is the Gerstenhaber bracket defined as follows. EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 21 Definition 6.3. Let f ∈ Hom k ( V ⊗ k m +1 , V ) and g ∈ Hom k ( V ⊗ k n +1 , V ) be twomultilinear maps. Write f • i g = f (id ⊗ k i ⊗ k g ⊗ k id ⊗ k m − i ) 0 ≤ i ≤ m and also f • g = X ≤ i ≤ m ( − ni f • i g. The Gerstenhaber bracket is then defined by[ f, g ] = f • g − ( − mn g • f. Remark . For later use we note that the Gerstenhaber bracket can easily be gen-eralized to Hom( ¯ A ⊗• +1 , A ), the (shifted) k Q -relative Hochschild cochain complex(see Remark 5.14). For simplicity, we still denote the bracket on Hom( ¯ A ⊗• +1 , A )by [ − , − ].Let us just for a moment assume that V (and thus A ) is finite-dimensional, saydim V = d . Choosing a k -basis { v , . . . , v d } of V , the bilinear map µ is determinedby its structure constants µ kij ∈ k defined by µ ( v i ⊗ v j ) = d X k =1 µ kij v k . Thus µ ∈ Hom k ( V ⊗ k , V ) ≃ k d , and the structure constants µ kij may be viewedas coordinate functions. Writing the Gerstenhaber bracket [ µ, µ ] in this basis, theobstruction to associativity of µ is given by structure constants ν lijk defined by12 [ µ, µ ]( v i ⊗ v j ⊗ v k ) = µ ( µ ( v i ⊗ v j ) ⊗ v k ) − µ ( v i ⊗ µ ( v j ⊗ v k )) = d X l =1 ν lijk v l where ν lijk = d X m =1 (cid:0) µ mij µ lmk − µ lim µ mjk (cid:1) ≤ i, j, k, l ≤ d (6.5)is quadratic in the µ kij ’s. In geometric terms we may view k d as the affine space A d , so that the elements (6.5) cut out an affine variety (or affine scheme)Alg V ⊂ A d ≃ Hom k ( V ⊗ k , V )of all associative structures on V given by d = dim Hom k ( V ⊗ k , V ) quadraticequations.Moreover, there is a natural action of GL( V ) on Alg V given by change of basis,i.e. for T ∈ GL( V ) we put( T · µ )( v ⊗ w ) = T − ( µ ( T ( v ) ⊗ T ( w )))and two algebras A = ( V, µ ) and A ′ = ( V, µ ′ ) are isomorphic if and only if theylie in the same GL( V )-orbit. In particular, the intuitive notion of “variation” ofassociative structures can be understood as varying an associative structure µ asa k -bilinear map in Alg V . Varying the associative structure inside a GL( V )-orbitwill give an isomorphic algebra (a “trivial deformation”) and passing to a differentorbit will give a non-isomorphic algebra (a “nontrivial deformation”). EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 22 Gabriel [55] showed that the scheme Alg V is always connected, so that any twoalgebras A, A ′ ∈ Alg V may be viewed as “actual deformations” of each other. In[55, 113] explicit descriptions of Alg V are given for dim V ≤ 6. For example, whendim V = 4, the variety Alg V has five irreducible components with 18 GL( V )-orbitsof dimensions between 6 and 15 and one 1-dimensional family of 11-dimensionalorbits [55]. An explicit description of Alg V becomes less and less accessible in higherdimensions. (In § Deformation theory is a way to study the local structure of the variety Alg V and thus the “local” properties of the variation of associative structures, where by“local” we mean an infinitesimal or formal neighbourhood of a point in Alg V . Yetthe formal theory makes sense for any associative algebra, not necessarily finite-dimensional.6.1.1. Infinitesimal and formal deformations. Let V be any k -vector space, notnecessarily finite-dimensional, and let A = ( V, µ ) for µ ∈ Hom k ( V ⊗ k , V ) be anassociative algebra (i.e. µ satisfies [ µ, µ ] = 0).A first-order deformation of A = ( V, µ ) (over k [ t ] / ( t )) is given by µ t = µ + µ t where µ : A ⊗ k A A is a linear map such that ( A [ t ] / ( t ) , µ t ) is an associativealgebra, where µ t is considered a multiplication on A [ t ] / ( t ) by t -linear extension. Afirst-order deformation is thus determined by the linear map µ and the associativitycondition (6.1) for µ t implies that µ is a Hochschild 2-cocycle. Two first-orderdeformations ( A [ t ] / ( t ) , µ t ) and ( A [ t ] / ( t ) , µ ′ t ) of ( A, µ ) are (gauge) equivalent ifthere is an k [ t ] / ( t )-algebra isomorphism T : ( A [ t ] / ( t ) , µ t ) ( A [ t ] / ( t ) , µ ′ t ) whichis the identity modulo ( t ). Such an isomorphism is thus of the form T = id + T t for some k -linear map T : A A and a straight-forward computation shows that µ = µ ′ + d ( T ), that is, two first-order deformations are equivalent if and onlyif they differ by a 2-coboundary. Thus the set of equivalence classes of first-orderdeformations can be identified as the cohomology group HH ( A ) (see [56, 60]). Remark . When A is finite-dimensional one can again interpret this geometri-cally: a first-order deformation of an algebra A ∈ Alg V gives a tangent vector toAlg V at the point A . The space of first-order deformations — which by the abovecoincides with the space of Hochschild 2-cocycles — is the Zariski tangent space ofAlg V at A and the subspace of 2-boundaries is the tangent space to the GL( V )-orbitpassing through the point A .In fact, this definition can be extended to all orders as follows. A formal defor-mation of ( A, µ ) (over k J t K ) is given by µ t = µ + µ t + µ t + · · · where the maps µ i are ( k J t K -linear extensions of) bilinear maps A ⊗ k A A making( A J t K , µ t ) into an associative algebra.It follows directly from Proposition 6.2 that the Gerstenhaber bracket also con-trols the formal deformation theory of associative algebras. Corollary 6.7. Let A = ( V, µ ) be an associative algebra and let e µ ∈ Hom k ( V ⊗ k V, V ) b ⊗ ( t ) , i.e. e µ = µ t + µ t + · · · . Then e A = ( V J t K , µ + e µ ) is associative if and EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 23 only if e µ satisfies the Maurer–Cartan equation d e µ + [ e µ, e µ ] = 0 . Indeed, one checks directly that the Hochschild differential d n : Hom k ( A ⊗ k n , A )Hom k ( A ⊗ k n +1 , A ), given via the Koszul sign rule by f − ( − n f ◦ d n +1 , coincideswith [ µ, − ], so that [ µ + e µ, µ + e µ ] = [ µ, µ ] | {z } =0 +2 [ µ, e µ ] | {z } d e µ +[ e µ, e µ ] . In fact, the Gerstenhaber bracket endows the (shifted) Hochschild cochain com-plex Hom k ( A ⊗ k • +1 , A ) with the structure of a DG Lie algebra and this structurecan be used to study both “actual” and formal deformations — not only over k J t K but over any complete local Noetherian k -algebra.6.2. Deformation theory via DG Lie and L ∞ algebras. Deformations of as-sociative algebras fit into the general framework of formal deformation theory viaDG Lie or L ∞ algebras. One has that deformations are controlled by a cochaincomplex, wherefirst-order deformations cochain complex ( C , d )higher-order deformations ( C , d, h− , −i , h− , − , −i , . . . | {z } “higher structures” ) . The “higher structures” are multilinear brackets defining a DG Lie or L ∞ algebra,the latter being a DG Lie algebra “up to homotopy” (see Definitions 6.9 and 6.10below). The fact that L ∞ algebras play a central role in deformation theory isillustrated in the following result, which might be called the Fundamental Theoremof Formal Deformation Theory. Theorem 6.8. ( i ) Let g and g ′ be two L ∞ algebras. An L ∞ -quasi-isomorphism Φ : g g ′ induces a natural transformation MC g MC g ′ and a naturalisomorphism of deformation functors Def g ≃ Def g ′ . ( ii ) For any (formal) deformation problem in characteristic with deformationfunctor D , there exists an L ∞ algebra g such that there is an isomorphismof deformation functors D ≃ Def g . For ( i ) see Getzler [61, Prop. 4.9]; ( ii ) was recently established by Lurie [89]and Pridham [106] as an equivalence of the ∞ -categories of formal moduli problemsand of DG Lie algebras, giving a formal proof of the “philosophy” formulated byDeligne [44] that deformation problems in characteristic 0 should be controlled byDG Lie algebras.In the remainder of this section we give the definitions of DG Lie and L ∞ al-gebras and their associated deformation functors. In § ∞ algebrastructure p ( Q, R ) on the cochain complex P associated to a reduction system R byhomotopy transfer and show that this L ∞ algebra controls the (formal) deforma-tions of reduction system, so that by Theorem 6.8 deformations of A = k Q/I canbe completely described as deformations of any chosen reduction system satisfying( ⋄ ) for I . Definition 6.9. A differential graded (DG) Lie algebra ( g , d, [ − , − ]) consists of agraded vector space g = Q n ∈ Z g n together with a linear map d : g g of degree 1 EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 24 satisfying d n d n − = 0, the differential , and a (graded) Lie bracket [ − , − ] : g ⊗ g g of degree 0 satisfying( i ) [ x, y ] = ( − | x || y | +1 [ y, x ] (graded) skew-symmetry ( ii ) d [ x, y ] = [ dx, y ] + ( − | x | [ x, dy ] (graded) Leibniz rule ( iii ) ( − | x || z | [ x, [ y, z ]] + ( − | y || x | [ y, [ z, x ]] + ( − | z || y | [ z, [ x, y ]] = 0 (graded) Jacobi identity. Definition 6.10. An L ∞ algebra ( g , { l n } n ≥ ) = ( g , h−i , h− , −i , h− , − , −i , . . . ) is agraded k -vector space g = Q m ∈ Z g m together with a collection of multilinear maps l n : g ⊗ n g of degree 2 − n satisfying for each n the identities( i ) l n ( x s (1) , . . . , x s ( n ) ) = χ ( s ) l n ( x , . . . , x n ) for any s ∈ S n skew-symmetry ( ii ) X i + j = n +1 i,j ≥ X s ∈ S i,n − i ( − i ( n − i ) χ ( s ) l j ( l i ( x s (1) , . . . , x s ( i ) ) , x s ( i +1) , . . . , x s ( n ) ) = 0 generalized Jacobi identities for homogeneous elements x , . . . , x n . Here • S n is the set of permutations of n elements • S i,n − i ⊂ S n is the set of shuffles , i.e. permutations s ∈ S n satisfying s (1) < · · · < s ( i ) and s ( i + 1) < · · · < s ( n ), and • χ ( s ) := sgn( s ) ǫ ( s ; x , . . . , x n ), where sgn( s ) is the signature of the permu-tation s and ǫ ( s ; x , . . . , x n ) is the Koszul sign of s , which also depends onthe degrees of the x i .We usually denote the n -ary multilinear maps l n ( − , . . . , − ) by h− , . . . , −i or some-times by h− , . . . , −i n to indicate the number of entries. Remark . If the n -ary brackets are identically zero for n ≥ 2, one obtains anordinary cochain complex with the differential given by the unary bracket; if thebrackets are zero for n ≥ 3, one obtains a DG Lie algebra (cf. Definition 6.9). Definition 6.12. A morphism of L ∞ algebras Φ : ( g , { l i } ) ( g ′ , { l ′ i } ) is given bygraded linear maps Φ n : g ⊗ n g ′ of degree 1 − n for all n ≥ n ( x s (1) , . . . , x s ( n ) ) = χ ( s ) Φ n ( x , . . . , x n )for any s ∈ S n and X i + j = n +1 i,j ≥ X s ∈ S i,n − i ( − i ( n − i ) χ ( s )Φ j ( l i ( x s (1) , . . . , x s ( i ) ) , x s ( i +1) , . . . , x s ( n ) )= X ≤ r ≤ ni + ··· + i r = n X t ( − u χ ( t ) l ′ r (Φ i ( x t (1) , . . . , x t ( i ) ) , . . . , Φ i r ( x t ( i + ··· + i r − +1) , . . . , x t ( n ) ) . where t runs over all ( i , . . . , i r )-shuffles for which t ( i + · · · + i l − + 1) < t ( i + · · · + i l + 1)and u = ( r − i − 1) + · · · + 2( i r − − 1) + ( i r − − ∞ algebras Φ is a quasi-isomorphism if Φ is a quasi-isomorphismof complexes. The Koszul sign of a transposition of two homogeneous elements x i , x j is defined by( − | x i || x j | , where | x i | denotes the degree of x i . This definition is then extended multiplicativelyto an arbitrary permutation using a decomposition into transpositions. EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 25 Remark . L ∞ algebras and their morphisms can also be viewed from a “dual”point of view as follows. Let V be a graded vector space and consider the gradedspace Λ c ( V ) = M n ≥ Λ n ( V ) := T( V ) /J where J is the graded subspace of T( V ) spanned by vectors of the form x ⊗ · · · ⊗ x n − χ ( s ) x s (1) ⊗ · · · ⊗ x s ( n ) for any x , . . . , x n ∈ V and s ∈ S n . Denote by x ∧· · ·∧ x n the image of x ⊗· · ·⊗ x n under the natural projection V ⊗ n Λ n V .Let Λ c ( V ) be the reduced cocommutative coalgebra, i.e. Λ c ( V ) = L n ≥ Λ n ( V )with the natural coproduct ∆ : Λ c ( V ) Λ c ( V ) ⊗ Λ c ( V ) given by∆( x ∧ · · · ∧ x n ) = n − X i =1 X s ∈ S i,n − i ( x s (1) ∧ · · · ∧ x s ( i ) ) ⊗ ( x s ( i +1) ∧ · · · ∧ x s ( n ) ) . Let g [1] be the 1-shifted graded space of g , i.e. g [1] i = g i +1 for any i ∈ Z .Then an L ∞ algebra structure on g is equivalent to a degree 1 coderivation Q onΛ c ( g [1]) satisfying Q ◦ Q = 0, in other words, (Λ c ( g [1]) , ∆ , Q ) is a DG coalgebra.Accordingly, a morphism of L ∞ algebras from g to g ′ is equivalent to a morphism ofDG coalgebras from (Λ c ( g [1]) , ∆ , Q ) to (Λ c ( g ′ [1]) , ∆ , Q ′ ). For more details we referto [84, 61]. Lemma 6.14 [36, Thm. 2.9] . Let F : g g ′ be a quasi-isomorphism betweenL ∞ algebras. Then F admits a quasi-inverse , i.e. there exists a quasi-isomorphism G : g ′ g which induces the inverse isomorphism on cohomology. Deformation functors. To a DG Lie or L ∞ algebra g one can associate a deformation functor Def g which captures all the (formal) information about thedeformation problem controlled by g . This deformation functor can be defined onthe category Art k of (commutative) local Artinian k -algebras (e.g. k [ t ] / ( t n +1 )) andthen by completion on the category c Art k of (commutative) complete local Noether-ian k -algebras (e.g. k J t K ). Deformations over a local Artinian k -algebra are usuallycalled (finite-order) infinitesimal deformations and deformations over a complete lo-cal Noetherian k -algebra are formal deformations. (See § k J t K or sometimes k J t , . . . , t n K and wewrite A J t , . . . , t n K = A b ⊗ k k J t , . . . , t n K = lim N A ⊗ k k [ t , . . . , t n ] / m N where m = ( t , . . . , t n ) is the (unique) maximal ideal of k J t , . . . , t n K so that b ⊗ k denotes the completion of the tensor product with respect to the m -adic topology.(Note that if A is finite-dimensional, one has A b ⊗ k k J t , . . . , t n K ≃ A ⊗ k k J t , . . . , t n K .)The deformation functor associated to a DG Lie algebra is defined as follows. EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 26 Definition 6.15. Let g be a DG Lie algebra over a field k of characteristic 0. Definethe gauge functor G g and the Maurer–Cartan functor MC g byG g : Art k Group ( B, m ) exp( g ⊗ m )MC g : Art k Sets ( B, m ) (cid:8) x ∈ g ⊗ m (cid:12)(cid:12) dx + [ x, x ] = 0 (cid:9) . The group exp( g ⊗ m ) is called the gauge group and is defined formally via theBaker–Campbell–Hausdorff formula ; the equation(6.16) dx + [ x, x ] = 0is called the Maurer–Cartan equation and elements satisfying (6.16) are called Maurer–Cartan elements .The deformation functor Def g associated to g is then defined by Def g = MC g (cid:14) G g ,where an element exp( w ) ∈ G g ( B, m ) acts on x ∈ MC g ( B, m ) byexp( w ) · x = x + X n ≥ ad n − w n ! (ad w x − dw ) ad w = [ w, − ] . Remark . More generally, (derived) deformation functors may be defined asfunctors on the category of not necessarily commutative Artinian local (DG) alge-bras or as functors on simplicial rings with values in the category of simplicial sets(see Pridham [106] for the equivalence between the various approaches), but for thisarticle we will content ourselves with the “classical” formulation.For an L ∞ algebra, one similarly obtains a Maurer–Cartan functor MC g bygeneralizing the Maurer–Cartan equation as follows. Definition 6.18. Given an L ∞ algebra g , a Maurer–Cartan element is an element x ∈ g satisfying the Maurer–Cartan equation ∞ X n =0 x h n i n ! = 0 x h n i = h x, . . . , x i n . (6.19) Remark . As the sum in (6.19) is infinite, the definition of a Maurer–Cartanelement only makes sense in the right context, i.e. if one can establish that the sumis in fact finite or at least converges in some topology.The former can sometimes be shown for elements in g which satisfy certaindegree conditions. The latter can always be achieved in the context of formaldeformations over any complete local Noetherian algebra ( B, m ) by working withthe m -adic topology. Given an L ∞ algebra ( g , { l n } n ), formal deformations over( B, m ) are described as Maurer–Cartan elements in the L ∞ algebra g b ⊗ m by linearlyextending the brackets, i.e. by setting l n ( x ⊗ m , . . . , x n ⊗ m n ) = l n ( x , . . . , x n ) ⊗ m · · · m n x i ∈ g , m i ∈ m . For the L ∞ algebra g b ⊗ m , the sum in (6.19) then converges in the m -adic topology. For a nilpotent Lie algebra n one defines the group law · on exp( n ) by exp( x ) · exp( y ) = exp( x + y + [ x, y ]+ · · · ) for x, y ∈ n , where the sum on the right-hand side is the Baker–Campbell–Hausdorffformula. Note that g ⊗ m is a nilpotent Lie algebra with Lie bracket [ x ⊗ a, y ⊗ b ] = [ x, y ] ⊗ ab . EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 27 In the more general setting of L ∞ algebras, equivalence of Maurer–Cartan ele-ments can no longer be described by the action of a gauge group . Rather, one saysthat two Maurer–Cartan elements x, y ∈ MC g are equivalent if they are homotopic (see Canonaco [36], Manetti [91, § 5] and Markl [92, Ch. 5] for nice expositions).This notion of homotopy between Maurer–Cartan elements is defined as follows.First note that for any L ∞ algebra g and any DG commutative algebra C , thereis a natural L ∞ algebra structure on the tensor product g ⊗ C given by h x ⊗ c , . . . , x n ⊗ c n i = ±h x , . . . , x n i ⊗ c · · · c n where the sign is determined by the Koszul sign rule.For the notion of homotopy between Maurer–Cartan elements consider the DGcommutative algebra Ω := k [ τ, d τ ]with | τ | = 0 and | d τ | = 1 and differential defined by d( τ ) = d τ and d(d τ ) = 0. (Ω is the de Rham algebra of simplicial differential forms on the unit interval.) We nowsay that two Maurer–Cartan elements x , x ∈ MC g are homotopic if there existsa Maurer–Cartan element x τ ∈ MC g ⊗ Ω such that x = ev ( x τ ) and x = ev ( x τ ),where ev λ : MC g ⊗ Ω MC g is given by evaluating τ λ ∈ k and d τ 0. Theabove notion of homotopy is a symmetric and reflexive relation and we denote by ∼ its transitive closure and say that x, y ∈ MC g are homotopic if x ∼ y . Remark . A useful necessary and sufficient condition is the following. An el-ement x τ = x + x ′ d τ for x, x ′ ∈ ( g ⊗ k [ τ ]) b ⊗ m is a Maurer–Cartan element of( g ⊗ Ω ) b ⊗ m if and only if x | τ = λ is a Maurer–Cartan element of g b ⊗ m for each λ ∈ k and moreover the following equation holds ∂x∂τ = X k ≥ k ! h x, . . . , x | {z } k , x ′ i . (6.22)The deformation functor Def g associated to an L ∞ algebra g can now be definedas Def g = MC g / ∼ . (Note that when g is a DG Lie algebra, two elements are homotopic if and only ifthey are related by the action of the gauge group, so that for DG Lie algebras thenotions of gauge equivalence and homotopy equivalence coincide [91, Thm. 5.5].) Remark . In light of the fact that equivalences between Maurer–Cartan elementsof an L ∞ algebra g can understood by studying Maurer–Cartan elements of g ⊗ Ω , the Maurer–Cartan equation is not only central for finding Maurer–Cartanelements, but also for determining equivalences between them. In § ∞ algebra p ( Q, R )controlling deformations of the reduction system R and this description can thusalso be used to determine the set of all formal deformations up to equivalence. (See § Algebraizations of formal deformations. Given a formal deformation over k J t K , say, one may ask to what extent t can be considered an “actual” parame-ter, i.e. if it is possible to evaluate t at some value λ ∈ k to obtain an “actual”deformation. EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 28 Given a formal deformation b A = ( A J t K , ⋆ ) of an associative algebra A , an al-gebraization of b A would be an algebra e A = ( A [ t ] , ⋆ ) such that b A is isomorphicto the ( t )-adic completion of e A as formal deformation, i.e. ϕ : b A ∼ lim n e A/ ( t n ),where ϕ restricts to the identity modulo t , so that ϕ = id + ϕ t + ϕ t + · · · for some linear maps ϕ i : A A . For example, given the commutative algebra A = k [ x, y ] / ( xy ), we have that b A = k [ x, y ] J t K (cid:14) ( xy − t ) b is a formal (commutative)deformation of A over k J t K , which coincides with the ( t )-adic completion of thealgebra e A = k [ x, y ][ t ] / ( xy − t ).When A = ( V, µ ) is a finite-dimensional algebra, the formal deformation theorystudies the “local” structure of the affine variety Alg V of all associative algebrastructures on V , and an algebraization of a formal deformation may be viewedas reconstructing an affine subvariety passing through A ∈ Alg V from its formalneighbourhood.When A is infinite-dimensional algebraizations may not always exist, but whenthey do one can consider the parameter t as an actual parameter and evaluate at allvalues of t , giving an “actual” deformation e A λ = e A/ ( t − λ ). For the above example, e A = { e A λ | λ ∈ k } is a family of commutative algebras of the same dimension with e A = A . In the context of deformations of A = k Q/I algebraizations can be shownto exist under some “degree conditions” (see § Deformations of path algebras of quivers with relations In this section we prove our main results about deformations of path algebras ofquivers with relations. We start with a summary of the results from the point ofview of deformations problems and introduce the notation used in the remainder ofthis section.7.1. Summary. In the case of formal deformations our results can be summarizedas follows. Theorem 7.1. Let A = k Q/I and let R be any reduction system satisfying ( ⋄ ) with respect to I .There is an equivalence of formal deformation problems between (1) deformations of the associative algebra structure on A (2) deformations of the reduction system R (3) deformations of the relations I . This equivalence can be used to give a rather explicit description of all of thesedeformations. The equivalence between the first two deformation problems followsfrom Theorems 7.9, 7.14 and 7.36 and the equivalence of the first two to the thirddeformation problem follows from Propositions 7.41 and 7.46.Moreover, under the assumption of reduction-finiteness, which can be establishedunder certain degree conditions, Theorem 7.1 also holds for actual deformations.This follows from Theorem 7.9, Remark 7.16 and Propositions 7.46.Our approach for studying the deformations of A = k Q/I is based on replacingthe bar resolution of A by the projective A -bimodule resolution P obtained from areduction system. The homotopy comparison maps between the bar resolution Barand the resolution P (see § 5) allow one to transfer the DG Lie algebra structure onthe Hochschild complex to an L ∞ algebra structure on a cochain complex associated EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 29 to P . It turns out that this L ∞ algebra, which we denote by p ( Q, R ), is ratherconvenient for concrete computations as one can give a combinatorial criterion foran element to satisfy the Maurer–Cartan equation (see § A = k Q/I for an arbitrary finite quiver Q and anarbitrary two-sided ideal I of relations. Already the case of a quiver with onevertex and n loops has an interesting application: we can give a combinatorialconstruction of a quantization of certain algebraic Poisson structures on A n (see § Outline and notation. Let k Q be the path algebra of a finite quiver Q , let I ⊂ k Q be a two-sided ideal of relations and let A = k Q/I denote the quotientalgebra under the natural projection π : k Q k Q/I .Fix a reduction system R = { ( s, f s ) } satisfying the condition ( ⋄ ) with respect to I (cf. Proposition 3.11). Recall from Remark 3.9 that R is determined by the set S = { s | ( s, f s ) ∈ R } and the k Q -bimodule map f ∈ Hom( k S, A ) with σf ( s ) = f s ,where σ : A k Q is the unique k -linear map such that πσ = id A and σπ ( u ) = u for all irreducible paths u .To keep the notation simple we usually phrase the results in the context of one-parameter deformations, i.e. as deformations over k J t K , but the results usually holdfor deformations over arbitrary complete local Noetherian k -algebras, so even in theone-parameter case we usually denote the maximal ideal ( t ) by m .For the rest of the section let(7.2) g ∈ Hom( k S, A ) b ⊗ m so that g can be written as g = g t + g t + g t + · · · g i ∈ Hom( k S, A )and let g ( n ) = g t + · · · + g n t n denote the image under tensoring by ⊗ k J t K k [ t ] / ( t n +1 ).Just as for f s = σf ( s ) characterizing R (see Remark 3.9), we set g s = σg ( s ) for any s ∈ S , where σ is extended k J t K -linearly. (The subscript on g is thus either a naturalnumber i or 1 , , . . . used as an index, or an element s ∈ S used as a shorthand,but the use should be clear both from the notation and from context, so we hopethis does not create any confusion.)The maps g (7.2) are candidates for deformations, deforming a reduction system R determined by f ∈ Hom( k S, A ) (cf. Remark 3.9) to a (formal) reduction systemdetermined by f + g . A priori we do not make any additional assumptions on g , butwe shall show that the objects (bilinear maps, reduction systems, ideals) that canbe associated to g have the intended nice properties precisely when g is a Maurer–Cartan element of the L ∞ algebra p ( Q, R ) b ⊗ m , which we construct via homotopytransfer in § g ∈ Hom( k S, A ) b ⊗ m we can associate(1) a collection of k J t K -bilinear maps A J t K ⊗ A J t K A J t K defining an algebra b A g = ( A J t K , ⋆ g )(2) a formal reduction system b R g for k Q J t K (3) an ideal b I g in k Q J t K such that EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 30 (1) b A g is a formal deformation of A (2) b R g is a formal deformation of R (3) b I g is a formal deformation of I if and only if g is a Maurer–Cartan element of the L ∞ algebra p ( Q, R ) b ⊗ m . In fact,to (1) and (2) we can associate (a priori different) star products ⋆ g and ⋆ g on A J t K (see Definitions 7.12 and 7.21) which coincide. When g is a Maurer–Cartan elementof p ( Q, R ) b ⊗ m , we can also associate a (generalized) Gutt star product ⋆ g on A J t K (see Definition 7.40). In this case, the three star products ⋆ g , ⋆ g and ⋆ g coincide.The above formal deformations are to be understood in the following sense. Formal deformations of associative algebras. Recall from § A is given by b A = ( A J t K , ⋆ ), where ⋆ is a k J t K -bilinearassociative product on A J t K = lim n A ⊗ k k [ t ] / ( t n +1 ) and b A/ ( t ) ≃ A as k -algebras.We say that two deformations ( A J t K , ⋆ ) and ( A J t K , ⋆ ′ ) are (gauge) equivalent if thereis an automorphism of k J t K -modules T : A J t K A J t K such that T ⊗ k J t K k = id A and T ( a ⋆ b ) = T ( a ) ⋆ ′ T ( b )for any a, b ∈ A . Formal deformations of reduction systems. Let k Q [ t ] := k Q ⊗ k k [ t ] and set k Q J t K :=lim n k Q [ t ] / ( t n +1 ). Recall that a reduction system R = { ( s, f s ) | s ∈ S } satisfying( ⋄ ) for I = ( s − f s ) ( s,f s ) ∈ R implies that A ≃ k Irr S as k Q -bimodules. A formaldeformation of R is given by b R g = { ( s, f s + g s ) } s ∈ S where g ∈ Hom( k S, k Irr S ) b ⊗ m such that for each n ≥ 1, the reduction system R ( n ) g = { ( s, f s + g ( n ) s ) } ( s,f s ) ∈ R for k Q [ t ] / ( t n +1 ) satisfies that every path is reductionunique. Indeed, b R g can be viewed as an inverse limit of reduction systems for thepath algebra of a finite quiver — which might be denoted Q [ t ] — obtained by addinga loop t i at each vertex i ∈ Q and adding relations t s( x ) x = x t t( x ) for each x ∈ Q .We say that two formal deformations b R g and b R g ′ are equivalent if there is anautomorphism of k J t K -modules T : k Irr S J t K k Irr S J t K satisfying T ⊗ k J t K k = id k Irr S such that T ( π ( u )) = red ( ∞ ) f + g ′ ( T ( u ) · · · T ( u m )) u = u · · · u m (7.3)for all irreducible paths u ∈ Irr S and that for every s ∈ Sf s + g s = T − (red ( ∞ ) f + g ′ ( T ( s ) · · · T ( s n ))) s = s · · · s n (7.4)(cf. Definition 3.13). Formal deformations of ideals of relations. Lastly, a formal deformation of an ideal I ⊂ k Q is given by a two-sided ideal b J = lim n J ( n ) of k Q J t K , which is complete asa k Q J t K -bimodule such that J (0) = I and there is an isomorphism of k J t K -modules ϕ : A J t K k Q J t K / b J (7.5)with ϕ (0) := ϕ ⊗ k J t K k = id A . We say that two formal deformations of ideals ( b J , ϕ )and ( b J ′ , ϕ ′ ) are equivalent if there is an automorphism of k J t K -modules T : A J t K EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 31 A J t K such that T ⊗ k J t K k = id A and ϕ ′ ◦ T ◦ ϕ − : k Q J t K / b J k Q J t K / b J ′ is an isomor-phism of k J t K -algebras. Actually a formal deformation ( b J , ϕ ) does not depend on thechoice of the isomorphism ϕ up to equivalence, since any two formal deformations( b J, ϕ ) and ( b J, ϕ ′ ) are equivalent under the automorphism T = ϕ − ◦ ϕ ′ . Let R = { ( s, f s ) | s ∈ S } be a reduction system satisfying ( ⋄ ) for I . Let g ∈ Hom( k S, A ) b ⊗ m . We define the two-sided ideal b I g of k Q J t K to be the completionof ideals I ( n ) g of k Q [ t ] / ( t n +1 ) generated by the set { s − f s − g ( n ) s | s ∈ S } , where g ( n ) s = g s t + · · · + g ns t n . Note that there is a natural k J t K -linear map ϕ g : A J t K k Q J t K / b I g sending π ( u ) to [ u ] for any irreducible path u , where [ u ] denotes the image of u under the natural map k Q J t K k Q J t K / b I g . We will show in Proposition 7.47 that( b I g , ϕ g ) is a formal deformation of I if and only if g is a Maurer–Cartan element of p ( Q, R ) b ⊗ m .7.1.2. Overview. We give a brief overview over the remainder of this section. In § § ∞ algebra p ( Q, R ) controlling deformations of reduction systems. In § p ( Q, R ) (see Theorem 7.36 in § all elements. Many of the computations appear in the proofs of various technicallemmas which can safely be skipped on a first reading.In § § § An L ∞ algebra structure on P • . In § P , ∂ ) (Bar , d ) FG h (7.6)between the bar resolution of A = k Q/I and the resolution P obtained from areduction system R satisfying the condition ( ⋄ ) for I .In this section, we apply the functor Hom A e ( − , A ) to (7.6) and get a homotopydeformation retract for the Hochschild cochain complex of A . By the HomotopyTransfer Theorem (see for example [86, § ∞ algebra structureon Hom A e ( P , A ).First note that for any k Q -bimodule M , there is a natural isomorphism(7.7) Hom A e ( A ⊗ M ⊗ A, A ) ≃ Hom k Q e0 ( M, A )given by e f f , where f ( v ) = e f (1 ⊗ v ⊗ EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 32 Set P i := Hom k Q e0 ( k S i , A ) and let ∂ i denote the image of Hom A e ( ∂ i +1 , A ) un-der the isomorphism (7.7). Similarly, we denote C i ( A, A ) := Hom k Q e0 ( ¯ A ⊗ i , A ) ≃ Hom A e ( A ⊗ ¯ A ⊗ i ⊗ A, A ) and denote by d i the image of Hom A e ( d i +1 , A ) under thenatural isomorphism.From (7.6), we get the following special homotopy deformation retract( P , ∂ ) (C ( A, A ) , d ) GF h (7.8)where F , G and h are respectively the duals of F , G and h under the naturalisomorphism (7.7) which satisfy F G − id = 0 G F − id = h d + d h F h = 0 h G = 0 h h = 0 . Recall that h ( A ) = (C ( A, A )[1] , d , [ − , − ]) is a DG Lie algebra (see § Theorem 7.9. There exists an L ∞ algebra (7.10) p ( Q, R ) = ( P , ∂ , h− , −i , h− , − , −i , . . . ) with underlying cochain complex ( P , ∂ ) such that the injection G : P C ( A, A ) extends to an L ∞ quasi-isomorphism G : p ( Q, R ) ∼ h ( A ) whose components are G k : Λ k P C ( A, A ) . The higher brackets of p ( Q, R ) can be computed from the DG Lie bracket [ − , − ]on (C ( A, A ) , d ), the comparison maps F and G , and the homotopy h . The higherbrackets will be described in § Remark . Note that if R ′ is any other reduction system satisfying ( ⋄ ) for I with associated cochain complex ( P ′ , ∂ ′ ), then the same construction gives anL ∞ algebra p ( Q, R ′ ) and there is an L ∞ quasi-isomorphism p ( Q, R ) ≃ p ( Q, R ′ ) (cf.Lemma 6.14), i.e. any reduction system can be used to study the deformation theoryof A . Definition 7.12. Given g ∈ Hom( k S, A ) b ⊗ m , one can use the quasi-isomorphism G of Theorem 7.9 to define a k J t K -bilinear map ⋆ g : A J t K ⊗ k J t K A J t K A J t K by(7.13) a ⋆ g b = ab + G ( g )( a ⊗ b ) + G ( g, g )( a ⊗ b ) + G ( g, g, g )( a ⊗ b ) + · · · for a, b ∈ A . Theorem 7.14. If g is a Maurer–Cartan element of p ( Q, R ) b ⊗ m , then ⋆ g is as-sociative. Moreover, if g and g ′ are gauge-equivalent Maurer–Cartan elements of p ( Q, R ) b ⊗ m , then ⋆ g and ⋆ g ′ are gauge equivalent star products, i.e. they are iso-morphic as deformations of the associative product on A = k Q/I .Proof. This follows straight from Theorems 6.8 ( i ) and 7.9 and Proposition 6.2. (cid:3) Remark . If the reduction system R has no overlaps, then P = 0 and theobstruction space HH ( A ) vanishes, so that any ⋆ g is associative for any g . In § EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 33 Remark . Let g be a Maurer–Cartan element of p ( Q, R ) such that G k ( g, . . . , g ) =0 for k ≫ 0. Then similar to Theorem 7.14 we have that ⋆ g is well-defined and as-sociative.7.3. Combinatorial star product. In this section we introduce a combinatorialstar product ⋆ g and prove that ⋆ g coincides with ⋆ g — even if g is not a Maurer–Cartan element of p ( Q, R ) b ⊗ m (see (7.13)). This gives a combinatorial interpretationof the star product ⋆ g obtained by homotopy transfer.Let g ∈ Hom( k S, A ), viewed as a degree 1 element in the L ∞ algebra p ( Q, R ),and denote by R g the reduction system R g = { ( s, f s + g s ) | s ∈ S } . At this point we do not assume that R g is reduction finite or reduction unique.Adapting the definition of the right-most reduction red f given in (3.6) we nowdefine a combinatorial k -bilinear operation ⋆ g on A .Let z be simply a formal (bookkeeping) central variable which will keep track ofhow many times g has been used in the reduction. Similar to (3.6), we definered gz : k Q [ z ] k Q [ z ]as the k [ z ]-linear extension of the following mapred gz ( p ) = ( q g s r z if p is a path such that g split R2 ( p ) = q ⊗ s ⊗ rp if p is an irreducible path in k Q. (In general, red gz = red g z since for an irreducible path u we have red gz ( u ) = u , butred g z ( u ) = uz .) We denote by red f + gz : k Q [ z ] k Q [ z ] the k [ z ]-linear extensionof red f + red gz and by red kf + gz the k th iterated composition of red f + gz .For any fixed n ≥ 0, the action of red f + gz on k Q [ z ] / ( z n +1 ) is stable , i.e. red kf + gz =red k +1 f + gz for k ≫ f is stable (because R is reduction finite) and red ngz =red n +1 gz = red n +2 gz = · · · . This induces a k -linear mapred ( n ) f + gz : k Q k Q [ z ] / ( z n +1 ) p [red kf + gz ( p )]where k ≫ kf + gz ( p )] denotes the image of red kf + gz ( p ) under the quotientmap k Q [ z ] k Q [ z ] / ( z n +1 ). In fact, the image of red ( n ) f + gz lies in k Irr S [ z ] / ( z n +1 )and we have the following commutative diagram k Q k Irr S [ z ] / ( z n +1 ) k Irr S [ z ] / ( z n ). red ( n ) f + gz red ( n − f + gz This induces a k -linear map red ( ∞ ) f + gz : k Q k Irr S J z K . (7.17) Lemma 7.18. Let R be a reduction system satisfying ( ⋄ ) for I and let g ∈ Hom( k S, A ) .If R g is reduction finite, then for any p ∈ k Q we have red ( ∞ ) f + gz ( p ) ∈ k Irr S [ z ] . EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 34 Proof. Note that red ( ∞ ) f + gz ( p ) is linear combination of irreducible paths obtained byright-most reduction of the element p with respect to the reduction system R g .Thus, if the reduction system R g is reduction finite, then red kf + gz ( p ) = red k +1 f + gz ( p )on k Q [ z ] for k ≫ 0. This yields that red ( ∞ ) f + gz ( p ) is a finite sum and in particularred ( ∞ ) f + gz ( p ) is in k Irr S [ z ]. (cid:3) Definition 7.19. Let g ∈ Hom( k S, A ). For each k ≥ 0, we define a k -linearoperation ⋆ g,k on A by a ⋆ g,k b = Res z =0 (cid:16) π (cid:0) red ( ∞ ) f + gz ( σ ( a ) σ ( b )) (cid:1) z − k − (cid:17) a, b ∈ A (7.20)where Res z =0 is the (algebraic) residue of the formal Laurent series in z , i.e. for h ( z ) = P n h n z n we have Res z =0 ( h ( z )) = h − . In other words, a ⋆ g,k b is the imageof the coefficient of z k in red ( ∞ ) f + gz ( σ ( a ) σ ( b )) under the projection π : k Q A andso a ⋆ g,k b is a linear combination of irreducible paths obtained by performing right-most reductions (with respect to R g ) on σ ( a ) σ ( b ) using g exactly k times.We have a ⋆ g, b = ab . Note that for any a, b the residue in (7.20) is well defined,since σ ( a ) σ ( b ) is reduction finite with respect to R . But, in general, the sum a ⋆ g b := X k ≥ a ⋆ g,k b = π (cid:0) red ( ∞ ) f + gz ( σ ( a ) σ ( b )) (cid:12)(cid:12) z =1 (cid:1) is not well defined since it may be an infinite sum of non-zero elements.If R g is reduction finite, then by Lemma 7.18, ⋆ g is well defined. In this case, a⋆ g b is the linear combination of irreducible paths obtained by right-most reductions of σ ( a ) σ ( b ) with respect to the reduction system R g .By k J t K -linear extension, the operation ⋆ g can also be defined for any formalreduction system R g , where g ∈ Hom( k S, A ) b ⊗ m . Definition 7.21. Let g ∈ Hom( k S, A ) b ⊗ m . The k J t K -bilinear operation ⋆ g on A J t K is given by a ⋆ g b = π (cid:0) red ( ∞ ) f + gz ( σ ( a ) σ ( b )) (cid:12)(cid:12) z =1 (cid:1) a, b ∈ A (7.22)where red ( ∞ ) f + gz is the k J t K -linear extension of (7.17): Write g = P i ≥ g i t i for g i ∈ Hom( k S, A ). Then red ( ∞ ) f + gz is defined as the inverse limit of(red f + P i ≥ red g i t i z ) ( n ) : k Q J t K k Irr S J t K [ z ] / ( z n +1 ) . Note that a ⋆ g b is the linear combination of irreducible paths obtained as right-most reductions of the element σ ( a ) σ ( b ). For any k ≥ 0, we denote by a ⋆ g,k b thesummands in a ⋆ g b obtained by performing right-most reductions using g exactly k times. In the formal case a ⋆ g,k b may also be given by the formula (7.20) and wehave a ⋆ g, b = ab and a ⋆ g b = X k ≥ a ⋆ g,k b. However, in the formal case a ⋆ g b is always well defined for any g ∈ Hom( k S, A ) b ⊗ m since the coefficient of t n is a finite sum. (Precisely, there are at most 2 n − termsfor the compositions of n into positive integers, e.g. for n + · · · + n k = n we have EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 35 a term obtained by doing right-most reductions using first g n , then g n , etc. up to g n k .)For general g , the bilinear map ⋆ g need not be associative. Nevertheless, thenext result shows that ⋆ g always coincides with ⋆ g . Theorem 7.23. Let a, b ∈ A and write G ( a ⊗ b ) := ab . ( i ) If g ∈ Hom( k S, A ) , then k ! G k ( g, . . . , g )( a ⊗ b ) = a ⋆ g,k b for any k ≥ . (7.24) In particular, if a ⋆ g,k b = 0 for k ≫ , then both a ⋆ g b := X k ≥ k ! G k ( g, . . . , g )( a ⊗ b ) and a ⋆ g b are well defined and a ⋆ g b = a ⋆ g b . ( ii ) If g ∈ Hom( k S, A ) b ⊗ m , then (7.24) still holds and moreover a ⋆ g b = a ⋆ g b i.e. the star product ⋆ g obtained from homotopy transfer can be describedpurely in terms of performing reductions with respect to R g . In the remainder of this subsection we give a proof of Theorem 7.23. Lemma 7.25. Let g ∈ Hom( k S, A ) or g ∈ Hom( k S, A ) b ⊗ m . Let u be an irreduciblepath. ( i ) If there exists x ∈ Q such that xu = sv for some s ∈ S , then for any k ≥ we have π ( x ) ⋆ g,k π ( u ) = π ( f s ) ⋆ g,k π ( v ) + π ( g s ) ⋆ g,k − π ( v ) . Similarly, if v is an irreducible path such that uv = u ′ s for some s ∈ S andsome irreducible path u ′ , then we have π ( u ) ⋆ g,k π ( v ) = π ( u ′ ) ⋆ g,k π ( f s ) + π ( u ′ ) ⋆ g,k − π ( g s ) . ( ii ) Let v be an irreducible path such that uv is irreducible. Then for any k > π ( u ) ⋆ g,k π ( v ) = 0 . ( iii ) Let v be an irreducible path and write u = u · · · u n for u i ∈ Q . Then π ( u ) ⋆ g,k π ( v ) = n − X i =1 k − X j =1 π ( u · · · u i − )( π ( u i ) ⋆ g,j ( π ( u i +1 · · · u n ) ⋆ g,k − j π ( v )))+ n X i =1 π ( u · · · u i − )( π ( u i ) ⋆ g,k π ( u i +1 · · · u n v )) . Proof. The first and second assertions follow straight from the definition.To see ( iii ), for any 1 ≤ i ≤ n − π ( u i · · · u n ) ⋆ g,k π ( v ) = k X j =0 π ( u i ) ⋆ g,j ( π ( u i +1 · · · u n ) ⋆ g,k − j π ( v )) . Multiplying π ( u · · · u i − ) and taking the sum for 1 ≤ i ≤ n − 1, we get ( iii ). Herewe need to use the fact that a ⋆ g, b = ab . (cid:3) EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 36 The following lemma is very useful to compute the higher morphisms. Denoteby e ψ the preimage of ψ under the isomorphism (7.7). Lemma 7.26. ( i ) Let g ∈ Hom( k S, A ) or g ∈ Hom( k S, A ) b ⊗ m . Then we havethe following recursive formula for any k ≥ G k ( g, . . . , g )(¯ u ⊗ ¯ v ) = k − X i =1 (cid:18) ki (cid:19) e G i,k − i ( g, . . . , g )( h (1 ⊗ ¯ u ⊗ ¯ v ⊗ where G i,k − i ( g, . . . , g ) := G i ( g, . . . , g ) • G k − i ( g, . . . , g ) is the Gerstenhabercircle product (cf. Definition 6.3) of the two elements G i ( g, . . . , g ) and G k − ( g, . . . , g ) in C ( A, A ) and the map h is given by (5.10). ( ii ) Let u and v be two irreducible paths such that uv is irreducible. Then forany k ≥ and any g ∈ Hom( k S, A ) or g ∈ Hom( k S, A ) b ⊗ m , we have G k ( g, . . . , g )(¯ u ⊗ ¯ v ) = 0 . Proof. We first prove ( i ). By the Homotopy Transfer Theorem G k ( g, . . . , g ) is givenby binary trees so that G k ( g, . . . , g ) = k − X i =1 h ([ G i ( g, . . . , g ) , G k − i ( g, . . . , g )]) . For example, the following is a tree appearing in the expression for G = h (cid:18) • (cid:19) with the two trees on the right-hand side appearing in the expressions for G and G , respectively. (Here denotes an input of G ( g ), the composition given bythe Gerstenhaber product • and a copy of h .)Next we prove ( ii ) by induction on k . It follows from Lemma 5.16 ( ii ) that G ( g )(¯ u ⊗ ¯ v ) = e g ( G (1 ⊗ ¯ u ⊗ ¯ v ⊗ . For k ≥ 2, we have G k ( g, . . . , g )(¯ u ⊗ ¯ v ) = k − X i =1 (cid:18) ki (cid:19) e G i,k − i ( ̟ F G (1 ⊗ ¯ u ⊗ ⊗ ¯ v )= k ! k − X i =1 n − X j =1 π ( u · · · u j − ) G i (¯ u j ⊗ G k − i ( u j +1 · · · u n ⊗ ¯ v ))= 0where the first identity follows from ( i ) and (5.9), the second identity from Lemma5.15, and the last identity from the induction hypothesis since u j +1 · · · u n v is stillirreducible. Here for simplicity we have written e G i,k − i for e G i,k − i ( g, . . . , g ). (cid:3) Proof of Theorem 7.23. For the proof, we shall use the relation (cid:22) introduced in [40, § 2] for any reduction system satisfying ( ⋄ ) for an ideal I . Here (cid:22) is a relation on theset { λp | λ ∈ k \ { } , p ∈ Q } defined as the least reflexive and transitive relationsuch that λp (cid:22) µq if there is a reduction r such that r ( µq ) = λp + r , where r is a EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 37 linear combination of paths and p does not appear as a summand of r . We write λp ≺ µq if λp (cid:22) µq and λp = µq . It follows from [40, Lem. 2.11] that (cid:22) satisfiesthe descending chain condition since R satisfies the condition ( ⋄ ). In this proof weagain simply write e G i,k − i instead of e G i,k − i ( g, . . . , g ).We now prove (7.24) for a = π ( u ) and b = π ( v ) by induction on k and byinduction on the length of u . (Note that for k = 0 (7.24) holds by definition so thebase case is k = 1.) We first consider a = π ( u ) and b = π ( v ) for u ∈ Q and v anirreducible path. If k = 1, it follows from Lemmas 7.25 and 5.16 that π ( u ) ⋆ g, π ( v ) = G ( g )(¯ u ⊗ ¯ v ) . For k ≥ ≺ for uv . By Lemma 7.26 ( ii ) andLemma 7.25 ( ii ), (7.24) holds if uv is irreducible. If uv is not irreducible, then wemay write uv = sw with s ∈ S and w irreducible. We have(7.27) π ( u ) ⋆ g,k π ( v ) = π ( g s ) ⋆ g,k − π ( w ) + π ( f s ) ⋆ g,k π ( w )= π ( g s ) ⋆ g,k − π ( w ) + 1 k ! G k ( g, . . . , g )( ¯ f s ⊗ ¯ w )where the first identity follows from Lemma 7.25 ( i ) and the second identity fromthe induction hypothesis since f s w ≺ uv .Let k ≥ 2. By Lemma 7.26 ( i ) we have G k ( g, . . . , g )(¯ u ⊗ ¯ v )(7.28)= k − X i =1 (cid:18) ki (cid:19) e G i,k − i (cid:16) ̟ F (1 ⊗ s ⊗ π ( w )) ⊗ ̟ F G (1 ⊗ ¯ f s ⊗ ¯ w ⊗ ⊗ (cid:17) = k − X i =1 (cid:18) ki (cid:19) e G i,k − i (cid:16) ̟ F (1 ⊗ s ⊗ π ( w )) ⊗ − ̟ F G (1 ⊗ ¯ f s ⊗ ⊗ ¯ w ⊗ (cid:17) + G k ( g, . . . , g )( ¯ f s ⊗ ¯ w )where in the first identity we use the formula (5.9) for h and Lemma 5.16 ( i ).Combining (7.27) and (7.28), we see that (7.24) is equivalent to(7.29) π ( g s ) ⋆ g,k − π ( w ) = 1 k ! k − X i =1 (cid:18) ki (cid:19) e G i,k − i (cid:18)(cid:24) − ̟ F G (1 ⊗ ¯ f s ⊗ ⊗ ¯ w ⊗ ̟ F (1 ⊗ s ⊗ π ( w )) ⊗ (cid:23)(cid:19) . (Here we use ⌈−⌋ simply to indicate a line break.) Since by Lemma 5.15 we have F (1 ⊗ s ⊗ π ( w )) = split ( s ) ⊗ ¯ w ⊗ F G (1 ⊗ ¯ f s ⊗ ⊗ ¯ w ⊗ π ( g s ) ⋆ g,k − π ( w ) = 1 k ! k − X i =1 (cid:18) ki (cid:19) e G i,k − i ( ̟ split ( s ) ⊗ π ( w ) ⊗ . For this, by Lemma 7.26 ( ii ) the right hand side of the identity equals1 k ! k − X i =1 (cid:18) ki (cid:19) e G i (cid:16) e G k − i ( ̟ split ( s ) ⊗ ⊗ π ( w ) ⊗ (cid:17) = 1( k − G k − (¯ g s ⊗ ¯ w )= π ( g s ) ⋆ g,k − π ( w ) EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 38 where in the first identity we note that all the summands vanish except for i = k − u is an arbitrary irreducible path. We prove(7.24) by induction on the length of u = u · · · u n where u i ∈ Q . From the abovewe have seen that this identity holds for n = 1, so now let n ≥ 2. If k = 1, (7.24)follows from Lemmas 5.16 ( iii ) and 7.25 ( iii ). For k ≥ i ) to obtain G k ( g, . . . , g )(¯ u ⊗ ¯ v )= k − X i =1 (cid:18) ki (cid:19) e G i,k − i ( ̟F G (1 ⊗ ¯ u ⊗ ⊗ ¯ v ⊗ ̟F G (1 ⊗ ¯ u ⊗ ¯ v ⊗ n − X j =1 k − X i =1 (cid:18) ki (cid:19) π ( u · · · u j − ) G i,k − i (¯ u j ⊗ u j +1 · · · u n ⊗ ¯ v )+ n X j =1 k − X i =1 (cid:18) ki (cid:19) π ( u · · · u j − ) e G i,k − i ( ̟ F G (1 ⊗ ¯ u j ⊗ u j +1 · · · u n v ⊗ iii ) and 5.15. Thus we get1 k ! G k ( g, . . . , g )(¯ u ⊗ ¯ v )= 1 k ! n − X j =1 k − X i =1 (cid:18) ki (cid:19) π ( u · · · u j − ) G i ( g, . . . , g )(¯ u j ⊗ G k − i ( g, . . . , g )( u j +1 · · · u n ⊗ ¯ v ))+ 1 k ! π ( u · · · u j − ) G k ( g, . . . , g )(¯ u j ⊗ u j +1 · · · u n v )= n − X i =1 k − X j =1 π ( u · · · u i − )( π ( u i ) ⋆ g,j ( π ( u i +1 · · · u n ) ⋆ g,k − j π ( v ))+ n X i =1 π ( u · · · u i − )( π ( u i ) ⋆ g,k π ( u i +1 · · · u n v ))= π ( u ) ⋆ g,k π ( v )where the first identity follows from the fact that ̟ F G (1 ⊗ ¯ u j ⊗ 1) = 0 whence h (1 ⊗ ¯ u j ⊗ u j +1 · · · u n v ⊗ ̟ F G (1 ⊗ ¯ u j ⊗ u j +1 · · · u n v ⊗ iii ). (cid:3) Combinatorial criterion for the Maurer–Cartan equation. The follow-ing lemma gives a combinatorial way to compute the higher brackets h g, . . . , g i which appear in the Maurer–Cartan equation for the L ∞ algebra p ( Q, R ) b ⊗ m . Lemma 7.30. Let g ∈ Hom( k S, A ) ( resp. g ∈ Hom( k S, A ) b ⊗ m ) . Then for k ≥ ,the bracket h g, . . . , g i k ∈ Hom( k S , A ) ( resp. h g, . . . , g i k ∈ Hom( k S , A ) b ⊗ m ) isgiven by k ! h g, . . . , g i k ( w )= π ( f s ′ ) ⋆ g,k π ( v ) + π ( g s ′ ) ⋆ g,k − π ( v ) − π ( u ) ⋆ g,k π ( f s ) − π ( u ) ⋆ g,k − π ( g s ) EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 39 where w ∈ S and we write w = us = s ′ v with s, s ′ ∈ S .Proof. For any w ∈ S , we have h g, . . . , g i k ( w ) = k − X i =1 (cid:18) ki (cid:19) e G i,k − i ( F (1 ⊗ w ⊗ . Thus, by Lemma 5.15 ( iii ) and Theorem 7.23 we have1 k ! h g, . . . , g i k ( w ) = π ( f s ′ ) ⋆ g,k π ( v ) − π ( u ) ⋆ g,k π ( f s )+ 1 k ! k − X i =1 (cid:18) ki (cid:19) e G i,k − i ̟ F (1 ⊗ s ′ ⊗ π ( v )) ⊗ ̟ F G (1 ⊗ ¯ u ⊗ ⊗ ¯ f s ⊗ − ̟ F G (1 ⊗ ¯ f s ′ ⊗ ⊗ ¯ v ⊗ . Recall from Lemma 5.15 ( i ) that ̟ F (1 ⊗ s ′ ⊗ π ( v )) = ̟ split ( s ′ ) ⊗ ¯ v + ̟ F G (1 ⊗ ¯ f s ′ ⊗ ⊗ ¯ v. Thus it remains to verify that(7.31) 1 k ! k − X i =1 (cid:18) ki (cid:19) e G i,k − i (cid:18)(cid:24) ̟ F G (1 ⊗ ¯ u ⊗ ⊗ ¯ f s ⊗ ̟ split ( s ′ ) ⊗ ¯ v ⊗ (cid:23)(cid:19) = π ( g s ′ ) ⋆ g,k − π ( v ) − π ( u ) ⋆ g,k − π ( g s ) . For this, we write w = w · · · w n for w i ∈ Q so that w = w w l w l +1 w m w m +1 w n u ss ′ v... ... ... for some 0 < l < m < n . That is, w = s ′ v = w · · · w m v and we have(7.32) 1 k ! k − X i =1 (cid:18) ki (cid:19) e G i,k − i ( ̟ split ( s ′ ) ⊗ ¯ v ⊗ k ! m − X j =1 k − X i =1 (cid:18) ki (cid:19) π ( w · · · w j − ) G i,k − i ( ¯ w j ⊗ w j +1 · · · w m ⊗ ¯ v )= 1( k − G k − ( g, . . . , g )( G ( g )( ¯ w ⊗ w · · · w m ) ⊗ ¯ v ) + m − X j =1 k − X i =1 π ( w · · · w j − ) G i ( g, . . . , g )( ¯ w j ⊗ G k − i ( g, . . . , g )( w j +1 · · · w m ⊗ ¯ v ))= π ( g s ′ ) ⋆ g,k − π ( v ) − π ( w · · · w l − )( π ( w l ) ⋆ g,k − π ( g s )) + l − X j =1 k − X i =1 π ( w · · · w j − )( π ( w j ) ⋆ g,i ( π ( w j +1 · · · w m ) ⋆ g,k − i π ( v )) EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 40 where the third identity follows from Lemma 7.25 ( i ) and ( ii ): m − X j = l +1 π ( w · · · w j − )( π ( w j ) ⋆ g,i ( π ( w j +1 · · · w m ) ⋆ g,k − i π ( v )) = 0 π ( w l +1 · · · w m ) ⋆ g,i π ( v ) = ( π ( g s ) if i = 10 otherwise . Similarly, we have(7.33) 1 k ! k − X i =1 (cid:18) ki (cid:19) e G i,k − i ( ̟ F G (1 ⊗ ¯ u ⊗ ⊗ ¯ f s ⊗ − l − X j =1 k − X i =1 π ( w · · · w j − )( π ( w j ) ⋆ g,i ( π ( w j +1 · · · w l ) ⋆ g,k − i π ( f s )) . By Lemma 7.25 ( i ), for any 1 ≤ j ≤ l − i ≥ π ( w j +1 · · · w m ) ⋆ g,i π ( v ) = π ( w j +1 · · · w l ) ⋆ g,i π ( f s ) + π ( w j +1 · · · w l ) ⋆ g,i − π ( g s ) . (7.34)Thus it follows from (7.32), (7.33) and (7.34) that1 k ! k − X i =1 (cid:18) ki (cid:19) e G i,k − i (cid:16) ̟ F G (1 ⊗ ¯ u ⊗ ⊗ ¯ f s ⊗ ̟ split ( s ′ ) ⊗ ¯ v ⊗ (cid:17) = π ( g s ′ ) ⋆ g,k − π ( v ) − l X j =1 π ( w · · · w j − )( π ( w j ) ⋆ g,k − π ( w j +1 · · · w l g s )) − l − X j =1 k − X i =1 π ( w · · · w j − )( π ( w j ) ⋆ g,i ( π ( w j +1 · · · w l ) ⋆ g,k − i − π ( g s ))= π ( g s ′ ) ⋆ g,k − π ( v ) − π ( u ) ⋆ g,k − π ( g s )where the second identity follows from Lemma 7.25 ( iii ). This verifies (7.31). (cid:3) Remark . In the context of formal deformations, we usually consider an element g ∈ Hom( k S, A ) b ⊗ m and the higher brackets h g, . . . , g i appearing in the Maurer–Cartan equation of the L ∞ algebra p ( Q, R ) b ⊗ m are then simply m -linear extensionsof the brackets given in Lemma 7.30.Recall from § ⋆ g admits a combinatorial description in terms of reduc-tions, i.e. ⋆ g = ⋆ g . The following theorem shows that this description can be usedto check the Maurer–Cartan equation. Theorem 7.36. Let g ∈ Hom( k S, A ) b ⊗ m . Then g satisfies the Maurer–Cartanequation for p ( Q, R ) b ⊗ m if and only if for any w = u u u ∈ S , where s ′ = u u and s = u u are in S , we have (7.37) ( π ( u ) ⋆ g π ( u )) ⋆ g π ( u ) = π ( u ) ⋆ g ( π ( u ) ⋆ g π ( u )) . Similarly, let g ∈ Hom( k S, A ) . Then g satisfies the Maurer–Cartan equation for p ( Q, R ) if and only if for any a, b ∈ A , we have a ⋆ g,k b = 0 for k ≫ and theidentity (7.37) holds for ⋆ g . EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 41 Note that to check if g satisfies the Maurer–Cartan equation, it suffices to checkthat ⋆ g is associative on elements w ∈ S . In the terminology of Definition 3.7,the condition (7.37) is equivalent to the overlap w ∈ S being resolvable usingright-most reductions with respect to R g . Proof of Theorem 7.36. The Maurer–Cartan equation for g is given by X k ≥ k ! h g, . . . , g i k = 0 ∈ Hom( k S , A ) b ⊗ m (see (6.19)). For any w ∈ S we have X k ≥ k ! h g, . . . , g i k ( w ) = X k ≥ & π ( u ) ⋆ g,k π ( f s ) + π ( u ) ⋆ g,k − π ( g s ) − π ( f s ′ ) ⋆ g,k π ( u ) − π ( g s ′ ) ⋆ g,k − π ( u ) % = π ( u ) ⋆ g π ( f s + g s ) − π ( f s ′ + g s ′ ) ⋆ g π ( u )= π ( u ) ⋆ g ( π ( u ) ⋆ g π ( u )) − ( π ( u ) ⋆ g π ( u )) ⋆ g π ( u )where the first identity follows from Lemma 7.30 and the third identity from Lemma7.25 ( i ). Here we set ⋆ g, − = 0. (cid:3) Remark . Note that the L ∞ quasi-isomorphism between h ( A ) and p ( Q, R ) im-plies that (up to equivalence) all formal deformations of A can be given as ( A J t K , ⋆ g )and the combinatorial star product ⋆ g then gives an explicit formula for the de-formed multiplication in terms of the k -basis given by irreducible paths. (Also see § § k Q , A ) b ⊗ m . Proposition 7.39. Two formal deformations b R g and b R g ′ are equivalent if and onlyif ( A J t K , ⋆ g ) and ( A J t K , ⋆ g ′ ) are gauge equivalent.Proof. Let T : A J t K A J t K be an equivalence between b R g and b R g ′ . Consider thealgebra homomorphism ψ : k Q J t K ( A J t K , ⋆ g ′ ) determined by ψ ( x ) = T ( x ) forarrows x ∈ Q . It follows from (7.3) that for any irreducible path u = u · · · u m ψ ( u ) = T ( π ( u )) ⋆ g ′ · · · ⋆ g ′ T ( π ( u m )) = T ( u )and (7.4) implies that for each s ∈ S we have ψ ( s − f s − g s ) = T ( π ( s )) ⋆ g ′ · · · ⋆ g ′ T ( π ( s m )) − T ( f s ) − T ( g s ) = 0 . This shows that ψ induces an algebra homomorphism ψ : k Q J t K / ( s − f s − g s ) s ∈ S ( A J t K , ⋆ g ′ ) . In Proposition 7.46 we will show that there is a natural algebra isomorphism( A J t K , ⋆ g ) ≃ k Q J t K / ( s − f s − g s ) s ∈ S . Note that the composition with ψ coincideswith T . This shows that T is a gauge equivalence between ( A J t K , ⋆ g ) and ( A J t K , ⋆ g ′ ). EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 42 On the other hand, if T : ( A J t K , ⋆ g ) ( A J t K , ⋆ g ′ ) is a gauge equivalence, thenfor any path u = u · · · u m we have T ( π ( u ) ⋆ g · · · ⋆ g π ( u m )) = T ( π ( u )) ⋆ g ′ · · · ⋆ g ′ T ( π ( u m ))= red ∞ f + g ′ ( T ( π ( u )) · · · T ( π ( u m ))) . This yields (7.3) and (7.4). (cid:3) Also observe that Theorems 7.14 and 7.23 imply that two formal deformations b R g and b R g ′ of the reduction system R are equivalent if and only if g and g ′ arehomotopy equivalent viewed as Maurer–Cartan elements of p ( Q, R ) b ⊗ m (cf. § Deformations of relations. In this section we will prove the equivalencebetween deformations of the ideal of relations I and deformations of reductionssystems for k Q/I . Definition 7.40. Let ( b J , ϕ ) be a formal deformation of the ideal I ⊂ k Q (cf. (7.5)),i.e. J (0) = I and there is a k J t K -module isomorphism ϕ : A J t K k Q J t K / b J .We associate to ( b J, ϕ ) a (generalized) Gutt star product ⋆ on b A = A J t K definedby a ⋆ b = ϕ − ( ϕ ( a ) ϕ ( b ))for any a, b ∈ A . (The original Gutt star product was introduced in Gutt [66] as aquantization of the Lie–Kirillov Poisson structure on the dual of a Lie algebra.) Proposition 7.41. Let ( b J, ϕ ) be a formal deformation of an ideal I ⊂ k Q . Then b A = ( A J t K , ⋆ ) is a formal deformation of A = k Q/I . Moreover, if ( b J, ϕ ) and ( b J ′ , ϕ ′ ) are equivalent formal deformations of I , then b A = ( A J t K , ⋆ ) and b A ′ =( A J t K , ⋆ ′ ) are equivalent formal deformations of the associative algebra A .Proof. By definition ⋆ is associative. Since ϕ (0) = id A we have b A/ ( t ) ≃ A .Set a⋆ ′ b = ϕ ′− ( ϕ ′ ( a ) ϕ ′ ( b )). By definition both ϕ : b A k Q J t K / b J and ϕ ′ : b A ′ k Q J t K / b J ′ become isomorphisms of k J t K -algebras. If ( b J, ϕ ) and ( b J ′ , ϕ ′ ) are equiva-lent, then by definition there is an isomorphism of k J t K -modules T : A J t K A J t K such that ϕ ′ ◦ T ◦ ϕ − is an isomorphism of k J t K -algebras. We infer that T inducesan k J t K -algebra isomorphism between b A and b A ′ . (cid:3) Remark . The Gutt star product ⋆ is defined by pulling back the algebrastructure on one vector space to another vector space and is as such bilinear andassociative by definition. Note that this definition thus works for any vector spaceisomorphism (e.g. also for an actual deformation).Let ( g , [ − , − ]) be a Lie algebra. Let U t ( g ) be the (formal) universal envelopingalgebra, i.e. U t ( g ) = T( g ) J t K / ( x ⊗ y − y ⊗ x − [ x, y ] t ) x,y ∈ g . In particular, when t = 1we get the usual universal enveloping algebra U( g ).The symmetrization map is the following k J t K -linear isomorphism ρ : S( g ) J t K U t ( g ) x x · · · x k k ! X s ∈ S k x s (1) x s (2) · · · x s ( k ) (7.43) EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 43 where S k is the set of permutations of k elements and S( g ) = T( g ) / ( x ⊗ y − y ⊗ x ) x,y ∈ g is the symmetric algebra over g . Then the classical Gutt star product on thesymmetric algebra S( g ) J t K is given by a ∗ G b = ρ − ( ρ ( a ) ρ ( b )) for a, b ∈ S( g ) J t K . (7.44)For the relation between ⋆ and the classical Gutt star product ∗ G see Remark 8.13( iii ).Let g ∈ Hom( k S, A ) b ⊗ m . We write g as g = g t + g t + g t + · · · g i ∈ Hom( k S, A )and let g ( n ) = g t + · · · + g n t n denote the image under tensoring by ⊗ k J t K k [ t ] / ( t n +1 ).We define the two-sided ideal b I g of k Q J t K to be the completion of ideals I ( n ) g of k Q [ t ] / ( t n +1 ) generated by the set { s − f s − g ( n ) s | s ∈ S } . There is a natural k J t K -linear map ϕ g : A J t K k Q J t K / b I g (7.45)sending π ( u ) to [ u ] for any irreducible path u . Proposition 7.46. ( i ) The element g ∈ Hom( k S, A ) b ⊗ m is a Maurer–Cartanelement of the L ∞ algebra p ( Q, R ) b ⊗ m if and only if ( b I g , ϕ g ) is a formaldeformation of I , i.e. the natural map ϕ g : A J t K k Q J t K / b I g is an isomor-phism. In this case, we have ⋆ g = ⋆ g . ( ii ) If R g is reduction finite, the pair ( I g , ϕ g ) is an actual deformation of I —i.e. the natural map ϕ g : A k Q/I g is an isomorphism — if and only if g ∈ Hom( k S, A ) is a Maurer–Cartan element of p ( Q, R ) . It follows from Proposition 7.46 and Remark 7.38 that any formal deformation( b J, ϕ ) of the ideal I is equivalent to ( b I g , ϕ g ) for some Maurer–Cartan element g . Proof of Proposition 7.46. Let us first prove the formal case. Let g be a Maurer–Cartan element of p ( Q, R ) b ⊗ m . It follows from Theorem 7.14 that ( A J t K , ⋆ g ) is anassociative k J t K -algebra. Since k Q is the (free) tensor algebra generated by Q over k Q , the k Q -bimodule map k Q A J t K x π ( x )extends uniquely to an algebra homomorphism k Q ( A J t K , ⋆ g ). By k J t K -bilinearextension, this yields a k J t K -algebra homomorphismΦ : k Q J t K ( A J t K , ⋆ g ) . We claim that for any irreducible path u we have(7.47) Φ( u ) = π ( u ) . We prove this claim by induction on the length of u . If u ∈ Q , then it is clear thatΦ( u ) = π ( u ). Assume that Φ( v ) = π ( v ) for any proper irreducible subpath v of u .Then for u = xv with x ∈ Q we haveΦ( u ) = Φ( xv ) = Φ( x ) ⋆ g Φ( v ) = π ( x ) ⋆ g π ( v ) = π ( x ) ⋆ g π ( v ) = π ( xv ) EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 44 where the third identity follows from the induction hypothesis, the fourth identityfrom Theorem 7.23, and the last identity from Lemma 7.25 ( ii ).Let us prove that the ideal b I g is sent to zero under the map Φ. That is, for any( s, f s ) ∈ R , we have Φ( s − f s − g s ) = 0 . Note that g s = σg ( s ) is irreducible (and πσ = id A ) so by (7.47), this is equivalentto proving Φ( s ) = π ( f s + g s ) . So let us write s = xu ∈ S with x ∈ Q and u ∈ Q ≥ . Then we haveΦ( s ) = Φ( x ) ⋆ g Φ( u ) = π ( x ) ⋆ g π ( u ) = π ( x ) ⋆ g π ( u ) = π ( f s + g s )where the first identity follows from the fact that Φ is an algebra homomorphismand the second identity from (7.47), the third identity from Theorem 7.23, and thelast identity from Lemma 7.25 ( i ). Thus, the morphism Φ induces a homomorphismof k -algebras Φ g : k Q J t K / b I g ( A J t K , ⋆ g ) . (7.48)Recall that k Q J t K = lim n k Q [ t ] / ( t n +1 ) and A J t K = lim n A [ t ] / ( t n +1 ). Note thatthe above map Φ : k Q J t K ( A J t K , ⋆ g ) induces an algebra homomorphismΦ ( n ) : k Q [ t ] / ( t n +1 ) ( A [ t ] / ( t n +1 ) , ⋆ g )for each n ≥ 0. Moreover, we have Φ = lim n Φ ( n ) . Note that Φ ( n ) sends I ( n ) g to zero,and thus inducing an algebra homomorphismΦ ( n ) g : k Q [ t ] / ( t n +1 ) (cid:14) I ( n ) g ( A [ t ] / ( t n +1 ) , ⋆ g ) . Moreover, the map Φ g in (7.48) can be written as Φ g = lim n Φ ( n ) g . It remains to prove that Φ g is bijective. We claim that Φ ( n ) g is bijective for each n ≥ 1. Indeed, note that k Q [ t ] / ( t n +1 ) (cid:14) I ( n ) g is k -spanned by the set k Irr S [ t ] / ( t n +1 ) = { u i t i | u i ∈ Irr S } ≤ i ≤ n . It follows from Theorem 3.8 that π ( k Irr S [ t ] / ( t n +1 )) = { π ( u i ) t i | u i ∈ Irr S } ≤ i ≤ n is a k -linear basis of A [ t ] / ( t n +1 ). Since Φ ( n ) g ( u i t i ) = π ( u i ) t i for all u i ∈ Irr S andall 0 ≤ i ≤ n , this yields that Φ ( n ) g is injective and moreover k Irr S [ t ] / ( t n +1 ) is a k -linear basis of k Q [ t ] / ( t n +1 ) (cid:14) I ( n ) g . Thus, Φ ( n ) g is bijective, proving the claim.Since Φ g = lim n Φ ( n ) g , we get that Φ g is bijective. Note that ϕ g is the inverse of Φ g .This shows that ( b I g , ϕ g ) is a formal deformation of I .Conversely, if ( b I g , ϕ g ) is a formal deformation of I , then we have for any irre-ducible paths u and vπ ( u ) ⋆ g π ( v ) = ϕ − g ( ϕ g ( π ( u )) ϕ g ( π ( v ))) = ϕ − g ([ uv ]) = π ( u ) ⋆ g π ( v )(7.49)where the second identity follows from the definition of ϕ g (see (7.45)) and thethird identity follows because reductions (using s f s + g s ) induce the identity onelements in the quotient k Q J t K / b I g . To show that g is a Maurer–Cartan element, it EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 45 suffices by Theorem 7.36 to check that for any w = u u u ∈ S with s ′ = u u , s = u u ∈ S , we have( π ( u ) ⋆ g π ( u )) ⋆ g π ( u ) = π ( u ) ⋆ g ( π ( u ) ⋆ g π ( u )) . But now π ( u ) ⋆ g ( π ( u ) ⋆ g π ( u )) − ( π ( u ) ⋆ g π ( u )) ⋆ g π ( u ) = π ( u ) ⋆ g ( π ( u ) ⋆ g π ( u )) − ( π ( u ) ⋆ g π ( u )) ⋆ g π ( u ) = 0where the first identity follows from (7.49) and the second identity from the factthat ⋆ g is associative.Let us prove the non-formal case. Let ( I g , ϕ g ) be a deformation of I . Using thesame argument as in (7.49), we get that ⋆ g = ⋆ g and in particular ⋆ g is associative.Thus, it follows from Theorem 7.36 that g is a Maurer–Cartan element of p ( Q, R ).Conversely, since R g is reduction finite and thus a ⋆ g,k b = 0 for k ≫ 0, it followsfrom (7.24) that k ! G k ( g, . . . , g )( a ⊗ b ) = 0 for k ≫ 0. Thus, the star product ⋆ g on A is well defined (cf. (7.13)). Since g is a Maurer–Cartan element, ⋆ g is associative.Then by an argument analogous to the formal case, we get an algebra isomorphismΦ g : k Q/I g ( A, ⋆ g ) which is the inverse of ϕ g . (cid:3) Remark . Let e ∈ A be an idempotent of A corresponding to some vertex of Q . Let g ∈ Hom( k S, A ) b ⊗ m be a Maurer–Cartan element of p ( Q, R ) b ⊗ m . Thenby Proposition 7.46, the subalgebra eA g e of A g is a deformation of eAe and ananalogous statement holds for actual deformations (cf. § Notation 7.51. Let ⋆ g denote the combinatorial star product ⋆ g and let ⋆ g,k denote ⋆ g,k for k ≥ ⋆ g = ⋆ g for any g ∈ Hom( k S, A ) b ⊗ m and by Proposition 7.46 we also have ⋆ g = ⋆ g when g is a Maurer–Cartan elementof p ( Q, R ) b ⊗ m .7.A. The Diamond Lemma from a deformation-theoretic viewpoint. Usingthe results of this section we can give a deformation-theoretic interpretation of theDiamond Lemma as follows.Given an algebra A = k Q/I , a reduction system satisfying the condition ( ⋄ ) for I is given by a set of pairs R = { ( s, f s ) } . One may then consider the reduction system R mon = { ( s, | ( s, f s ) ∈ R } and the associated monomial algebra A mon = k Q/ ( S ),where S = { s | ( s, f s ) ∈ R } . Note that the irreducible paths only depend on the set S , so that the notion of irreducible paths is the same for R and R mon . Indeed, R is determined by S and a function f ∈ Hom( k S, A mon ) (cf. Remark 3.9). Clearly,the irreducible paths form a k -basis of A mon (cf. Example 3.4 ( i )) and R mon clearlysatisfies ( ⋄ ) for the ideal ( S ).Now Theorem 7.36 and Proposition 7.46 show that the equivalent conditions( i )–( iii ) in the Diamond Lemma 3.8 have the following deformation-theoretic in-terpretation: EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 46 Diamond Lemma ( i ) all overlap ambiguities of R areresolvable( ii ) R satisfies ( ⋄ ) for I ( iii ) the image of the set Irr S of ir-reducible paths under the pro-jection k Q π A is a k -basis of A . Deformation-theoretic viewpoint ( i ) f satisfies the Maurer–Cartanequation of p ( Q, R mon )( ii ) R is a deformation of R mon ( iii ) A is a (flat) deformation of A mon .Here ( i ) of the deformation-theoretic viewpoint says that if R = { ( s, f s ) } sat-isfies ( ⋄ ), then f can be viewed as a Maurer–Cartan element of the L ∞ algebra p ( Q, R mon ). Under the identification A ≃ k Irr S ≃ A mon one may now obtain theL ∞ algebra structure of p ( Q, R ) from the L ∞ algebra structure on p ( Q, R mon ) by Maurer–Cartan twisting (see for example [41, Thm. 2.6]). This process may begiven by the following explicit formulae: denoting the n -ary bracket of p ( Q, R mon )by h− , . . . , −i mon , the n -ary bracket h− , . . . , −i of p ( Q, R ) may be given by h x , . . . , x n i = h x , . . . , x n i mon + X m ≥ m ! h f, . . . , f | {z } m , x , . . . , x n i mon . In practice one usually deals with algebras which are not monomial and we havegiven all formulae in §§ Remark . After posting our preprint to the arXiv, the L ∞ algebra structurefor the monomial case also appeared in independent work by M.J. Redondo andF. Rossi Bertone [108] and a similar “homotopical” interpretation for the DiamondLemma was given by V. Dotsenko and P. Tamaroff [49].7.B. Computing Hochschild cohomology in degree . In § A the space of first-order deformations up to equivalence coin-cides with the second Hochschild cohomology group HH ( A, A ), where the associa-tive algebra structure is given by an element µ ∈ Hom k ( A ⊗ k , A ). It may be usefulto point out that as a consequence of Theorem 7.9 this still holds true if we identifydeformations of the associative multiplication on A = k Q/I with deformations of areduction system (cf. Theorem 7.1). Corollary 7.53. Let A = k Q/I and let R be any reduction system satisfying ( ⋄ ) for I . Then HH ( A, A ) is isomorphic to the space of first-order deformations of R modulo homotopy equivalence. We record the direct computation in the following lemma. Lemma 7.54. Let k [ t ] / ( t ) be the algebra of dual numbers with maximal ideal ( t ) . ( i ) For g ∈ Hom( k S, A ) the following are equivalent: (a) g is a -cocycle of ( P , ∂ )(b) gt ∈ Hom( k S, A ) ⊗ ( t ) is a Maurer–Cartan element of p ( Q, R ) ⊗ ( t )(c) for any w = u u u ∈ S , where s ′ = u u and s = u u are in S , wehave ( π ( u ) ⋆ gt π ( u )) ⋆ gt π ( u ) = π ( u ) ⋆ gt ( π ( u ) ⋆ gt π ( u )) mod t . ( ii ) For two -cocycles g, g ′ ∈ Hom( k S, A ) the following are equivalent: (a) g and g ′ are cohomologous EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 47 (b) there is a homotopy between gt and g ′ t viewed as Maurer–Cartan ele-ments of p ( Q, R ) ⊗ ( t )(c) the star products ⋆ gt and ⋆ g ′ t are gauge equivalent.Proof. This follows directly from the L ∞ quasi-isomorphism between p ( Q, R ) ⊗ ( t )and h ( A ) ⊗ ( t ) given in Theorem 7.9, but we give a direct proof for the equivalencesbetween (a) and (b).For ( i ) the element gt ∈ Hom( k S, A ) ⊗ ( t ) is a Maurer–Cartan element of p ( Q, R ) ⊗ ( t ) if and only if it satisfies the Maurer–Cartan equation ∂ ( gt ) + 12 h gt, gt i + 13! h gt, gt, gt i + · · · = 0 . Since the multibrackets h− , . . . , −i are k [ t ] / ( t )-linear and t = 0, this equationreduces to ∂ ( g ) t = 0.For ( ii ) two 2-cocycles g, g ′ ∈ Hom( k S, A ) are cohomologous if and only if thereexists h ∈ Hom( k Q , A ) such that g − g ′ = ∂ ( h ). Let Ω = k [ τ, d τ ] and define g τ ∈ p ( Q, R ) ⊗ Ω ⊗ ( t ) by g τ = g ′ t + ( g t − g ′ t ) τ + ht d τ (see Remark 6.23). Then g τ is a Maurer–Cartan element of p ( Q, R ) ⊗ Ω ⊗ ( t ) since d ( g τ ) + 12 h g τ , g τ i + · · · | {z } =0 = (cid:0) ∂ ( h ) − ( g − g ′ ) (cid:1) t d τ = 0where d is the differential of p ( Q, R ) ⊗ Ω ⊗ ( t ) and ev ( g τ ) = g and ev ( g τ ) = g ′ so that g t and g ′ t are homotopy equivalent Maurer–Cartan elements. (cid:3) Of course Lemma 7.54 is nothing but a deformation-theoretic reinterpretationof the formulae of the differential in p ( Q, R ), but it certainly makes these formulamore intuitive and the formulation of Corollary 7.53 may be a useful mnemonic forcomputing second Hochschild cohomology HH ( A, A ) for path algebras of quiverswith relations. 8. Application to deformation quantization In this section we give applications to deformation quantization of Poisson struc-tures on n -dimensional affine space A n . Throughout the section we denote thedeformation parameter by ~ and work with deformations over k J ~ K with maximalideal m = ( ~ ).8.1. Background. We give a very brief overview of deformation quantization andrefer to [82, 78, 38] for more details. Definition 8.1. Let ( X, {− , −} ) be a Poisson manifold and let A = C ∞ ( X ) bethe algebra of smooth functions. A star product ⋆ is an R J ~ K -bilinear associativeproduct on A J ~ K of the form f ⋆ g = f g + X n ≥ B n ( f, g ) ~ n such that( i ) each B n is a bidifferential operator which is a differential operator withrespect to each argument( ii ) 1 ⋆ f = f = f ⋆ f ∈ C ∞ ( X ). EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 48 We say that ⋆ is a quantization of {− , −} (i.e. a deformation quantization of( X, {− , −} )) if f ⋆ g − g ⋆ f ~ (cid:12)(cid:12)(cid:12) ~ =0 = B ( f, g ) − B ( g, f ) = { f, g } i.e. if the skew-symmetrization of the first-order term B of ⋆ coincides with thePoisson bracket.Deformation quantization was proposed in Bayen–Flato–Frønsdal–Lichnerowicz–Sternheimer [18] as viewing quantum mechanics as a formal deformation of classicalmechanics. The existence of a deformation quantization was first shown for sym-plectic manifolds (i.e. the non-degenerate case) by De Wilde–Lecomte [45] with adifferent explicit construction given by Fedosov [52]. The deformation quantizationof arbitrary Poisson structures was solved by Kontsevich [82], giving the followinggeneral result. Theorem 8.2 (Kontsevich [82]) . Any Poisson manifold ( X, {− , −} ) admits a de-formation quantization. Kontsevich’s proof relies on the construction of an L ∞ quasi-isomorphism, the Kontsevich formality morphism , between the DG Lie algebra v ( X ) of polyvectorfields on X (with the Schouten–Nijenhuis bracket and trivial differential) and theDG Lie algebra d ( X ) of multi-differential operators (viewed as a DG Lie subalgebraof the Hochschild DG Lie algebra h ( C ∞ ( X ))). This quasi-isomorphism gives anequivalence of Maurer–Cartan elements up to gauge equivalence in these two DGLie algebras (cf. Definition 6.15 and Theorem 6.8), which for v ( X ) are precisely(formal) Poisson structures and for d ( X ) star products.This formality morphism was constructed using an explicit universal formula forany Poisson structure on R d ; the statement for arbitrary Poisson manifolds followsfrom a globalization argument (see [82, § 7] and [39]). In particular, to each Poissonstructure on R d , given by a Maurer–Cartan element of v ( R d ), one can thus associatean explicit star product .Kontsevich’s explicit formula for Poisson structures on R d takes the followingform(8.3) f ∗ K g = f g + X k ≥ ~ k k ! X Γ ∈ G k, w Γ B Γ ( f, g )where G k, is a (finite) set of admissible graphs with k vertices in the upper half-plane and 2 vertices on the real line. To each graph Γ ∈ G k, one can associatea certain bidifferential operator B Γ , which is built from k copies of the Poissonbivector field.The subtlety in the formula (8.3) arguably lies in the choice of weights w Γ whichmake ∗ K an associative product. These Kontsevich weights are obtained by lookingat geodesic embeddings of Γ into the upper half plane (with the hyperbolic met-ric) and integrating a certain differential form (associated to the angles ϕ e of theindividual edges) over the compactification of a configuration space of points: w Γ = 1(2 π ) k Z H k ^ e ∈ Γ d ϕ e . EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 49 These weights are rather mysterious and some of them are conjecturally irrational(cf. Felder–Willwacher [53]), but the integrals have recently been computed asinteger-linear combinations of multiple zeta values in Banks–Panzer–Pym [12]. Remark . Locally, the same formula can also be used to quantize holomorphicPoisson structures on C d . However, the globalization proceduce works rather dif-ferently as one should replace the algebra of smooth functions C ∞ ( X ) by the struc-ture sheaf O X . In particular, one should consider deformations of O X as (twisted)presheaf, which naturally include “commutative” deformations of O X correspond-ing to deformations of the complex structure of X (see § Combinatorial deformation quantization. The combinatorial star prod-uct constructed in § A n . From the quiverpoint of view, one should look at the quiver with one vertex and n loops and imposethe commutativity relations giving the polynomial ring(8.5) A = k [ x , . . . , x n ] = k (cid:18) x , . . . , x n ... (cid:19).(cid:0) x j x i − x i x j (cid:1) ≤ i Let Q be the quiver with one vertex and n loops x , . . . , x n andlet R = { ( x j x i , x i x j ) } ≤ i In particular, if g = P i ≥ g i ~ i ∈ p ( Q, R ) b ⊗ m is a Maurer–Cartan element, then η = P ≤ i A, A ) are given as follows. For any ψ ∈ C k ( A, A ), we have F k ( ψ ) = X i < ···
Let η be any algebraic Poisson structure on A n for n ≥ and let g ∈ Hom( k S, A ) be the corresponding element. If g = g ~ satisfies x k ⋆ g ( x j ⋆ g x i ) =( x k ⋆ g x j ) ⋆ g x i for any ≤ i < j < k ≤ n , then ⋆ g gives an explicit deformationquantization of η .In particular, the combinatorial star product ⋆ g gives an explicit formula for thequantization of any algebraic Poisson structure on A .Proof. This follows from Theorem 7.36 and Proposition 8.8. (cid:3) EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 51 Let us describe this result for Poisson structures on A in a little more detail.Any (algebraic) Poisson structure on A is of the form η = a ∂∂y ∧ ∂∂x = − a ∂∂x ∧ ∂∂y and thus determined by its (polynomial) coefficient function a = P i,j a ij x i y j ∈ A .For k [ x, y ] = k h x, y i / ( yx − xy ) the reduction system (8.6) is given by R = { ( yx, xy ) } , so that S = { yx } and we can thus view the coefficient function a ∈ A of the Poisson bivector field η as an element g ∈ Hom( k S, A ) ≃ A , i.e. g ( yx ) = a .Since P = 0, it follows that g = g ~ ∈ Hom( k S, A ) b ⊗ m is a Maurer–Cartanelement of p ( Q, R ) b ⊗ m (cf. Remark 7.15), and by Theorem 7.14 ⋆ g is associative.To see that ⋆ g is a quantization of η , one checks that(8.10) x k y l ⋆ g x m y n = x k + m y l + n + lm a x k + m − y l + n − ~ + · · · so that (cid:8) x k y l , x m y n (cid:9) η = x k y l ⋆ g x m y n − x m y n ⋆ g x k y l ~ (cid:12)(cid:12)(cid:12)(cid:12) ~ =0 = ( lm − kn ) ax k + m − y l + n − = (cid:10) a ∂∂y ∧ ∂∂x , d x k y l ⊗ d x m y n (cid:11) where d denotes the exterior derivative and h− , −i the pairing between vector fieldsand forms.To verify the second (first-order) term on the right-hand side of (8.10), one shouldcalculate x k y l ⋆ g, x m y n (see § x k y l x m y n replacing yx by g ( yx ) = g ( yx ) ~ = a ~ exactly once and by xy allother times so that x k y l ⋆ g, x m y n = l X p =1 m X q =1 x k y p − x q − ax m − q y n + p ~ = lmax k + m − y n + l − ~ where in the summands a appears by commuting the p th y past the q th x . Of course a itself contains terms x i y j , but in the remaining reductions we can only replace yx by xy since we have used g once already, i.e. for the remaining reductions we maysimply treat x and y as commuting variables.8.2.1. The combinatorial star product via graphs. Similar to the definition of theKontsevich star product (8.3), there is a graphical description of combinatorial starproduct.For this let G ∗ k, denote the set of “admissible” graphs for the combinatorial starproduct, which is in bijection with the set of pairs (cid:0) Λ , { < i } i ∈ V (Λ) (cid:1) where Λ ∈ G k, is a Kontsevich graph without oriented cycles (“wheels”), V (Λ)denotes the set of vertices of the graph Λ, and < i is a well-order on the set ofincoming edges at the vertex i ∈ V (Λ). Then we have the following statement. Theorem 8.11. The combinatorial star product ⋆ g can be given as (8.12) a ⋆ g b = X k ≥ X Π ∈ G ∗ k, C Π ( a, b ) where C Π are bidifferential operators corresponding to performing reductions. EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 52 If g = g ~ + g ~ + · · · is a Maurer–Cartan element of p ( Q, R ) b ⊗ m , then ⋆ g gives an explicit formula for the deformation quantization of the Poisson structurecorresponding to g . (Details of this graphical calculus shall be given in a future article.) Note alsothat in the formula for the combinatorial star product the “weights” for each graphcan be taken as 1. Remarks . ( i ) Rational deformation quantization. The formula (8.12) ap-pears to be a rather natural way to define a star product for any Maurer–Cartan element of p ( Q, R ) b ⊗ m since the expression does not need any weightsand thus works over Z . It is not yet clear how to define the higher correctionterms g ~ + · · · in case g ~ does not satisfy the Maurer–Cartan equation.However, in examples this can be done by considering graphs with cyclesand we hope that a general combinatorial formula to achieve this can befound, which would give a rational universal formula.( ii ) “Loopless” deformation quantization. Dito [47] and Willwacher [128] haveshown that a universal formula without cycles (“wheels”) cannot exist. The“smallest” example of a Poisson structure which by itself does not satisfythe Maurer–Cartan equation is a quadratic Poisson structure on A (seeExample 8.17). It appears that the higher order terms to “correct” a Poissonstructure to a Maurer–Cartan element can indeed be found by consideringgraphs with cycles.( iii ) Linear Poisson structures and the Gutt star product. The combinatorialstar product ⋆ g makes use of the ordering of the monomials x ≺ · · · ≺ x n .One can define a “symmetrized” version of ⋆ g , which is still an associativequantization of the corresponding Poisson structure. A linear Poisson struc-ture on A n can be viewed as a Lie algebra structure and the symmetrizedstar product coincides precisely with the Gutt star product quantizing theKirillov–Lie Poisson bracket. Indeed, Poisson structures with only linearand constant terms are always Maurer–Cartan elements of p ( Q, R ) b ⊗ m .We give several examples. Example 8.14. Let η = P ≤ i By Proposition 8.9 any algebraic Poisson structure on A can be quantized usingthe formula of the combinatorial star product (8.12) and this is also true for linearPoisson structure on A n (cf. Remark 8.13 ( iii )). We now give two concrete examplesquantizing nonlinear Poisson structures in higher dimensions.The following first example is a quadratic Poisson structure on A which is aMaurer–Cartan element and thus can be quantized using the combinatorial starproduct. Example 8.16. Consider the Poisson structure η = − ( x + λyz ) ∂∂z ∧ ∂∂y on A with λ ∈ k . Note that the associated element g ∈ p ( Q, R ) b ⊗ m where g ( yx ) = 0 = g ( zx ) and g ( zy ) = − ( x + λyz ) ~ is a Maurer–Cartan element since ( z ⋆ g y ) ⋆ g x = xyz − ( x + λxyz ) ~ = z ⋆ g ( y ⋆ g x ). Itfollows that ( A J ~ K , ⋆ g ) is a deformation quantization of η . Note that by Manchon–Masmoudi–Roux [90, Prop. III.2] the Poisson structure η is not the image of aclassical r -matrix when λ = 0 and thus cannot be quantized using Drinfel’d twists.The following example is an exact Poisson structure of degree k ≥ 2, whichis not a Maurer–Cartan element of p ( Q, R ) b ⊗ m , but which may be corrected byadding higher-order terms. (We obtained the higher-order terms by consideringgraphs with cycles and leave a precise description of this process for future work.)After adding these higher-order terms, the combinatorial star product gives againan explicit formula for the quantization of this Poisson structure. Example 8.17. Consider the exact Poisson structure η associated to the polyno-mial − xyz − k +1 x k +1 for k ≥ 2, i.e. η = ( yz + x k ) ∂∂z ∧ ∂∂y − xz ∂∂z ∧ ∂∂x + xy ∂∂y ∧ ∂∂x on A . Let g ∈ Hom( k S, A ) be the associated element to η : g ( yx ) = xy, g ( zx ) = − xz, and g ( zy ) = yz + x k . We note that g = g ~ is not a Maurer–Cartan element of p ( Q, R ) b ⊗ m since( z ⋆ g y ) ⋆ g x = xyz (1 + ~ − ~ − ~ ) + x k +1 ~ z ⋆ g ( y ⋆ g x ) = xyz (1 + ~ − ~ − ~ ) + x k +1 ( ~ − ~ )and thus ( z ⋆ g y ) ⋆ g x = z ⋆ g ( y ⋆ g x ). However, by adding higher-order correctionterms to g we obtain a Maurer–Cartan element e g = g ~ + g ~ + · · · of p ( Q, R ) b ⊗ m given as e g ( yx ) = xy ( ~ − ~ + ~ − ~ + ~ − · · · ) e g ( zx ) = xz ( ~ + 2 ~ − ~ + 8 ~ − ~ + · · · ) e g ( zy ) = ( yz + x k ) ~ . Indeed one easily verifies that ( z ⋆ e g y ) ⋆ e g x = z ⋆ e g ( y ⋆ e g x ) and thus ( A J ~ K , ⋆ e g ) is adeformation quantization of the Poisson structure η .8.A. Quantum affine space and hypersurfaces. In noncommutative geometry, quantum affine space A dq is usually defined via its coordinate ring k q [ x , . . . , x d ] := k h x , . . . , x d i / ( x j x i − q ij x i x j ) ≤ i Let η be the bivector field η = X ≤ i 2. Since (8.19) also holds when replacing g by e g , it follows that e g isanother Maurer–Cartan element quantizing the same Poisson structure.Note that the quantum affine space A dq only “sees” the evaluation q ij at some ~ and thus A dq may be viewed as a quantization of many different Poisson structuresof the form (8.18). In the following example we consider the question of choosingan appropriate Poisson structure to quantize a hypersurface. Example 8.20 (Quantum hypersurfaces of type A n ) . Let Q be the quiver withthree loops at one vertex and let A = k [ x, y, z ] as in (8.5).Let F = xz − y n +1 ∈ A and denote by X F = { x ∈ A | F ( x ) = 0 } ⊂ A the cor-responding hypersurface, i.e. the toric type A n surface singularity, with coordinatering A F = A/ ( F ).Consider the Poisson structure η = yz ∂∂z ∧ ∂∂y + xy ∂∂y ∧ ∂∂x + ( n + 1) xz ∂∂z ∧ ∂∂x on A so that for all a ∈ A , { F, a } = h η, d F ⊗ d a i lies in the ideal ( F ), i.e. η is tangentto the hypersurface X F . This implies that η induces a Poisson structure η F on X F .Let us consider the question of quantizing the Poisson structure η F on the singularspace X F . The usual reduction system R of A = k [ x, y, z ] (cf. (8.6)) obtainedfrom x ≺ y ≺ z can be extended to a reduction system of A F by setting R F = R ∪ { (tip( F ) , F − tip( F )) } (cf. § reduction system for A reduction system for A F R = { ( yx, xy ) , ( zy, yz ) , ( zx, xz ) } R F = R ∪ { ( y n +1 , xz ) } S = { yx, zy, zx } S F = S ∪ { y n +1 } and R F satisfies ( ⋄ ) for the ideal I F = ( s − f s ) ( s,f s ) ∈ R F (where k Q/I F = A F ). Theoverlaps of element in S F are zyx , y n +1 x and zy n +1 .Consider the element g ∈ Hom( k S F , A F ) b ⊗ m given by g ( yx ) = xy ~ , g ( zy ) = yz ~ , g ( zx ) = xz (1 + ~ ) n +1 − xz, g ( y n +1 ) = 0 . This also follows from Proposition 9.16 for the degree condition ( ≺ ) with respect to the order x ≺ · · · ≺ x n to be introduced in § EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 55 It is easy to check that the following identities hold( z ⋆ g y ) ⋆ g x = z ⋆ g ( y ⋆ g x )( z ⋆ g y ) ⋆ g y n = z ⋆ g ( y ⋆ g y n )( y n ⋆ g y ) ⋆ g x = y n ⋆ g ( y ⋆ g x ) . Thus, g is a Maurer–Cartan element of p ( Q, R F ) b ⊗ m and we obtain a deformationquantization ( A F J ~ K , ⋆ g ) of η F . Finally, it is straight-forward to see that productsare again polynomial in ~ , so that ~ can be evaluated for any value in k and F defines a hypersurface in quantum affine space with coordinate ring k h x, y, z i / ( yx − q xy, zy − q yz, zx − q n +1 xz, xz − y n +1 )where q = (1 + ~ ) ∈ k .We will revisit this example in § e A n preprojective algebras.9. Algebraizations of formal deformations In this section we define certain subspaces Hom( k S, A ) < (resp. Hom( k S, A ) ≺ ) ofHom( k S, A ) such that for any (algebraic) element g = g t + · · · + g n t n ∈ Hom( k S, A ) ⊗ ( t )with g i ∈ Hom( k S, A ) < (resp. in Hom( k S, A ) ≺ ), the associated formal deformationadmits an algebraization.Here Hom( k S, A ) < and Hom( k S, A ) ≺ are the subspaces of elements in Hom( k S, A )satisfying certain degree conditions ( < ) resp. ( ≺ ) to be defined in Definitons 9.2 and9.12 below.Note that in the context of algebraizations it is natural to look at “algebraic”Maurer–Cartan elements lying in the uncompleted tensor product p ( Q, R ) ⊗ m ⊂ p ( Q, R ) b ⊗ m . Even though p ( Q, R ) ⊗ m is still an L ∞ algebra, the Maurer–Cartan equation (beingan infinite sum) does not necessarily make sense for all degree 1 elements (cf. Remark6.20) and the degree conditions ensure that for the elements in the uncompletedtensor products Hom( k S, A ) < ⊗ m and Hom( k S, A ) ≺ ⊗ m it does.9.1. Poincar´e–Birkhoff–Witt deformations. The Gutt star product, obtainedby pulling back the associative multiplication on U( g ) to S( g ) via the classicalPoincar´e–Birkhoff–Witt isomorphism, can be viewed as a deformation of the com-mutative product on the graded algebra S( g ) and is the classical example of a Poincar´e–Birkhoff–Witt (or simply PBW ) deformation . In this classical setup,S( g ) ≃ k [ x , . . . , x n ] is an algebra with quadratic relations and the Gutt star prod-uct can be viewed as a quantization of the linear Lie–Kirillov Poisson structure (cf. § iii )).Here we study PBW deformations in the general setting of path algebras ofquivers with relations, i.e. quotients of tensor algebras over the semisimple ring k r for any r ≥ k Q by path length also gives A = k Q/I the structureof a filtered algebra , i.e. A = S m ≥ A ≤ m for A ≤ ⊂ A ≤ ⊂ A ≤ ⊂ · · · ⊂ A and EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 56 A ≤ m · A ≤ n ⊂ A ≤ m + n , by setting A ≤ m = π ( k Q ≤ m ). The associated graded algebra of a filtered algebra is gr A = M m A ≤ m /A ≤ m − . If I is a homogeneous ideal of relations, then A = k Q/I is a graded algebra bysetting A m = π ( k Q m ), i.e. A = L m A m and A m · A n ⊂ A m + n . Definition 9.1. Let A = k Q/I be a graded algebra where I is a homogeneous idealof relations, i.e. I = ( r , . . . , r n ) where r i ∈ A are homogeneous elements. A PBWdeformation of A is given by an ideal I ′ = ( r − r ′ , . . . , r n − r ′ n ) where r ′ i ∈ A < | r i | for 1 ≤ i ≤ n such that gr( k Q/I ′ ) ≃ A .The condition gr( k Q/I ′ ) ≃ A ensures that the algebra does not get any “smaller”and indeed this condition is only satisfied for certain choices for r ′ i (cf. Example9.9). PBW deformations of A = k Q/I can be studied from the point of view ofreduction systems, by fixing any (homogeneous) reduction system satisfying ( ⋄ ) for I and considering Maurer–Cartan elements of the L ∞ algebra p ( Q, R ) which satisfycertain degree conditions which we define next. Indeed, all PBW deformations of A can be obtained in this way (see Proposition 9.4). Definition 9.2. Let R = { ( s, f s ) | s ∈ S } be a reduction system satisfying thecondition ( ⋄ ) for an ideal I of k Q and let A = k Q/I as usual.For any h ∈ Hom( k S, A ) we introduce the following two degree conditions: h s = σh ( s ) ∈ k Q < | s | for any s ∈ S ( < ) h s = σh ( s ) ∈ k Q ≤| s | for any s ∈ S ( ≤ )which are to be understood as conditions on h . Recall that σ is defined in (3.10).Let us also denote Hom( k S, A ) < ⊂ Hom( k S, A ) ≤ ⊂ Hom( k S, A ) the subspacesof elements satisfying ( < ) and ( ≤ ), respectively. Theorem 9.3. Let A = k Q/I be a graded algebra for I a homogeneous ideal ofrelations. ( i ) Let g ∈ Hom( k S, A ) < ⊗ m . If g is a Maurer–Cartan element of p ( Q, R ) b ⊗ m ,then gλ is a Maurer–Cartan element of p ( Q, R ) for any λ ∈ k . In particular,the formal deformation ( A J t K , ⋆ g ) admits ( A [ t ] , ⋆ g ) as an algebraization. ( ii ) Let g ∈ Hom( k S, A ) < . The algebra A g is a PBW deformation of A if andonly if g is a Maurer–Cartan element of p ( Q, R ) .Proof. Let us prove the first assertion. Note that R g is reduction finite since R isreduction finite and reductions using g decrease the length of the path. By Lemma7.18 it follows that for all a, b we have that a ⋆ g | t = λ ,k b = 0 for k ≫ 0. Thus, a ⋆ g | t = λ b is well defined and moreover a ⋆ g | t = λ b = ( a ⋆ g b ) | t = λ for any a, b ∈ A . If g is a Maurer–Cartan element of p ( Q, R ) b ⊗ m , then by Theorem 7.36 we get that g | t = λ is a Maurer–Cartan element of p ( Q, R ).Let us prove the second assertion. If g is a Maurer–Cartan element of p ( Q, R ),then by Proposition 7.46 ( ii ) there is an algebra isomorphism ϕ g : ( A, ⋆ g ) A g sending π ( u ) to [ u ] for any irreducible path u . Clearly, ϕ g is compatible with thefiltrations, thus it induces an algebra isomorphism between gr( A, ⋆ g ) and gr A g .Note that gr( A, ⋆ g ) ≃ A as graded algebras, thus A g is a PBW deformation of A . EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 57 Now if A g is a PBW deformation, then we have that ϕ g is a k -linear isomorphismand it follows from Proposition 7.46 ( ii ) that g is a Maurer–Cartan element. (cid:3) The following proposition shows that all PBW deformations of A can be foundas Maurer–Cartan elements satisfying the degree condition ( < ). Proposition 9.4. Let A = k Q/I be a graded algebra. Then there is a one-to-onecorrespondence between PBW deformations of A (up to isomorphism) and Maurer–Cartan elements of p ( Q, R ) satisfying ( < ) (up to gauge equivalence) for any choiceof reduction system R satisfying ( ⋄ ) for I .Proof. This follows from Proposition 7.46 and Theorem 9.3. (cid:3) The following theorem gives a necessary and sufficient condition for PBW defor-mations of N -Koszul algebras. (This theorem was proved for Koszul algebras, i.e. for N = 2, in Braverman–Gaitsgory [28], Positselski [105] and Polishchuk–Positselski[104].) Theorem 9.5 (Berger–Ginzburg [21], Fløystad–Vatne [54]) . Let A = T( V ) /I bean N -Koszul algebra where I = ( e S ) is a homogeneous ideal generated by a set ofrelations e S ⊂ V ⊗ N . For each s ∈ e S let g s = g ( s )+ · · · + g N ( s ) , where g i ( s ) ∈ V ⊗ N − i and let I g = ( s − g s ) s ∈ e S . Then A g = T( V ) /I g is a PBW deformation of A if andonly if I g ∩ T ≤ N ( V ) = k { s − g s } s ∈ e S . Moreover, this condition is equivalent to the following two conditions: ( i ) im (cid:0) ( k e S ⊗ k V ) ∩ ( V ⊗ k k e S ) [id V , g ] V ⊗ k N (cid:1) ⊂ k e S ( ii ) g i ◦ [id V , g ] = [id V , g i +1 ] for ≤ i ≤ N (and we set g N +1 = 0 )where [id V , g i ] = id V ⊗ g i − g i ⊗ id V as maps ( k e S ⊗ V ) ∩ ( V ⊗ k e S ) V ⊗ N − i +1 . (See also Cassidy–Shelton [37] for a criterion for PBW deformations of graded al-gebras using central extensions.)As a corollary of Theorem 9.3 we can prove the following analogous result withthe observation that the conditions ( i ) and ( ii ) of Theorem 9.5 correspond preciselyto the Maurer–Cartan equation for p ( Q, R ). Corollary 9.6. Let A = k Q/I = T k Q ( k Q ) /I be a finitely presented graded algebraover k Q ≃ k r and let R = { ( s, f s ) } be a reduction system satisfying ( ⋄ ) for I suchthat S ⊂ Q N and f s ∈ k Q N for all s ∈ S and assume that S ⊂ Q N +1 .Let g ∈ Hom( k S, A ) < . Then A g = k Q/I g is a PBW deformation of A if andonly if I g ∩ k Q ≤ N = k { s − f s − g s } s ∈ S . Similarly, this condition is equivalent to ( i ) im (cid:0) ( k { s − f s } ⊗ k Q ) ∩ ( k Q ⊗ k { s − f s } ) [id , g ] k Q N (cid:1) ⊂ k { s − f s } ( ii ) g i ◦ [id , g ] = [id , g i +1 ] for ≤ i ≤ N , where g i : k { s − f s } k Q N − i isdefined by g i ( s − f s ) = π N − i ( g s ) . Here π N − i : k Q k Q N − i is the naturalprojection.Remark . For N = 2 the condition S ⊂ Q N +1 is automatic. In general, if thereduction system R is induced from a reduced noncommutative Gr¨obner basis (withrespect to some admissible order), then by [65, Prop. 11] the algebra A = k Q/I is EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 58 N -Koszul if and only if S ⊂ Q N +1 . In this case, A is called a strong N -Koszulalgebra (cf. [63]). Proof of Corollary 9.6. If A g is a PBW deformation of A , then the set Irr S ofirreducible paths forms a basis of A g . Since any monomial u ∈ Q ≤ N which isnot in S is irreducible, we have I g ∩ k Q ≤ N = k { s − f s − g s } s ∈ S . Let us prove that the condition I g ∩ k Q ≤ N = k { s − f s − g s } s ∈ S implies theconditions ( i ) and ( ii ). Let p be in ( k { s − f s } ⊗ k Q ) ∩ ( k Q ⊗ k { s − f s } ). Considerthe following two elements in I gN X i =1 (id ⊗ g i )( p ) + p and N X i =1 ( g i ⊗ id)( p ) + p. The difference of the two elements is in I g ∩ k Q ≤ N and thus N X i =1 (id ⊗ g i − g i ⊗ id)( p ) ∈ k { s − f s − g s } s ∈ S . Comparing degrees, we have [id , g ]( p ) ∈ k { s − f s } and g i ◦ [id , g ]( p ) = [id , g i +1 ]( p )for i = 1 , . . . , N .We need to prove that the conditions ( i ) and ( ii ) imply that g is a Maurer–Cartan element of p ( Q, R ). By Theorem 7.36, it suffices to check the identity(7.37). Since S ⊂ Q N +1 , we note that the identity (7.37) is equivalent to ( i ) and( ii ) by comparing degree components. (cid:3) Remark . In Corollary 9.6 we assume the existence of a homogeneous reductionsystem, which for example exists for strong N -Koszul algebras (cf. Remark 9.7).For a general N -Koszul algebra we believe it would be more natural to work withthe Koszul resolution K (rather than working with P , which depends on the choiceof a particular reduction system). Similarly, one obtains an L ∞ algebra structureon K = Hom A e ( K , A ) by homotopy transfer and we expect the conditions ( i ) and( ii ) in Theorem 9.5 to be precisely the Maurer–Cartan equation for this L ∞ algebrasplit into degree components. Example 9.9 (A very simple example) . Let A = k (cid:0) ab (cid:1)(cid:14) ( ab, ba ) . Then A is a self-injective algebra of dimension 4 with gldim A = ∞ . We illustratehow deformations of A can be described as Maurer–Cartan elements of the L ∞ algebra p ( Q, R ).The ideal ( ab, ba ) admits a reduction system R = { ( ab, , ( ba, } EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 59 so that Q =: S = { e , e } Q =: S = { a, b } S =: S = { ab, ba } S = { aba, bab } S = { abab, baba } ...Note that the only irreducible paths parallel to ab and ba are the idempotents, soa general element of Hom( k S, A ) is of the form ab λe ba µe For λ = µ = 1 we obtain the map g : k S A defined by g ( ab ) = e g ( ba ) = e which is a Maurer–Cartan element of the L ∞ algebra p ( Q, R ) since for aba ∈ S wehave a ⋆ g ( b ⋆ g a ) = a ⋆ g e = a ( a ⋆ g b ) ⋆ g a = e ⋆ g a = a and similarly for bab ∈ S . (That these conditions are equivalent to the Maurer–Cartan equation is the content of Theorem 7.36.) Writing I g = ( ab − e , ba − e )we have that A g = k Q/I g ≃ M ( k ) is the 4-dimensional algebra of 2 × g ∈ Hom( k S, A ) b ⊗ m defined by g ( ab ) = 0 g ( ba ) = e t is not a Maurer–Cartan element and the algebra e A = k Q J t K / ( ab, ba − e t ) is not aflat formal deformation of A . Indeed, e A has t -torsion as using the deformed relationswe have for example at = aba = 0. Here g satisfies the degree condition ( < ), but k Q/ ( ab, ba − e ) ≃ k { e } is not a PBW deformation of A . Remark . Any finite-dimensional algebra over an algebraically closed field k isMorita equivalent to the path algebra of a quiver modulo an admissible ideal I (i.e. k Q ≥ N ⊂ I ⊂ k Q ≥ for some N ≥ A g is a PBWdeformation of A and thus has the same dimension, but the ideal ( ab − e , ba − e )is no longer admissible since it contains constant terms and A g ≃ M ( k ) is Moritaequivalent to k , so gldim A g = gldim k = 0.In other words, deformations of path algebras of quivers need not preserve theadmissibility of the ideal and may be Morita equivalent to the path algebra of aquiver with less vertices and less arrows. (See also Theorem 11.8 for a geometric,infinite-dimensional example of this phenomenon.)Note that for the quiver in Example 9.9 we have k ≃ Hom( k S, A ) < ⊂ Hom( k S, A )and the space of deformations may be viewed as an algebraic variety in A = { ( x, y ) } cut out by x − y . This point of view is the topic of the following section. EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 60 Algebraic varieties of algebras. If we assume that R is a finite reductionsystem (which we hereby do), we may view the set of “actual” deformations (ob-tained from evaluating an algebraization at some values of the parameters) as anaffine algebraic variety whose defining equations are given by the Maurer–Cartanequation. The “actual” deformations can be obtained under degree conditions, com-ing either from the path length or from the order in a noncommutative Gr¨obnerbasis.9.2.1. PBW deformations. Let A = k Q/I and let R = { ( s, f s ) } be a finite reductionsystem satisfying ( ⋄ ) for I such that f ∈ Hom( k S, A ) ≤ . Then A is filtered by pathlength and both A and gr A can be viewed as points in a finite-dimensional algebraicvariety of PBW deformations. Theorem 9.11. The subspace Hom( k S, A ) < ⊂ Hom( k S, A ) consisting of elements g satisfying the degree condition ( < ) is a finite-dimensional affine space and theMaurer–Cartan equation for p ( Q, R ) cuts out an affine variety V < of PBW defor-mations of gr A .Proof. It follows from Theorem 9.3 that the PBW deformations correspond toMaurer–Cartan elements of p ( Q, R ) satisfying the degree condition ( < ). Since S isa finite set, the space Hom( k S, A ) < is finite dimensional and the Maurer–Cartanequation gives algebraic equations in Hom( k S, A ) via the identity (7.37). (cid:3) Degree conditions from Gr¨obner bases. A similar idea may be used to definealgebraic varieties of algebras admitting a finite noncommutative Gr¨obner basis,where instead of requiring that the length g s be strictly smaller than the length of s one requires that g s ≺ s , where ≺ is an admissible order on Q (see § ∞ algebra p ( Q, R ) can now be viewed as the explicit equations defining this variety. Definition 9.12. Similar to the degree condition ( < ) which is a condition on pathlength (see Definition 9.2), we may give the following degree condition g s = σg ( s ) ∈ k Q ≺ s for any s ∈ S ( ≺ )where k Q ≺ s is the k -linear span of all paths which are “smaller” than s with re-spect to some chosen admissible order ≺ . Let us also denote by Hom( k S, A ) ≺ ⊂ Hom( k S, A ) the subspace of elements g satisfying ( ≺ ).Let Q be a finite quiver and let Q denote the set of all paths in Q . Fix anadmissible order ≺ on Q (see § S ⊂ Q ≥ be a finite subset such that s is not a subpath of s ′ for any s = s ′ ∈ S . Define a set of algebras (cf. [64, Def. 4.1])Alg ≺ = { k Q/I | I ideal of k Q and h tip ≺ ( I ) i = h S i} (see § I )).Let Irr S be the set of irreducible paths in Q . For any s ∈ S , denoteIrr S, ≺ s = { u ∈ Irr S | u is parallel to s and u ≺ s } . Suppose that the set Irr S, ≺ s is finite for each s ∈ S . Denote N = P s ∈ S | Irr S, ≺ s | and consider the affine space A N . To each point g = ( λ s,u ) s ∈ S,u ∈ Irr S, ≺ s of A N we EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 61 associate a reduction system for k QR g = n ( s, g s ) (cid:12)(cid:12)(cid:12) s ∈ S and g s = P u ∈ Irr S, ≺ s λ s,u u o . Let V ≺ ⊂ A N denote the set of points(9.13) V ≺ = (cid:8) g ∈ A N (cid:12)(cid:12) R g satisfies the condition ( ⋄ ) (cid:9) . Then we have the following results. Theorem 9.14 (Green–Hille–Schroll [64] Thm. 4.3 & Thm. 4.4) . ( i ) There is a one-to-one correspondence between the sets V ≺ and Alg ≺ . ( ii ) V ≺ admits the structure of an affine algebraic variety. The following result shows that the algebraic equations for the variety V ≺ aregiven by the Maurer–Cartan equation. Theorem 9.15. Assume that S ⊂ Q is finite and let ≺ be any admissible ordersuch that Hom( k S, A ) ≺ is finite dimensional. Let A = k Q/ ( S ) be the monomialalgebra associated to S . Then the variety V ≺ ⊂ A N is the zero locus of finitelymany algebraic equations obtained by evaluating the Maurer–Cartan equation onelements in S , i.e. for each w ∈ S we obtain an equation X k ≥ k ! h g, . . . , g i k ( w ) = 0 which for w = u u u with u u , u u ∈ S can also be written explicitly as ( u ⋆ g u ) ⋆ g u = u ⋆ g ( u ⋆ g u ) where ⋆ g is the combinatorial star product. (See Example § V ≺ .) Theorem 9.15 will follow from thefollowing result. Proposition 9.16. Let R be a reduction system obtained from a finite noncommu-tative Gr¨obner basis of k Q/I and let g ∈ Hom( k S, A ) ≺ ⊗ m . ( i ) For any elements a, b ∈ A we have that a ⋆ g,k b = 0 for k ≫ . ( ii ) If g is a Maurer–Cartan element of p ( Q, R ) b ⊗ m , then the formal deformation ( A J t K , ⋆ g ) ≃ k Q J t K / b I g admits k Q [ t ] /I g as an algebraization. This proposition is very useful in practice, since it allows one to establish thepassage from formal to actual deformations in contexts where one deals with defor-mations which are homogeneous and thus do not satisfy the PBW condition, butstill satisfy the degree condition ( ≺ ) for some suitable choice of admissible order.In geometric contexts this can for example be used to obtain algebraizations ofquantizations of quadratic Poisson structures, see for example § § Proof of Proposition 9.16. Since the order ≺ is a well-order, we get that for any a, b the product a ⋆ g,k b = 0 for k ≫ 0. Then it follows from Lemma 7.30 that h g, . . . , g i k ( w ) = 0 for w ∈ S and k ≫ 0. Since S is a finite set (because S isfinite), we get that h g, . . . , g i k = 0 for k ≫ 0. This proves ( i ), and ( ii ) follows fromProposition 7.46. (cid:3) EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 62 Proof of Theorem 9.15. It follows from Proposition 9.16 that there exists a positiveinteger k such that h g, . . . , g i k = 0 for any k > k . It follows from Proposition7.46 that an element g ∈ A N is in the variety V ≺ if and only if g satisfies theMaurer–Cartan equation X ≤ k ≤ k k ! h g, . . . , g i k = 0 . By Theorem 7.36, this gives the following equations( u ⋆ g u ) ⋆ g u = u ⋆ g ( u ⋆ g u )for any w = u u u ∈ S with u u , u u ∈ S . (cid:3) Note that if the Maurer–Cartan equation holds for all 2-cochains, for example if P = 0, then V ≺ is simply the affine space A n .As is apparent from the notation, the affine variety V ≺ depends on the chosenadmissible order. However, if we restrict our attention to a certain class of admissibleorders, we may consider a variety of algebras that does not depend on the particularadmissible order in this class.We say that an admissible order ≺ is compatible with the path length if for anytwo paths p, q ∈ k Q such that | p | < | q | , we have p ≺ q . Theorem 9.17. Let A = k Q/ ( S ) . Let T be the set of admissible orders compatiblewith the path length. ( i ) S ≺∈ T V ≺ is an algebraic variety. ( ii ) T ≺∈ T V ≺ = V < is the variety of PBW deformations.Proof. Let N = dim Hom( k S, A ) ≤ . To prove ( i ) note that for each ≺ ∈ T we haveHom( k S, A ) ≺ ⊂ Hom( k S, A ) ≤ , so that W := [ ≺∈ T Hom( k S, A ) ≺ ⊂ Hom( k S, A ) ≤ ≃ A N is an arrangement of subspaces which may be viewed as an affine variety in A N . ByTheorem 9.15 the Maurer–Cartan equation is well defined on each Hom( k S, A ) ≺ and thus also on W , cutting out an affine variety S ≺∈ T V ≺ ⊂ A N .To prove ( ii ), note that since ≺ ∈ T is compatible with the path length, we havethat Hom( k S, A ) < ⊂ Hom( k S, A ) ≺ for all ≺ ∈ T . Thus the variety V < of PBWdeformations given in Theorem 9.11 is contained in the intersection. Conversely,given any order on arrows x i ∈ Q , the corresponding degree–lexicographic order iscompatible with the path length, so for each s ∈ S and each irreducible path u ∈ Q | s | parallel to s , there is always some (degree–lexicographic) admissible order such that s ≺ u . Thus T ≺∈ T Hom( k S, A ) ≺ = Hom( k S, A ) < and consequently the Maurer–Cartan equation cuts out precisely the variety V < of PBW deformations. (cid:3) More generally one may consider the sub set MC ⊂ Hom( k S, A ) of all elements whose associated reduction system is reduction finite satisfying the Maurer–Cartanequation of p ( Q, R ). (If a reduction system is not reduction finite, the Maurer–Cartan equation does not necessarily make sense, since in the non-formal settingthe Maurer–Cartan equation involves an infinite sum. In the formal case one maywork with the notion of convergence in the m -adic topology, which is not availablein the non-formal context.) EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 63 Moreover, Hom( k S, A ) may be an infinite-dimensional vector space and it is notimmediately clear what geometric structure (if any) V ⊂ Hom( k S, A ) has. Thedegree conditions ( < ) and ( ≺ ) ensure that the Maurer–Cartan equation does makesense and then Theorems 9.11, 9.15 and 9.17 identify concrete affine varieties ofactual deformations(9.18) V < ⊂ [ ≺∈ T V ≺ ⊂ V. (See Fig. 5 for a concrete example.)The degree conditions ( < ) and ( ≺ ) are very useful in practice to obtain actualdeformations. However, thus far we have not considered the notion of equivalence.As Green–Hille–Schroll [64] already note, there does not appear to be a naturalaction of a group on the variety V ≺ . However, there is a natural groupoid action byrestricting the groupoid G MC in Definition 3.13. Proposition 9.19. The groupoid G MC restricts to a groupoid G ≺ V ≺ andtwo reduction systems in V ≺ are equivalent if and only if they lie in the same orbitof G ≺ V ≺ . Recall that the algebras corresponding to equivalent reduction systems are iso-morphic, so the orbits of the variety under this groupoid action can be viewedas a low-dimensional blueprint of the orbits of GL( V ) acting on the variety of allassociative algebra structures on a k -vector space V .Without the degree conditions ( < ) or ( ≺ ) one can still always work with formaldeformations, where equivalences between Maurer–Cartan elements are built intothe (formal) deformation functor MC p ( Q,R ) / ∼ . (See § Finite-dimensional algebras. Deformations of finite-dimensional algebras canin principle be determined already from the classical theory, i.e. as Maurer–Cartanelements in the Hochschild complex with its DG Lie algebra structure given bythe Gerstenhaber bracket. From a practical perspective, however, the approachvia reduction systems (in particular, those from a finite Gr¨obner basis) can beconsidered a substantial improvement.For instance, in Example 9.9 we recovered the 2 × ( k ) asa deformation of a certain 4-dimensional self-injective algebra A . Describing thealgebra deformations using the approach via the reduction system, the space of2-cochains Hom k Q e0 ( k S, A ) is 2-dimensional, whereas the space of Hochschild 2-cochains Hom k ( A ⊗ k A, A ) is 64-dimensional. (See also Remark 9.24 below for amore compelling dimension count.)Recall that any finite-dimensional algebra B is Morita equivalent to A = k Q/I ,where Q is a finite quiver and I is an admissible ideal, i.e. k Q ≥ N ⊂ I ⊂ k Q ≥ for some N ≥ 2. Since Morita-equivalent algebras have the same deformationtheory (this is even true for derived Morita-equivalent algebras, see Keller [77]) thedeformation theory of B can be described as the deformation theory of A = k Q/I .Using the following theorem one can now find a finite noncommutative Gr¨obnerbasis for I . Theorem 9.20 [62, Prop. 2.11] . If A = k Q/I is finite-dimensional, then I admitsa finite Gr¨obner basis. EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 64 In this case a finite noncommutative Gr¨obner basis may be computed using thenoncommutative version of Buchberger’s algorithm (see [62, § § Corollary 9.21. If A = k Q/I is finite-dimensional, then there exists a finite re-duction system R for k Q which satisfies the condition ( ⋄ ) for I . Deformations of Brauer tree algebras. In this section we describe defor-mations of certain Brauer tree algebras which are special biserial algebras and areexamples of Khovanov arc algebras which appear in various different contexts fromlow dimensional topology and topological quantum field theory to representationtheory and symplectic geometry (see e.g. [79, 121, 30, 2, 23]). Deformations of theseBrauer tree algebras were studied in Mazorchuk–Stroppel [93] in the context of therepresentation theory of sl n . Here we illustrate how to recover the deformationsof these Brauer tree algebras using the approach via reduction systems and thecombinatorial criterion for the Maurer–Cartan equation given in Theorem 7.36. Wecompute both the variety of actual deformations as well as the equivalence classesof formal deformations.Let n ≥ Q be the quiver · · · a a a a n − b b b b n − The Brauer tree algebra is the quotient A = k Q/I where I is the ideal generated bythe set { a i a i +1 , b i +1 b i , a i +1 b i +1 − b i a i } ≤ i ≤ n − which is a finite-dimensional algebra of dimension 4 n − R = { ( b a b , , ( a b a , } ∪ { ( a i a i +1 , , ( b i +1 b i , , ( a i +1 b i +1 , b i a i ) } ≤ i 0) and ( a b a , 0) are addedto the reduction system by resolving the overlap ambiguities a a b and a b b ,respectively, in accordance with Buchberger’s algorithm. We have S = { a b a , b a b } ∪ { a i a i +1 , b i +1 b i , a i +1 b i +1 } ≤ i ≤ n − S = { a b a b , b a b a , a b a a , b b a b }∪ { a i +1 b i +1 b i , a i a i +1 b i +1 } ≤ i ≤ n − ∪ { a i a i +1 a i +2 , b i +2 b i +1 b i } ≤ i ≤ n − . Note that the irreducible path parallel to a b a is a , the irreducible path par-allel to b a b is b , the irreducible paths parallel to a i +1 b i +1 are e i +1 and b i a i , andthere are no irreducible paths parallel to a i a i +1 or b i +1 b i . This immediately givesthe following lemma. Lemma 9.22. We have that Hom( k S, A ) ≃ k n − with an element g ∈ Hom( k S, A ) of the following general form g a b a = λ a g b a b = µ b g a i +1 b i +1 = λ i +1 e i +1 + µ i +1 b i a i for ≤ i ≤ n − g a i a i +1 = 0 g b i +1 b i = 0 EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 65 for λ j , µ j ∈ k for ≤ j ≤ n − .Moreover, the subspace Hom( k S, A ) < ⊂ Hom( k S, A ) is given by the elements g of the form (9.23) with µ = · · · = µ n − = 0 .Remark . Note that in the classical picture, we have dim k Hom k ( A ⊗ k A, A ) =8(2 n − so that n · · · dim A 10 14 18 22 · · · dim Hom( k S, A ) 4 6 8 10 · · · dim Hom( A ⊗ , A ) 1000 2744 5832 10648 · · · dim Hom( A ⊗ , A ) 10000 38416 104976 234256 · · · In principle, the deformations of the 10-dimensional Brauer tree algebra can beobtained by studying a certain irreducible component of the variety of all associativealgebra structures on a 10-dimensional vector space — which is cut out by 10000equations in A (cf. § surface of actual deformations of R which contains an affine line alreadycapturing all nontrivial deformations (see Fig. 5).9.A.1. The Maurer–Cartan equation. The Maurer–Cartan equation of p ( Q, R ) al-lows us to compute the second Hochschild cohomology group HH ( A, A ) (see § A is essentially contained in a simplecomputation which we record in the following lemma. Lemma 9.25. Let g ∈ Hom( k S, A ) be given by λ i , µ j ∈ k for ≤ i, j ≤ n − as inLemma 9.22. Then the Maurer–Cartan equation for g is well defined for all λ i , µ j in k and is equivalent to λ = µ and λ i = ( − i +1 µ (1 + µ ) · · · (1 + µ i ) for < i ≤ n − .The same equation is valid for formal deformations over any complete local Noe-therian k -algebra ( B, m ) by viewing λ i , µ j ∈ m so that g ∈ Hom( k S, A ) b ⊗ m .Proof. By Theorem 7.36 the Maurer–Cartan equation is equivalent to the condition u ⋆ g ( u ⋆ g u ) = ( u ⋆ g u ) ⋆ g u for every u u , u u ∈ S and u u u ∈ S . Forexample the condition λ = µ is equivalent to ( a ⋆ g b a ) ⋆ g b = a ⋆ g ( b a ⋆ g b ).The other equations follow from computing associativity of ⋆ g for other elements in S . (For all n ≥ g ∈ Hom( k S, A ) followsalready from the above direct computation, which shows that ⋆ g,k = 0 for all k ≥ (cid:3) Corollary 9.26. HH ( A, A ) ≃ k . EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 66 V ≺ V < Figure 5. The variety of actual deformations of the reduction sys-tem for the 10-dimensional Brauer tree algebra and its subvarietyof PBW deformations. Proof. Let g ∈ Hom( k S, A ) be any element of the general form (9.23). By Lemmas7.54 and 9.25 we have that g is a 2-cocycle if and only if λ i = ( − i +1 µ for1 ≤ i ≤ n − 2. Note that µ i for 2 ≤ i ≤ n − k .That is, there is a one-to-one correspondence between 2-cocycles and the tuples( µ , . . . , µ n − ) for µ i ∈ k .We claim that the 2-cocycle g corresponding to ( µ , . . . , µ n − ) is cohomologousto the 2-cocycle g ′ corresponding to ( µ , , . . . , h ∈ Hom( k Q , A ) given by h ( a ) = 0 = h ( b j ) for 1 ≤ j ≤ n − h ( a j ) = ν j a j for 1 < j ≤ n − 2, where ν j = µ + · · · + µ j . Then we have g = g ′ + ∂ ( h ). (cid:3) The varieties of actual deformations. To determine the varieties of actualdeformations, observe that the reduction system R can be obtained from the non-commutative Gr¨obner basis induced by the degree–lexicographic order ≺ whichextends a ≻ · · · ≻ a n − ≻ b n − ≻ · · · ≻ b (cf. Theorem 9.20) and for this orderwe have Hom( k S, A ) ≺ = Hom( k S, A ) ≃ k n − .The equations in Lemma 9.25 cut out the variety V ≺ ⊂ A n − which contains thevariety V < ≃ A of PBW deformations (cf. (9.18)). When n = 4, V ≺ is isomorphicto a saddle k [ x, y, z ] / ( x − yz ) in 3-dimensional space (see Fig. 5). For all n ≥ 4, thesubvariety V < of PBW deformations is parametrized by λ ∈ k , where λ = µ = λ = − λ = · · · = ( − n − λ n − and µ = · · · = µ n − = 0 . The algebras A λ in this family are given by A λ = k Q/ ( a b a − λa , b a b − λb , a i +1 b i +1 − ( − i λe i +1 , a i a i +1 , b i +1 b i ) ≤ i EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 67 have found an algebraizable deformation whose first order term is a cocycle rep-resenting this cohomology class. The other deformations come from cobound-aries and it can be shown directly that the algebra corresponding to the point( λ , . . . , λ n − , µ , . . . , µ n − ) ∈ V ≺ is isomorphic to the algebra determined by µ with µ = · · · = µ n − = 0. In fact, it is straight forward to construct an ex-plicit equivalence between the corresponding reduction systems giving the followingtheorem. Proposition 9.27. There is a bijection between the (closed) points of V < and theorbits of the groupoid G MC . (Note that here Hom( k S, A ) ≺ = Hom( k S, A ), so that all corresponding reductionsystems are reduction finite and we have V ≺ = MC.)In this example all formal deformations admit algebraizations and Proposition9.27 explicitly determines the orbit space of actual deformations. For completenesswe shall also compare this to the point of view of formal deformations.9.A.3. Formal deformations up to equivalence. Intuitively, the tangent space of the“deformation space” should be given by the second Hochschild cohomology, whichfor the Brauer tree algebras is HH ( A, A ) ≃ k (Corollary 9.26). We found abovethat the variety V < ≃ A is a geometric model of the orbit space of the groupoid ofactual deformations and it gives an explicit family of nontrivial deformations whosetangent space at 0 ∈ A (corresponding to the undeformed Brauer tree algebra) isindeed 1-dimensional.The formal deformation theory is encoded in the deformation functor Def p ( Q,R ) =MC p ( Q,R ) / ∼ which we simply defined on the category c Art k of (commutative) com-plete local Noetherian k -algebras (with values in Sets ), so that from the formalpoint of view we expect to find the formal neighbourhood of 0 ∈ A whose ring offunctions is just the formal power series ring k J t K with maximal ideal ( t ). Indeed,this can be shown directly by determining the Maurer–Cartan elements up to (ho-motopy) equivalence in the formal setting giving the following result, which is theformal analogue of Proposition 9.27. Proposition 9.28. The deformation functor Def A encoding the formal deformationtheory of the Brauer tree algebra A is prorepresented by b O A , ≃ ( k J t K , ( t )) ∈ c Art k .Proof. We have that Def A ≃ Def h ( A ) ≃ Def p ( Q,R ) , where Def p ( Q,R ) = MC p ( Q,R ) / ∼ is the deformation functor encoding deformations of the reduction system R .Let ( B, m ) be any complete local Noetherian k -algebra. Also in the formal case,the Maurer–Cartan equation of p ( Q, R ) b ⊗ m is equivalent to the equations in Lemma9.25 for λ i , µ j ∈ m . In other words, there is a one-to-one correspondence betweenMaurer–Cartan elements of p ( Q, R ) b ⊗ m and the tuples ( µ , . . . , µ n − ) for µ j ∈ m .We claim that the Maurer–Cartan element g corresponding to ( µ , . . . , µ n − ) ishomotopy equivalent to the Maurer–Cartan element g ′ corresponding to ( µ , , . . . , g τ ∈ ( p ( Q, R ) ⊗ Ω ) b ⊗ m such One could equivalently work with the notion of equivalence of formal reduction systems asin § EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 68 that g = ev ( g τ ) and g ′ = ev ( g τ ) as follows. Let g τ = e g + ∞ X k =0 h k τ k d τ where e g ∈ ( p ( Q, R ) ⊗ k [ τ ]) b ⊗ m is given by e g a b a = µ a e g b a b = µ b e g a i +1 b i +1 = ( − i µ (1 + µ τ ) · · · (1 + µ i +1 τ ) e i +1 + µ i +1 τ b i a i e g a i a i +1 = 0 e g b i +1 b i = 0so that e g is a Maurer–Cartan element of p ( Q, R ) b ⊗ m for each value of τ and h k ∈ Hom( k Q , A ) b ⊗ m is given by h k ( a ) = 0 = h k ( b i ) for 1 ≤ i ≤ n − h k ( a i ) = ( − k ( µ k +12 + · · · + µ k +1 i ) a i for 1 < i ≤ n − . From (6.22) it remains to verify the following equation(9.29) ∂ e g∂τ = ∞ X k =0 (cid:16) ∂ ( h k ) τ k + h e g, h k τ k i + 12 h e g, e g, h k τ k i + · · · | {z } =0 (cid:17) . Applying (9.29) to the elements a b a , b a b , a i a i +1 , b i +1 b i ∈ S we obtain thatthe left and right hand side of the identity are zero. Applying (9.29) to the element a i +1 b i +1 ∈ S and comparing the coefficients of τ k for each k ≥ 0, the equality essen-tially follows from Newton’s identities on symmetric polynomials of µ , . . . , µ i +1 .Thus any Maurer–Cartan element is homotopy equivalent to the one determinedby ( µ , , . . . , p ( Q,R ) ( B, m ) = m . But this means that Def p ( Q,R ) is(pro)represented by b O A , ≃ ( k J t K , ( t )) as Hom d Art k (( k J t K , ( t )) , ( B, m )) = m , becauseany such morphism is determined by the image of t — which may take any valuein the maximal ideal m . (cid:3) Deformations of type e A n preprojective algebras. The McKay corre-spondence (see McKay [94] and Wemyss [125]) relates the geometry of a ratio-nal surface singularity C / Γ for some Γ < GL ( C ) and its (minimal) resolutionto the representation theory of the corresponding reconstruction algebra. WhenΓ < SL ( C ), then C / Γ is an ADE singularity whose minimal resolution containsa chain of P ’s with self-intersection − preprojective algebra associated to the extended Dynkin diagramof type e A e D e E. The preprojective algebras are also related to the symplectic reflec-tion algebras (see [43, 50]). Naturally, this beautiful interplay between algebra andgeometry also extends to the deformation theory, so that deformations of the pre-projective algebra are seen to induce deformations of the singularity correspondingto commutative or noncommutative deformations of the minimal resolution.We show how to give an explicit description of deformations for the preprojectivealgebra associated to the extended Dynkin diagram of type e A n — correspondingunder the McKay correspondence to the A n surface singularity. In this case, the EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 69 preprojective algebra admits a particularly simple reduction system and the degreeconditions ( < ) and ( ≺ ) introduced in Definitions 9.2 and 9.12 allow us to recoverthe deformed preprojective algebras as well as the so-called deformed quantum pre-projective algebras as algebraizations of formal deformations (see §§ §§ Reduction system for the preprojective algebras of type e A n . Let Q be theextended Dynkin quiver of type e A n (with n + 1 vertices) and let Q be the doublequiver of Q ... α α α α n ¯ α ¯ α ¯ α ¯ α n obtained by adding for each (clockwise) arrow α i ∈ Q a new (counter-clockwise)arrow ¯ α i with s(¯ α i ) = t( α i ) and t(¯ α i ) = s( α i ).The preprojective algebra associated to Q is defined asΠ( Q ) = k Q (cid:14) (cid:0)P ni =0 α i ¯ α i − ¯ α i − α i − (cid:1) where the index i is to be understood cyclically, i.e. i ∈ Z / ( n +1) Z so that α − = α n etc.We have an obvious reduction system R = { ( α i ¯ α i , ¯ α i − α i − ) } i with S = { α i ¯ α i } i satisfying ( ⋄ ) for the ideal I = ( P i α i ¯ α i − ¯ α i − α i − ). Note that there are no overlapambiguities and thus no higher k -ambiguities, i.e. S k +2 = ∅ for k ≥ 1. In particular, S = ∅ so that all elements in Hom( k S, Π( Q )) are Maurer–Cartan elements andcan be used to define formal deformations. We now look at those deformationsadmitting algebraizations by considering the Maurer–Cartan elements satisfyingthe degree conditions ( < ) and ( ≺ ) with respect to the admissible order uniquelydetermined by α ≻ α ≻ · · · ≻ α n ≻ ¯ α ≻ ¯ α ≻ · · · ≻ ¯ α n .9.B.2. Deformed preprojective algebras. The deformed preprojective algebras can beviewed as PBW deformations of preprojective algebras. For PBW deformations,we should look at 2-cocycles g ∈ Hom( k S, A ) < . Since the only parallel irreduciblepaths of length strictly less than the length of the elements in S are the idempotents e i at the vertices, PBW deformations of Π( Q ) are given by g : k S Π( Q ) suchthat g ( α i ¯ α i ) = λ i e i where λ i ∈ k for 0 ≤ i ≤ n , i.e. all PBW deformations are parametrized by λ =( λ , . . . , λ n ) ∈ k n +1 . We denote the corresponding deformed algebra by Π λ ( Q ) = EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 70 (Π( Q ) , ⋆ g ). It follows from Theorem 9.3 that Π λ ( Q ) is isomorphic, as (filtered)algebras, to the deformed preprojective algebra Π λ ( Q ) ≃ k Q (cid:14) (cid:0)P ni =0 α i ¯ α i − ¯ α i − α i − − λ i e i (cid:1) studied in Crawley-Boevey–Holland [43].We now describe the deformed preprojective algebras in terms of reductions. Notation 9.30. To simplify the formulae we use the following shorthands for pathsin k Q : α i...j := α i α i +1 · · · α j ¯ α j...i := ¯ α j ¯ α j − · · · ¯ α i − n < i ≤ j ≤ n We also set α i...i − := e i and ¯ α i − ...i := e i . Furthermore, we define the followingconstants from the λ i ’s λ i,j := λ i + λ i +1 + · · · + λ j − n < i < j ≤ n Λ kli := X n − l +1 ≤ m i < ··· The product ⋆ g on Π( Q ) is determined by the following formu-lae. For any ≤ l ≤ k ≤ n + 1 α n − l +1 ...n ⋆ g ¯ α n...n − k +1 = l X i =0 Λ kli ¯ α n − l...n − l − k + i +1 α n − l − k + i +1 ...n − k α n − k +1 ...n ⋆ g ¯ α n...n − l +1 = l X i =0 Λ kli ¯ α n − k...n − k − l + i +1 α n − k − l + i +1 ...n − l In particular, for any ≤ k, l ≤ n we have ¯ α n...k α k...n ⋆ g ¯ α n...l α l...n = ¯ α n...l α l...n ⋆ g ¯ α n...k α k...n . Proof. Let us prove the first formula. Recall that a ⋆ g b = ab + P i ≥ a ⋆ g,i b . Let a = α n − k +1 ...n and b = ¯ α n...n − l +1 . Note that a ⋆ g,i b = 0 for any i > l . For 1 ≤ i ≤ l and n − l + 1 ≤ m i < · · · < m < m ≤ n , we perform the right-most reduction forthe path ab using g exactly i times involving the elements α m , . . . , α m i in a . Thenwe get( λ m − k +1 ,m )( λ m − k +2 ,m ) · · · ( λ m i − k + i,m i )¯ α n − k...n − k − l + i +1 α n − k − l + i +1 ...n − l . Thus, taking the sum over n − l + 1 ≤ m i < · · · < m < m ≤ n we get a ⋆ g,i b .The second formula can now be deduced from the symmetry of the two formulae:since ⋆ g is associative we may perform left -most reductions for reducing the path ab ,giving the same constants Λ kli . Of course, the second formula can also be computedvia right-most reductions as done for the first formula. (cid:3) EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 71 Relation to the A n singularity. The preprojective algebra Π( Q ) can be viewedas a noncommutative resolution of the (commutative) A n singularity, which maybe identified with e Π( Q ) e (cf. Remark 11.11). It follows from Remark 7.50 that e Π λ ( Q ) e ≃ ( e Π( Q ) e , ⋆ g ) is a deformation of e Π( Q ) e and we may describe( e Π( Q ) e , ⋆ g ) explicitly in terms of reductions.Let ω := α ...n and ¯ ω := ¯ α n... be the elements in e Qe corresponding to theclockwise and counterclockwise cycles of length n + 1 and note that the followingirreducible paths form a k -basis of e Π( Q ) e : (cid:8) ¯ ω k ω l , ¯ ω k ¯ α n α n ω l , . . . , ¯ ω k ¯ α n... α ...n ω l (cid:9) k,l ≥ . (9.32)Note that since ¯ ωω is an irreducible path we have ¯ ω ⋆ g ω = ¯ ωω .The explicit expressions of the product given in Proposition 9.31 may be used toshow the following result. The first two assertions are contained in [43, Thm. 0.4]and the third assertion in [98, § Corollary 9.33. ( i ) The subalgebra e Π λ ( Q ) e is commutative if and only if λ ,n = 0 . In this case, we have an isomorphism of algebras k [ x, y, z ] (cid:14) (cid:0) xz − n Q i =0 ( y − λ i,n ) (cid:1) e Π λ ( Q ) e x ¯ ωy ¯ α n α n z ω. ( ii ) e Π λ ( Q ) e is Morita equivalent to Π λ ( Q ) if and only if λ i,j = 0 for all < i ≤ j ≤ n . ( iii ) Consider λ i = 0 for < i ≤ n and set λ := ~ . We have an k J ~ K -algebraisomorphism k h x, y, z i J ~ K /I ~ e Π λ ( Q ) e J ~ K where I ~ is the two-sided ideal generated by xz − y n +1 = 0 , [ y, x ] = x ~ , [ z, y ] = z ~ , [ z, x ] = ( y + ~ ) n +1 − y n +1 . In particular, k h x, y, z i J ~ K /I ~ is a deformation quantization of the symplec-tic structure given by { y, x } = x, { z, y } = z, { z, x } = ( n + 1) y n on the A n singularity k [ x, y, z ] / ( xz − y n +1 ) .Remark . From a geometric perspective ( i ) describes precisely the commutativedeformations of the A n singularity, ( ii ) describes the (homologically) smooth de-formations (cf. [43, Thm. 0.4]) which in the commutative case describe the smoothdeformations of the A n singularity, and ( iii ) describes a special noncommutativedeformation which may be viewed as a quantization of a non-degenerate Poissonstructure. In § degenerate Poisson struc-ture (cf. Remark 9.39). Proof of Corollary 9.33. Let us illustrate the proof of ( i ). If ( e Π λ ( Q ) e , ⋆ g ) iscommutative, then in particular we have ω ⋆ g ¯ ω = ¯ ω ⋆ g ω = ¯ ωω . By Proposition9.31 we have ω ⋆ g ¯ ω = ¯ ωω + λ ,n (¯ α n... α ...n + lower-length terms) . EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 72 This yields that λ ,n = 0 . Conversely, let λ ,n = 0. Then from the above we have ω ⋆ g ¯ ω = ¯ ωω = ¯ ω ⋆ g ω (since λ ,n = 0) and by Proposition 9.31 again for any1 ≤ k, l ≤ n ¯ α n...k α k...n ⋆ g ¯ α n...l α l...n = ¯ α n...l α l...n ⋆ g ¯ α n...k α k...n . This implies that a ⋆ g b = b ⋆ g a for any elements a, b in the basis (9.32) of e Π( Q ) e .We get that ( e Π λ ( Q ) e , ⋆ g ) is commutative. Thus, the algebra map f : k h x, y, z i ( e Π( Q ) e , ⋆ g )(9.35)determined by f ( x ) = ¯ ω, f ( y ) = ¯ α n α n and f ( z ) = ω induces an algebra map¯ f : k [ x, y, z ] ( e Π( Q ) e , ⋆ g ). It remains to prove(9.36) ¯ f ( xz − y ( y − λ ,n ) · · · ( y − λ n,n )) = 0 . Since ¯ f is an algebra homomorphism we have¯ f ( y ( y − λ n,n ) · · · ( y − λ ,n ))= (¯ α n α n ) ⋆ g (¯ α n α n − λ n,n ) ⋆ g (¯ α n α n − λ n − ,n ) ⋆ g · · · ⋆ g (¯ α n α n − λ ,n )= ¯ α n...n − α n − ...n ⋆ g (¯ α n α n − λ n − ,n ) ⋆ g · · · ⋆ g (¯ α n α n − λ ,n )= · · · = ¯ α n... α ...n where in the second identity we compute the first ⋆ g and the third identity meansthat we compute ⋆ g from left to right. This shows (9.36) and we obtain an inducedhomomorphism¯ f : k [ x, y, z ] / ( xz − y ( y − λ n,n ) · · · ( y − λ ,n )) e Π λ ( Q ) e . which is seen to be an isomorphism by looking at the bases given by irreduciblepaths.To prove ( ii ) one may use a criterion from Buchweitz [31, Prop. 1.9] whichstates that Π λ ( Q ) is Morita equivalent to e Π λ ( Q ) e if and only if the restrictedmultiplication map µ e : Π λ ( Q ) e ⊗ e Π λ ( Q ) e e Π λ ( Q ) Π λ ( Q ) a ⊗ b ab is surjective. This is seen to be equivalent to e i being in the image of µ e for any0 ≤ i ≤ n . Using the formulae of Proposition 9.31 it is then straight forward tofind elements in Π λ ( Q ) e ⊗ e Π λ ( Q ) e e Π λ ( Q ) mapping to e i precisely under thecondition that λ i,j = 0 for all 0 < i ≤ j ≤ n .Let us prove the third assertion ( iii ). Consider the k J ~ K -linear extension (stilldenoted by f ) of the map f in (9.35). Using Proposition 9.31 we may provethat f sends the ideal I ~ to zero and thus it induces an algebra homomorphism¯ f : k h x, y, z i J ~ K /I ~ ( e Π( Q ) e J ~ K , ⋆ g ) . Note that we have a reduction system( zx, xz + ( y + ~ ) n +1 − y n +1 ) , ( zy, yz + y ~ ) , ( yx, xy + x ~ ) , ( y n +1 , xz )satisfying ( ⋄ ) for the ideal I ~ . Thus, the irreducible paths { x i y j z l ~ m } i,l,m ≥ , ≤ j ≤ n are k -linearly independent in k h x, y, z i J ~ K /I ~ . It follows from (9.32) that ¯ f is bi-jective. Since e Π( Q ) e ≃ k [ x, y, z ] / ( xz − y n +1 ), we get that k h x, y, z i J ~ K /I ~ is aformal deformation of k [ x, y, z ] / ( xz − y n +1 ) via the isomorphism ¯ f . (cid:3) EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 73 Deformed quantum preprojective algebras of type e A n . Let n ≥ 2. Thereis another class of deformations of type e A n preprojective algebras for which theexistence of an algebraization follows from the degree condition ( ≺ ), i.e. g s ≺ s (seeProposition 9.16) with respect to the admissible order determined by α ≻ α ≻· · · ≻ α n ≻ ¯ α ≻ ¯ α ≻ · · · ≻ ¯ α n .Consider the 2-cocycle g : k S Π( Q ) given by g ( α i ¯ α i ) = λ i e i + µ i ¯ α i − α i − (9.37)where λ i , µ i ∈ k for 0 ≤ i ≤ n . Note that g satisfies the condition ( ≺ ). Recall fromLemma 5.15 that the differential ∂ : Hom( k Q , Π( Q )) Hom( k S, Π( Q )) is givenby ∂ ( h )( α i ¯ α i ) = h ( α i )¯ α i + α i h (¯ α i ) − h (¯ α i − ) α i − − ¯ α i − h ( α i − ) . Take h ( α i ) = ν i α i and h (¯ α i ) = 0 where ν i ∈ k for 0 ≤ i ≤ n . Then we have ∂ ( h )( α i ¯ α i ) = ( ν i − ν i − )¯ α i − α i − where we set ν − = ν n . Note that P ni =0 ( ν i − ν i − ) = 0. Thus a 2-cocycle g of theform (9.37) is a 2-coboundary if and only if P ni =0 µ i = 0. Up to equivalence (cf.Definition 3.13), we may take µ i = 0 for 1 ≤ i ≤ n and write q = 1 + µ . It followsfrom Proposition 9.16 that we get a family of deformations of Π( Q ):Π λq ( Q ) := k Q (cid:14) (cid:0) α ¯ α − q ¯ α n α n − λ e , P ni =1 α i ¯ α i − ¯ α i − α i − − λ i e i (cid:1) which are called deformed quantum preprojective algebras of type e A n . The quantumpreprojective algebras , i.e. the case λ = · · · = λ n = 0, were studied in [111, 81] andthe general case in [72, 42].Similar to Corollary 9.33 we have the following result. Proposition 9.38. Let k q [ x, y, z ] denote the quantum affine space k h x, y, z i / ( zx − q n +1 xz, yx − q xy, zy − q yz ) .For λ i = 0 for ≤ i ≤ n , the algebra e Π λq ( Q ) e is isomorphic to the quantumhypersurface k q [ x, y, z ] / ( xz − y n +1 ) . Proof. Consider the algebra map f : k h x, y, z i ( e Π q ( Q ) e , ⋆ g ) determined by f ( x ) = ¯ ω , f ( y ) = ¯ α n α n and f ( z ) = ω . By a straight-forward computation, we havethat f sends the elements zx − q n +1 xz , yx − q xy , zy − q yz and xz − y n +1 to zero.Thus f induces an algebra map¯ f : k h x, y, z i / ( zx − q n +1 xz, yx − q xy, zy − q yz, xz − y n +1 ) e Π q ( Q ) e which is seen to be an isomorphism by comparing the bases given by irreduciblepaths on both sides. (cid:3) Remark . Note that Proposition 9.38 recovers Example 8.20 and describes thequantization of a degenerate Poisson structure on the A n surface singularity. Thequantization of the (nondegenerate) symplectic structure was described in Corollary9.33 ( iii ). EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 74 Applications to geometry In this section we give applications to algebraic geometry and show how defor-mations of the Abelian category Qcoh( X ) can be described as deformations of thealgebra k Q/I . In this context, there are two main sources of path algebras of quiv-ers with relations: one is defined from an affine open cover U of a projective scheme X and the other is defined from a tilting bundle on X . The former works for anyquasi-compact and semi-separated scheme X , but when X admits a tilting bundle,the latter is computationally more convenient.In § X ) and explain how to describethis using path algebras of quivers with relations. In § k (1 , 1) surface singularities, will be described in § Deformations of Abelian categories of (quasi)coherent sheaves. Let X be a variety over an algebraically closed field k of characteristic 0 (or moregenerally a quasi-compact semi-separated scheme). Then X admits a finite affineopen cover U which is closed under intersections.The restriction O X | U of the structure sheaf to the cover U can be viewed as a diagram of algebras , which is a contravariant functor (a presheaf) O X | U : U Alg k ,where U can be viewed as a finite subcategory of the category Open ( X ) of open setswith morphisms given by inclusion. Deformations of diagrams of algebras were firststudied by Gerstenhaber–Schack [60] and higher structures on the Gerstenhaber–Schack complex were given in Dinh Van–Lowen [46]. (See also [15] for a differentconstruction of higher structures via higher derived brackets.)The following result establishes an equivalence of different deformation problems. Theorem 10.1 (Lowen–Van den Bergh [87, 88]) . Let ( X, O X ) be a quasi-compactand semi-separated scheme over an algebraically closed field k of characteristic .There is an equivalence of formal deformations between deformations of ( i ) O X | U as twisted presheaf ( ii ) O X | U ! as associative algebra ( iii ) Qcoh( X ) as Abelian category ( iv ) Mod( O X ) as Abelian category.Moreover, if X is Noetherian, then the above deformations are also equivalent todeformations of ( v ) coh( X ) as Abelian category. (We shall mainly focus on the case of varieties, which are special cases of quasi-compact and semi-separated schemes.) The different types of deformations in The-orem 10.1 are parametrized by what are essentially various versions of Hochschildcohomology with first-order deformations parametrized by(10.2)HH ( X ) ≃ H ( O X | U ) ≃ HH ( O X | U !) ≃ H (Qcoh( X )) ≃ H (Mod( O X ))and obstructions in HH ( X ) ≃ · · · ≃ H (Mod( O X )). Here HH ( A ) = Ext A e ( A, A )is the usual Hochschild cohomology for associative algebras, and the Hochschild EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 75 cohomology of a scheme may be defined analogously as(10.3) HH ( X ) := Ext O X × X ( δ ∗ O X , δ ∗ O X )where δ ∗ O X is the pushforward of the structure sheaf along the diagonal map δ : X X × X [60, 82, 122]. Also, H GS is a cohomology theory for diagrams(or prestacks) of algebras (see [46, 60]) and H Ab a cohomology theory for Abeliancategories (see [87, 88]).First we shall consider ( ii ) of Theorem 10.1 which concerns deformations of theso-called diagram algebra O X | U ! which is defined as follows. Definition 10.4. Let A be a diagram of k -algebras over a small category U , i.e. acontravariant functor A : U Alg k .The diagram algebra of A , denoted A !, is given as k -module by A ! = Y U ∈ U M U ϕ V A ( U ) x ϕ where the sum is over all morphisms in U and x ϕ is simply a formal (bookkeeping)symbol. The multiplication of elements a ∈ A ( U ) and b ∈ A ( V ) is defined by( ax ϕ )( bx ψ ) = ( a A ( ϕ )( b ) x ψ ◦ ϕ if ψ ◦ ϕ is defined0 otherwisewhere a A ( ϕ )( b ) is the product of a and A ( ϕ )( b ) in the algebra A ( U ) U V W A ( U ) A ( V ) A ( W ). ϕ ψ A ( ϕ ) A ( ψ ) Theorem 10.1 states that studying deformations of Qcoh( X ) as Abelian cate-gory is equivalent to studying deformations of the diagram algebra O X | U !. Theconstruction summarized in the following proposition can be used to study defor-mations of Qcoh( X ) for X any projective variety via the methods developed in § A = k Q/I . Proposition 10.5. Let X ⊂ P n be a projective variety and let U be a cover of X obtained by restricting (a subset of ) the standard affine charts of P n to X andtaking its closure under intersections.The diagram algebra O X | U ! can be written as the path algebra of a finite quiver Q on U vertices modulo a finitely generated ideal I of relations.Moreover, there is a noncommutative Gr¨obner basis for I giving rise to a reduc-tion system R satisfying ( ⋄ ) for I .Proof. We give a general proof. For concrete examples see Examples 10.9 and 10.33.We first show how to obtain Q and I for X = P n . Let U , . . . , U n be the standardaffine coordinate charts of P n with U i = { [ x : · · · : x n ] ∈ P n | x i = 0 } and let U = { U i ··· i m | ≤ i < · · · < i m ≤ n } ≤ m ≤ n where U i ··· i m = U i ∩ · · · ∩ U i m i.e. U is the closure of { U , . . . , U n } under taking intersections. EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 76 The diagram O P n | U can be viewed as the 1-skeleton of an ( n + 1)-hypercube withone vertex (corresponding to P n ) and the incident edges removed. The cover U hascardinality 2 n +1 − O P n | U ! can then be written as thepath algebra A = k Q/I of a quiver Q with relations I as follows.Let Q be the quiver on 2 n +1 − U i ··· i m ∈ U and let each vertex be labelled by i · · · i m , say. For each label i · · · i m there are m + 1 inclusions U i ··· ˆ ı k ··· i m ⊂ U i ··· i m where ˆ ı k indicates the omission ofthe k th index and for each inclusion we add an arrow i · · · i m i · · · ˆ ı k · · · i m inthe opposite direction since O P n | U is contravariant. For example, for n = 1 , , 01 01 012 010212 012 0123 012013023123 010203121323 0123 For each square(10.7) i · · · i m i · · · ˆ ı k · · · i m i · · · ˆ ı l · · · i m i · · · ˆ ı k · · · ˆ ı l · · · i m ψ ik ψ il ϕ il ϕ ik we add the relation ϕ i l ψ i k − ϕ i k ψ i l . (Here the subscripts of ϕ, ψ correspond toadding the subscript to the index.)Now at each vertex i · · · i m we have a commutative algebra O P n ( U i ··· i m ) ≃ k (cid:2) x j x ik (cid:3) ≤ j ≤ n ≤ k ≤ m of Krull dimension n whose Spec is just k n − m × ( k × ) m . This algebra can be writtenas the path algebra of a quiver of a single vertex with n + m loops z , . . . , z n + m modulo commutativity relations z i z j = z j z i and m further relations z i k + k − z i k + k =1 for 1 ≤ k ≤ m , corresponding to the fact that on U i ··· i m , m of the generators x j x i are invertible (i.e. those for which j = i , . . . , i m ).More precisely, we may choose the loops z , z , . . . , z n + m at the vertex i · · · i m as corresponding to x x i , . . . , c x i x i , . . . , x i x i , x i x i , x i +1 x i , . . . , x i x i , x i x i , x i +1 x i , . . . , x i m x i , x i x i m , x i m +1 x i , . . . , x n x i . We add one last set of relations involving the loops at each vertex and the arrowsof the underlying acyclic quiver. Let ϕ be a morphism i · · · i m i · · · ˆ ı k · · · i m ϕ ...... ...z ,..., z n + m y ,..., y n + m − in the underlying acyclic quiver. If k = 0, we add relations y p ϕ − ϕz p for 1 ≤ p ≤ i k + k − y p ϕ − ϕz p +1 for i k + k ≤ p ≤ n + m − 1. Otherwise weadd relations y p ϕ − ϕz q for y p = x i x i = z q and relations y p ϕ − ϕz q z r whenever y p = x i x il = x i x i x i x il = z q z r for 1 < l ≤ m or y p = x j x i = x j x i x i x i = z q z r for j = i . EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 77 Fix the (commutative) Gr¨obner basis for k [ z , . . . , z n ] / ( z i − z i − ≤ i ≤ n at thevertex 01 · · · n with respect to the lexicographical ordering which extends the order z ≺ z ≺ · · · ≺ z n . This gives a natural ordering on the n + m generators at eachvertex i · · · i m , and the natural ordering on the vertices gives a natural ordering onthe compositions of the morphisms as in (10.7).We thus obtain a reduction system R with entries • ( ( z i k + k − z i k + k , , ( z i k + k z i k + k − , 1) for 1 ≤ k ≤ m ( z j z i , z i z j ) for i < j, ( i, j ) = ( i k + k − , i k + k ) at each vertex • ( y p ϕ, ϕz q ) or ( y p ϕ, ϕz q z r ) along each morphism • ( ϕ i l ψ i k , ϕ i k ψ i l ) i k
Deformations of a genus curve. Consider the smooth genus 3curve X ⊂ P = { [ x , x , x ] } cut out by the quartic equation F = x x + x x + x = 0 . Note that the point [0 , , 1] does not lie in X so that X ⊂ P \ { [0 , , } = U ∪ U where U = { [ x , x , x ] | x = 0 } = { [1 , z, u ] } ≃ k U = { [ x , x , x ] | x = 0 } = { [ ζ, , v ] } ≃ k . Let U, V denote the restrictions of U , U to X . Setting U = { U, V, U ∩ V } we canfollow the proof of Proposition 10.5 and write the diagram algebra O X | U ! as the EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 78 path algebra of the following quiver zu x yw ζvϕ ψ U U ∩ V V with ideal of relations I generated by s − f s for ( s, f s ) ∈ R where R is the reductionsystem consisting of the following pairs • ( uz, zu ), ( vζ, ζv ), ( wx, xw ), ( wy, yw ) commutativity of charts • ( zϕ, ϕx ), ( uϕ, ϕw ), ( ζψ, ψy ), ( vψ, ψyw ) compatibility with morphisms • ( xy, yx, mutually inverse coordinates • ( u , − z u − z ), ( w , − x w − x ), ( v , − ζ − v ) equation of curve. In particular, we have that S = { uz, vζ, wx, wy, xy, yx, zϕ, uϕ, ζψ, vψ, u , v , w } . To show that R satisfies ( ⋄ ) for I we apply Theorem 7.36 which implies that itsuffices to show that the overlaps uzϕ , u ϕ , vζψ , v ψ , xyx , yxy , wxy , wyx , w x , w y , u z , v ζ in S are resolvable. For each overlap this is a short and straight-forward computation, for example uzϕzuϕ uϕxzϕw ϕwxϕxw w yyw − x wy − xy − yx w − yx − x yw − − x w − wxy xwyxyww. Now let g ∈ Hom( k S, A ) be any 2-cochain and as usual we denote g s = σg ( s ).Write F U , F V and F W for the restrictions of F to the charts U , U and U = U ∩ U of P , i.e. F U = u + z u + zF V = v + ζ + vF W = w + x w + x. In order for g to satisfy the Maurer–Cartan equation, so that k Q/I g is a defor-mation of A , we need only check that R g is reduction finite and reduction unique.Again, this can be checked on the elements in S , giving the conditions(10.10) g wx = g wy = g uz = g vζ = 0 and g xy = g yx as well as(10.11) g v ψ = ψy g w + ∂F V ∂ζ g ζψ + ∂F V ∂v g vψ − ψ ∂F W ∂x y g yx g u ϕ = ϕg w + ∂F U ∂u g uϕ + ∂F U ∂z g zϕ . EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 79 Here (10.10) signifies that the commutativity relations of the charts should notbe changed and (10.11) imposes compatibility conditions between changing themorphisms and changing the equation of X in the individual charts.Since X is a degree 4 plane curve, we can choose natural representatives of thecocycles g s by using the vector space isomorphism J F V ≃ H ( T X ) ≃ k where J F V = k [ ζ, v ] (cid:14) (cid:0) ∂F V ∂ζ , ∂F V ∂v (cid:1) = k [ ζ, v ] / (4 v + 1 , ζ )is the Jacobian ring of F V with k -basis { , ζ, v, ζv, v , ζv } .In other words, for X it is enough to deform the equation F , which correspondsto choosing g v = λ + λ ζ + λ v + λ ζv + λ v + λ ζv then setting g zϕ = g uϕ = g ζψ = g vψ = g yx = g xy = 0 and solving (10.11) giving g w = λ x + λ x + λ x w + λ x w + λ x w + λ xw g u = λ z + λ z + λ z u + λ z u + λ z u + λ zu . Note that for a genus g curve C one has HH ( C ) ≃ H ( T C ) ≃ k g − . Here X is a genus 3 curve and indeed we obtain a 6-dimensional family of non-trivialdeformations of X parametrized by λ = ( λ , . . . , λ ) ∈ k with fibres X λ = { F λ =0 } ⊂ P , where F λ = x x + x + λ x + λ x x + (1 + λ ) x x + λ x x x + λ x x + λ x x x . This example shows that the deformation theory of Qcoh( X ) can be studiedrather explicitly. Indeed, each element ( s, f s ) in the reduction system has a cleargeometric meaning: it corresponds either to a commutativity relation of the localcoordinates, to the identification of coordinates across charts, to the defining equa-tions, or to the commutativity of the coordinate changes across charts. That is,the geometric meaning of the modifications to the reduction system is very muchtransparent.Note that in general, a deformation of O X | U (equivalently of Qcoh( X )) may bea not necessarily commutative diagram of not necessarily commutative algebras, soit is indeed natural to look at deformations of O X | U ! in this general context of pathalgebras of quivers with relations.10.1.1. Geometry. In Theorem 10.1 we saw that the deformation theory of theAbelian category Qcoh( X ) admits several equivalent algebraic descriptions — namelydeformations of O X | U as a twisted presheaf, or deformations of the diagram algebra O X | U ! as an associative algebra.The isomorphisms (10.2) showed that the cohomology groups parametrizing thesedeformations are isomorphic to the Hochschild cohomology HH ( X ) of the scheme,providing the following geometric interpretation.If X is smooth, then the Hochschild–Kostant–Rosenberg theorem (see [129]) givesa decomposition(10.12) HH n ( X ) ≃ M p + q = n H p (Λ q T X )where Λ q T X is the sheaf of sections in the q th exterior power of the (algebraic)tangent bundle. (See [32, 33] for a decomposition in the singular case.) Infinitesimal EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 80 deformations of Qcoh( X ) are thus parametrized byHH ( X ) ≃ H (Λ T X ) ⊕ H ( T X ) ⊕ H ( O X )where( i ) H (Λ T X ) is the space of bivector fields, which for Poisson bivector fieldsparametrize a (noncommutative) algebraic quantization of O X ( ii ) H ( T X ) is well known to parametrize algebraic deformations of X as ascheme (which over k = C correspond to classical deformations of the com-plex structure), and( iii ) H ( O X ) parametrize “twists”, i.e. deformations of the (trivial) O ∗ X -gerbestructure of O X .Hence deformations of Qcoh( X ) can be thought of as a combination of these threetypes of deformations.Note that if X is a curve, then HH ( X ) ≃ H ( T X ) and HH ( X ) = 0 so thatall deformations of Qcoh( X ) are induced by classical deformations of the curve (cf.Example 10.9).10.1.2. Tilting bundles and their endomorphism algebras. By Theorem 10.1 defor-mations of the Abelian category Qcoh( X ) admit a description as deformations of thediagram algebra A = O X | U ! which by Proposition 10.5 can be written as A ≃ k Q/I .In case the variety admits a tilting bundle (e.g. projective spaces [19], Grass-mannians, quadrics [73, 74], rational surfaces [69], hypertoric varieties [118]) — forexample the tilting bundle obtained from a strong full exceptional collection —there is a much more economical description of this deformation theory by meansof a much smaller quiver: we haveEnd E ≃ A = k Q/I for some finite quiver Q and some ideal of relations I . Here Q is constructed byputting a vertex for each direct summand of E and adding arrows to generate themorphism spaces between the direct summands.The tilting bundle E induces a derived equivalence between the Abelian cate-gory of coherent sheaves on X and the Abelian category of finitely generated rightmodules for the endomorphism algebra A = k Q/I ≃ End E D b (mod A ) ≃ D b (coh X )(10.13)given by the functors R Hom( E , − ) and − ⊗ L E and (10.13) induces an isomorphismof Hochschild cohomologies HH ( A ) ≃ HH ( X ) . (See for example [11, 26, 70].)10.2. Noncommutative projective planes. We now explain how a tilting bun-dle on the projective plane P can be used to describe noncommutative projectiveplanes via deformations of the relations of the Be˘ılinson quiver (see [1] for a similarapproach) illustrating the role of the reduction system throughout. Noncommuta-tive projective planes have been studied by many authors (see e.g. [5, 6, 7, 8, 27])using both algebraic and geometric methods and we also relate the point of view EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 81 of deformations of path algebras of quivers with relations to quantizations of Pois-son structures, Artin–Schelter regular algebras, and to deformations of the diagramalgebra O P | U !.10.2.1. Noncommutative projective geometry. The Gabriel–Rosenberg reconstruc-tion theorem implies that a quasi-separated scheme can be recovered from itsAbelian category of quasi-coherent sheaves and the idea of noncommutative pro-jective geometry is to work with Abelian categories with similar properties tothose of the category of quasi-coherent sheaves on projective varieties. By theresults of Serre’s seminal paper [116], there is an equivalence of Abelian categoriescoh( X ) ≃ qgr( B ) where X = Proj( B ) is a projective variety and B is its homoge-neous coordinate ring.In the case of P this equivalence is given as coh( P ) ≃ qgr( B ), where B = k [ x , x , x ] is the graded polynomial ring with generators x , x , x in degree one.Instead of the (commutative) homogeneous coordinate ring of a projective variety,one may consider the Abelian categories qgr( S ) for S belonging to some class of non-commutative graded algebras (see e.g. [9, 10, 112]). Following [119, Def. 11.2.1], a noncommutative projective plane is then an Abelian (Grothendieck) category qgr( S )for some Artin–Schelter regular algebra S of dimension 3 with Hilbert series (1 − t ) − (i.e. the same Hilbert series as the homogeneous coordinate ring of P ).10.2.2. The Be˘ılinson quiver. It goes back to Be˘ılinson [19] that the projective space P n admits a tilting bundle E = O P n ⊕ O P n (1) ⊕ · · · ⊕ O P n ( n ) whose endomorphismalgebra can be written as the path algebra of the quiver Q O O (1) · · · O ( n − O ( n ) x ,..., x n x ,..., x n x n ,..., x nn ... ... ... ... modulo the ideal I generated by (cid:0) x i,k x i +1 ,j − x i,j x i +1 ,k (cid:1) ≤ i The relations (10.16) for the Be˘ılinson quiver Q of P can be obtained from the reduction system R = (cid:8) ( x y , x y ) , ( x y , x y ) , ( x y , x y ) (cid:9) with S = (cid:8) x y , x y , x y (cid:9) and the deformation theory of A can be described via the L ∞ algebra p ( Q, R ) givenin Theorem 7.9.The 2-cochains are thus arbitrary k Q -bimodule maps k S A . Expressingthese maps in the basis Irr S of irreducible paths, the compatibility with the k Q -bimodule structure implies that 2-cochains are thus arbitrary linear maps k S k Irr S, where Irr S, = { x y , x y , x y , x y , x y , x y } are the irreducible paths of length 2. Since R has no overlaps, it moreover followsthat any 2-cochain is also a 2-cocycle (cf. Remark 7.15).The space of 2-cocycles can thus be identified as Hom( k S, A ) ≃ M × ( k ). Thisspace is of dimension 18 and it is straight forward to determine the space of 2-coboundaries, which is 8-dimensional. (This gives a direct computation of the iso-morphism HH ( A ) ≃ k using the resolution obtained from the reduction system.)Concretely, 2-cocycle representatives for the cohomology classes can be given by theelements in Hom( k S, A ) defined by x y λ x y + λ x y + λ x y + λ x y + λ x y + λ x y x y λ x y + λ x y + λ x y x y λ x y (10.18)where λ , . . . , λ ∈ k and one immediately obtains the following result. Proposition 10.19. There is a -dimensional algebraic family of deformations of A = k Q/I = End O P ( O ⊕ O (1) ⊕ O (2)) over k ≃ HH ( A ) with fibres A g = k Q/I g where I g is the ideal generated by x y − ( λ + 1) x y − λ x y − λ x y − λ x y − λ x y − λ x y x y − x y − λ x y − λ x y − λ x y x y − x y − λ x y . EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 83 x x x Figure 6. The degeneracy loci of the basis elements. Proof. Clearly, R g is reduction finite. Then the result follows from Proposition 7.46( ii ) since S = ∅ . (cid:3) In particular, we may work directly with actual deformations rather than formaldeformations.In the remainder of the section we relate this description of noncommutativeprojective planes to quantizations of Poisson structures, to Artin–Schelter regularalgebras and to moduli spaces of quiver representations.10.2.4. Poisson structures and quantizations. By (10.17) each element in HH ( A )can be viewed as a bivector field on P , which (since Λ T P = 0) defines a Poissonstructure on P whose degeneracy locus can be an arbitrary cubic curve, i.e. thedegeneracy locus is given by the zero locus of a homogeneous polynomial of degree3. Choosing homogeneous coordinates [ x , x , x ] for P , the Hochschild cohomology(10.20) HH ( P ) ≃ H ( O (3)) ≃ k { x i x j x k } ≤ i,j,k ≤ i + j + k =3 has a basis given by the degree 3 monomials and we shall write a general elementas(10.21) η = X i,j,k η ijk x i x j x k . In the standard coordinate charts U i = { [ x , x , x ] | x i = 0 } ≃ { ( z i , w i ) ∈ k } , thebivector field corresponding to η is then given by η U i ∂∂z i ∧ ∂∂w i where η U i ∈ O ( U i ) is the dehomogenization of some homogeneous cubic polynomialcorresponding to η under the isomorphism H (Λ T P ) ≃ H ( O P (3)). The associatedPoisson structure is then given locally by { f, g } η = η U i (cid:18) ∂f∂z i ∂g∂w i − ∂f∂w i ∂g∂z i (cid:19) for f, g ∈ O ( U i ).For each basis element of (10.20), the degeneracy locus of the associated Poissonstructure consists of three lines (a configuration of the lines “at infinity” for the localcharts) as shown in Fig. 6 and taking linear combinations, the degeneracy locus can EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 84 be any cubic curve which may be a smooth, nodal or cuspidal irreducible cubic curve,the union of a conic and a secant or tangent line, or any configuration of three linescounted with multiplicity. The degeneracy loci of Poisson structures thus fall intonine types illustrated in Fig. 7, stratifying the space of cubic curves P (H (Λ T P )) ≃ P into nine components of dimension 9 , , , , , , , , 2, respectively. Figure 7. The degeneracy loci of Poisson structures on P In view of the equivalence coh( P ) ≃ qgr( k [ x , x , x ]) deformations of P shouldcorrespond to graded deformations of the homogeneous coordinate ring B (cf. Kon-tsevich [83, § quadratic Poisson structures on A inthe sense of § Proposition 10.22 (Polishchuk [103, Thm. 12.2]) . Let η ∈ H (Λ T P ) be a Poissonstructure on P . There exists a unique lifting of η to a unimodular quadratic Poissonstructure e η = P ≤ i We now describe the corre-spondence between quantizations of a Poisson structure and deformations of therelations in the Be˘ılinson quiver, by relating the coefficients η ijk (10.21) of a Poissonstructure η ∈ H (Λ T P ) to the coefficients λ , . . . , λ (10.18) of an k Q -bimodulemap g ∈ Hom( k S, A ). This relationship will be given in Proposition 10.30 below.We shall use Be˘ılinson’s “resolution of the diagonal” [19]. Notation 10.23. Let δ : P P × P be the diagonal embedding and write∆ := δ ( P ) ⊂ P × P for the diagonal and O ∆ = δ ∗ O P for the structure sheaf ofthe diagonal. Let p, q : P × P P be the natural projections p ( x, y ) = x and q ( x, y ) = y . For two sheaves E and F on P write E ⊠ F := p ∗ ( E ) ⊗ O P × P q ∗ ( F )for their external tensor product, which is a sheaf on P × P . (Note that O P ⊠ O P = O P × P .) For ease of reading we shall drop the subscript P on O or Ω i . Lemma 10.24. For any i, j, k, l ≥ we have the following isomorphism Hom O P × P ( O ( − i ) ⊠ O ( j ) , O ( − k ) ⊠ O ( l )) ≃ H ( O ( i − k )) ⊗ k H ( O ( l − j )) . Proof. This follows straight-forwardly from the adjunctions between ⊗ and Hom and between p ∗ and p ∗ , the projection formula, and the base change formula. (cid:3) EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 85 Recall now from Be˘ılinson [19] that we have the following resolution of the diag-onal O ∆ (10.25) 0 O ( − ⊠ Ω (2) O ( − ⊠ Ω (1) O P × P O ∆ (2) := Ω ⊗ O O (2) ≃ O ( − ≃ O ( − 3) is the canonical bundle on P . Consider the Euler exact sequence (e.g. [67, Thm. II.8.13])0 Ω O ( − ⊕ O O ( − ⊕ should be understood as O ( − ⊗ V ∨ where x , x , x are homo-geneous coordinates of P = P ( V ) and V ∨ = k { x ∗ , x ∗ , x ∗ } is the dual space. TheEuler sequence induces a morphism q ∗ (Ω (1)) q ∗ ( O ⊕ ) ∼ ( O P × P ) ⊕ ∼ p ∗ ( O ⊕ ) p ∗ ( O (1)) . Thus by tensoring with p ∗ O ( − 1) we get the first differential in the Be˘ılinson reso-lution O ( − ⊠ Ω (1) O ( − ⊠ O (1) ∼ O P × P whose cokernel is isomorphic to O ∆ . The Hochschild cohomology (10.3) may nowbe computed from the local-to-global spectral sequence E p,q = H p ( P × P , Ext q O P × P ( O ∆ , O ∆ )) = ⇒ Ext p + q O P × P ( O ∆ , O ∆ )which is concentrated in the q = 0 row and thus already collapses on the E -pagegiving(10.26) HH ( P ) ≃ Hom O P × P ( O ( − ⊠ Ω (2) , O ∆ ) ≃ Hom O ( δ ∗ ( p ∗ ( O ( − ⊗ O q ∗ (Ω (2))) , O ) ≃ Hom O ( δ ∗ p ∗ ( O ( − ⊗ O δ ∗ q ∗ (Ω (2)) , O ) ≃ Hom O ( O ( − , O ) ≃ H ( O (3))where the second isomorphism follows from the adjunction between δ ∗ and δ ∗ , thethird isomorphism is due to the fact that the pullback functor δ ∗ commutes with ⊗ , and the fourth isomorphism follows from p ◦ δ = id = q ◦ δ and Ω (2) ≃ O ( − E = O ⊕ O (1) ⊕ O (2) be the tilting bundle of P . It follows from Buchweitz–Hille [34, Prop. 2.6] that E ∨ ⊠ E is a tilting bundle on P × P , where E ∨ := Hom ( E , O ). In other words, the derived category D b ( P × P ) is generated by objects { O ( − i ) ⊠ O ( j ) } ≤ i,j ≤ . By Lemma 10.24 we have Hom O P × P ( E ∨ ⊠ E , E ∨ ⊠ E ) = A e .Resolving each term in the Be˘ılinson resolution (10.25) one obtains the followingdiagram O ( − ⊠ O ( − O ( − ⊠ Ω (1) O P × P O ∆ O ( − ⊠ O ⊕ O ( − ⊠ O ⊕ O P × P O ( − ⊠ O (1) ⊕ O ( − ⊠ O (1) O ( − ⊠ O (2) EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 86 so that O ∆ can be seen to be quasi-isomorphic to the total complex P of the doublecomplex obtained by deleting the first row of this diagram. Now P consists of directsums of O ( − i ) ⊠ O ( j ) for 0 ≤ i, j ≤ 2. Applying the functor Hom O P × P ( E ∨ ⊠ E , − )to this complex P , we get a projective bimodule resolution Hom O P × P ( E ∨ ⊠ E , P )of Hom O P × P ( E ∨ ⊠ E , O ∆ ) ≃ A which is isomorphic to the complex P A ⊗ k S ⊗ A A ⊗ k Q ⊗ A A ⊗ k Q ⊗ A R .It follows from Lemma 10.24 that we have an isomorphismHom O P × P ( E ∨ ⊠ E , O ( − ⊠ O ⊕ ) ≃ Hom O P × P ( O ( − ⊠ O , O ( − ⊠ O ⊕ ) ≃ A ⊗ k S ⊗ A. (10.27)Consider the following diagramHom O P × P ( O ( − ⊠ O ⊕ , O ∆ ) Hom O P × P ( O ( − ⊠ O ( − , O ∆ )Hom A e ( A ⊗ k S ⊗ A, A ) H ( O (3)) ≃ ≃ (10.28)where the horizontal map at the top is induced by the natural map O ( − O ⊕ ,the right vertical map is given by (10.26) and the left vertical map is induced bythe isomorphism (10.27) and Hom O P × P ( E ∨ ⊠ E , O ∆ ) ≃ A (cf. [34, Thm. 3.4]). Wedefine the map ρ : Hom( k S, A ) H ( O (3))(10.29)by making the above diagram commute under the isomorphism Hom A e ( A ⊗ k S ⊗ A, A ) ≃ Hom( k S, A ) (7.7). Therefore, we have the following result. Proposition 10.30. The map ρ : Hom( k S, A ) H ( O (3)) induces the isomor-phism HH ( A ) ≃ HH ( P ) ≃ H ( O (3)) in (10.17) . Moreover, ρ is given by ρ ( g ) = ( x g x y − x g x y + x g x y ) | y i = x i where | y i = x i means y , y , y in g x m y n should be replaced by x , x , x , respectively.In particular, g , g ∈ Hom( k S, A ) represent the same element in HH ( A ) if andonly if ρ ( g ) = ρ ( g ) .Proof. The first part follows from the above analysis. Let us prove the second part.In (10.27) the second isomorphism is given byHom O P × P ( O ( − ⊠ O , O ( − ⊠ O ) ⊗ V ∨ ∼ A ⊗ k { x y , x y , x y } ⊗ A which sends id ⊗ x ∗ e ⊗ x y ⊗ e , id ⊗ x ∗ e ⊗ x y ⊗ e and id ⊗ x ∗ e ⊗ x y ⊗ e . The third part follows since the composition of the following mapsis zero Hom( k Q , A ) ∂ Hom( k S, A ) ρ H ( O (3))where we recall that ∂ is given by x y h ( x ) y + x h ( y ) − h ( x ) y − x h ( y ) x y h ( x ) y + x h ( y ) − h ( x ) y − x h ( y ) x y h ( x ) y + x h ( y ) − h ( x ) y − x h ( y )for any h ∈ Hom( k Q , A ). (cid:3) EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 87 The map ρ thus sends the elements (10.18) in Hom( k S, A ) to η = x (cid:0) λ x x + λ x x + λ x x + λ x + λ x + λ x (cid:1) − x ( λ x x + λ x + λ x ) + λ x . By Proposition 10.22 the corresponding unimodular Poisson structure on A is e η = P ≤ i Consider the Poisson structure η with degeneracy locus λ x x x + λ x on P for λ , λ ∈ k not both zero. Note that for λ = 0 the degeneracy locusis a union of three lines pairwise intersecting in a point, for λ = 0 the degeneracylocus is a line (with multiplicity 3) and for λ , λ both non-zero the degeneracylocus is a conic and a line intersecting the conic in two points (cf. Fig. 7).By Proposition 10.22 the corresponding unimodular quadratic Poisson structure e η on A is given by (cid:18) λ x x + λ x (cid:19) ∂∂x ∧ ∂∂x − λ x x ∂∂x ∧ ∂∂x + 13 λ x x ∂∂x ∧ ∂∂x . One checks that e η satisfies the condition of Proposition 8.9. Then the algebra k h x , x , x i J ~ K / ( x x − e λ ~ x x − λ ~ x , x x − e − λ ~ x x , x x − e λ ~ x x )is a formal deformation quantization of e η on A . Moreover, e η satisfies the degreecondition ( ≺ ) with respect to the order x ≺ x ≺ x and thus by Proposition 9.16admits an algebraization, so that by evaluating ~ 1, the algebra(10.32) B e η = k h x , x , x i / ( x x − qx x − px , x x − q − x x , x x − qx x )is an actual deformation of k [ x , x , x ]. (Here q = e λ and p = λ are constants.)The algebra B e η is an Artin–Schelter regular algebra (see e.g. [97]). By Belmans–Presotto [20, Lem. 23] we have a triangle equivalence D b (qgr B e η ) ≃ D b ( A ′ ), where A ′ := k Q/ ( x y − qx y − px y , x y − q − x y , x y − qx y ).Let g ∈ Hom( k S, A ′ ) be the element of the form (10.18) with ρ ( g ) = η under themap ρ in (10.29). That is, g x y = λ x y + λ x y , g x y = 0 , g x y = 0 . Then k Q J ~ K / ( x y − e λ ~ x y − λ ~ x y , x y − x y , x y − x y ) is gauge equivalentto A ′ b g = k Q J ~ K / ( x y − e λ ~ x y − λ ~ x y , x y − e − λ ~ x y , x y − e λ ~ x y )where b g x y = P n ≥ λ n ~ n n ! x y + λ x y , b g x y = 0 , b g x y = 0 . Note that by evaluating ~ A ′ b g | ~ =1 = A ′ . EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 88 Relation to the diagram algebra. Finally we shall illustrate how the variousdescriptions of noncommutative projective planes can be seen as deformations of thediagram algebra O P | U !, where U is the closure of the standard open cover U , U , U under intersections.Following the construction of the proof of Proposition 10.5, we have that O P | U ! ≃ k Q/I , where Q is a quiver with 7 vertices and 28 arrows and I an ideal generated by48 relations, giving rise to a reduction system R with 48 entries and, as it happens,also 48 overlaps.Not only is the diagram algebra infinite-dimensional, the corresponding quiver isalso much more elaborate. However, abstractly we already know that HH ( O P | U ) ≃ HH ( P ) ≃ k (see (10.2) and (10.17)). In the following example we use thedescription obtained in Example 10.31 to find a Maurer–Cartan element for thedeformation of the diagram algebra. Example 10.33. Consider the Poisson structure η with degeneracy locus λ x x x + λ x on P as in Example 10.31.We may deduce the corresponding deformation of the diagram algebra O P | U !directly from the Artin–Schelter regular algebra (10.32) B e η = k h x , x , x i / ( x x − qx x − px , x x − q − x x , x x − qx x )where q = e λ and p = λ . For x = 0 we may rewrite the second and thirdrelation as(10.34) x − x − q − x x − and x − x − qx x − by multiplying by x − from the left and the right. At the vertex 0 (cf. the middlequiver of (10.6)) we may then identify the two loops at this vertex (see the proofof Proposition 10.5) with z = x x − and z = x x − and calculate the relationbetween z and z by using the first relations of B e η and the relations (10.34), giving z z = x x − x x − = q z z + pq. One may modify the reduction system for O P | U ! at the vertex 0 from ( z z , z z )to ( z z , q z z + pq ) . The cases for x = 0 and x = 0 are completely analogous and by modifyingonly the commutativity relations at the vertices, one obtains a deformation of thediagram algebra O P | U !. The corresponding noncommutative plane thus appears tobe “glued” from three “noncommutative affine planes” • k h z , z i / ( z z − q z z − pq ) at the vertex 0 • k h z , z i / ( z z − q z z − pqz ) at the vertex 1 • k h z , z i / ( z z − q z z − pqz ) at the vertex 2.Although the computation is a little cumbersome, the diagram algebra is of amore geometrical nature than the approach via the tilting bundle. By Proposition10.5 deformations of the diagram algebra can be computed for any projective variety— even if it does not admit any tilting bundle — and using the degree conditions( < ) or ( ≺ ) it should be possible to obtain explicit actual deformations for morecomplicated examples. EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 89 Moduli spaces of quiver representations. We have seen in § E can be described rather conveniently andpurely algebraically as deformations of the endomorphism algebra A = End E ≃ k Q/I for some quiver Q with relations I .Even though the noncommutative geometry of noncommutative algebraic geom-etry is still somewhat mysterious, the effectiveness of geometric methods as seenin the construction of noncommutative projective planes is striking. Much of thegeometry (e.g. in the existing constructions and classification of Artin–Schelter reg-ular algebras) can also be recovered from the point of view of quivers, namely byconsidering representations of the quiver Q with relations I and their moduli spacesof (semi)stable representations . We briefly recall the main constructions here andrefer to [80, 109, 127] for more details.A (finite-dimensional) representation of a quiver Q is given by V = (cid:16)(cid:0) V i (cid:1) i ∈ Q , (cid:0) V s( x ) V ( x ) V t( x ) (cid:1) x ∈ Q (cid:17) where for each vertex i ∈ Q one specifies a finite-dimensional k -vector space V i andfor each arrow x ∈ Q a k -linear map V ( x ) : V s( x ) V t( x ) between the correspond-ing vector spaces of the source and target vertices. The tuple (dim V i ) i ∈ Q ∈ N Q is called the dimension vector of the representation V . The set of representationswith a fixed dimension vector α ∈ N Q can be viewed as an affine variety which wedenote by Rep α ( Q ).Similarly, a representation of a quiver Q with ideal of relations I ⊂ k Q is arepresentation of Q such that the linear maps V ( x ) : V s( x ) V t( x ) also satisfy therelations I and Rep α ( Q, I ) shall denote the variety of representations of ( Q, I ) withdimension vector α .A stability function is a Z -linear map ϑ : Z Q Z and a representation V of Q with dimension vector α is said to be ϑ -semistable (resp. ϑ -stable ) if ϑ ( α ) = 0and every subrepresentation V ′ ⊂ V with dimension vector α ′ satisfies ϑ ( α ′ ) ≥ ϑ ( α ′ ) > ϑ and a dimension vector α are coprime if ϑ ( α ) = 0.Finally, one defines the moduli space of ϑ -semistable representations of a quiverwith relations as the quotient M ss ϑ α ( Q, I ) = Rep α ( Q, I ) // χ ϑ G := Proj (cid:16) M n ≥ k [Rep α ( Q, I )] G,χ nϑ (cid:17) where on the right-hand side k [Rep α ( Q, I )] denotes the coordinate ring of the affinevariety Rep α ( Q, I ) and G = Q i ∈ Q GL α i ( k ) is a reductive group with character χ ϑ : G k × ( g i ) i Q i det( g i ) ϑ i determined by ϑ = ( ϑ i ) i ∈ Q and // χ ϑ denotes the GIT quotient. (This definition wasintroduced by King [80], who also proved that ϑ -(semi)stability for representationsof quivers is equivalent to χ ϑ -(semi)stability in the sense of geometric invarianttheory (GIT).)Moduli spaces of quiver representations can often be shown to recover a varietyfrom the endomorphism algebra of a tilting bundle (see Karmazyn [75]). Karmazyn EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 90 [76] also showed that (under some conditions on the base of deformations) tiltingbundles lift to a (classical) deformation of the variety. If E is a tilting bundle on asmooth variety X , the Hochschild–Kostant–Rosenberg isomorphism (10.12) gives adecomposition HH (End E ) ≃ H (Λ T X ) ⊕ H ( T X ) ⊕ H ( O X )and the direct summand H ( T X ) corresponds to “geometric” deformations of X .Indeed, the deformed variety can sometimes be recovered as a moduli space of quiverrepresentations for the quiver with the corresponding deformed relations (see § ( T X )one looks at deformations corresponding to H (Λ T X ), the moduli spaces of rep-resentations for the deformed relations are no longer geometric deformations, butthey are still rich in geometry, as the case of projective planes nicely illustrates.10.3.1. Relation to noncommutative projective planes. Let Q be the Be˘ılinson quiverfor P given in (10.15) with ideal of relations I (10.16) and let k Q/I g be a defor-mation of A = k Q/I for some g ∈ Hom( k S, A ) as in Proposition 10.19. Lemma 10.35. For each g , the moduli space M ss ϑ α ( Q, I g ) with dimension vector α = (1 , , and stability function ϑ = ( − , , can be viewed as a subvariety of P × P given by the three equations in Proposition 10.19 for ([ x , x , x ] , [ y , y , y ]) ∈ P × P . Proof. We calculate the moduli space M ss ϑ α ( Q, I g ) for α = (1 , , 1) and ϑ = ( − , , Q with dimension vector α is just given by assigning to each x , x , x and each y , y , y a linear map k k , i.e. x i , y j are just assigned arbi-trary constants in k giving(10.36) k k k . x , x , x y , y , y A representation of Q is ϑ -semistable if it admits no subrepresentations with di-mension vector α ′ such that ϑ ( α ′ ) < 0. For α = (1 , , 1) the possible dimension vec-tors α ′ of subrepresentations such that ϑ ( α ′ ) < α ′ ∈ { (1 , , , (1 , , , (1 , , } .For example, a subrepresentation with dimension vector α ′ = (1 , , 0) is given by acommutative diagram(10.37) k k kk k . x , x , x y , y , y x , x , x Thus for a representation given by constants x i , y j to be ϑ -semistable, it shouldnot admit (10.37) as a subrepresentation. This is the case when y j = 0 for at leastone j since then the right square cannot commute. Similarly one checks that for α ′ = (1 , , 1) or (1 , , ϑ -semistability imposes the condition that also x i = 0 forat least one i .We thus have that representations of Q are given by ( x , x , x ) , ( y , y , y ) ∈ k \ { } and the action of G = k × × k × × k × identifies M ss ϑ α ( Q, I ) ≃ ( k \ { } ) × ( k \ { } ) /G ≃ P × P . EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 91 Finally, (10.36) is a ϑ -semistable representation of ( Q, I g ) if it is a ϑ -semistablerepresentation of Q and the constants x i , y j satisfy the relations generating the ideal I g (see Proposition 10.19). (cid:3) Noncommutative projective planes correspond to three-dimensional regular qua-dratic Z -algebras which are determined by triples ( C, L , L ). Here the triple iseither ( P , O (1) , O (1)) or a curve C embedded in P as a divisor of degree 3 bythe global sections of two line bundles L and L (see Bondal–Polishchuk [27, § L and L embed C as a completeintersection of three bidegree (1 , 1) hypersurfaces in P × P . If C = dgn( η ) is thedegeneracy locus of some Poisson structure η , let g ∈ Hom( k S, A ) be the corre-sponding element (10.18). Then M ss ϑ α ( Q, I g ) is cut out by three such bidegree (1 , L , L , so that the moduli space M ss ϑ α ( Q, I g ) is isomorphic to thedegeneracy locus of the Poisson structure (see also § η ) is reducible (i.e. contains a line), then this can be seen explicitlyfrom the following calculation. Proposition 10.38. Let η be a Poisson structure on P whose degeneracy locus dgn( η ) is reducible. Let g ∈ Hom( k S, A ) be the representative of η of the form (10.18) and let M ss ϑ α ( Q, I g ) be the moduli space of semistable representations of A g = k Q/I g with ϑ = ( − , , and α = (1 , , .Then M ss ϑ α ( Q, I g ) ≃ dgn( η ) as algebraic varieties.Proof. If the degeneracy locus of η is reducible, then by the classification of cubicplane curves η can be written as x f ( x , x , x ) for some f ( x , x , x ) = a x + a x x + a x x + a x + a x x + a x where a i ∈ k . Thus, the representative g in (10.18) of η is given by g x y = e f ( x , x , x ) and g x y = 0 = g x y . Here e f ( x , x , x ) is the multilinearization of f ( x , x , x ), i.e. e f ( x , x , x ) = a x y + a x y + a x y + a x y + a x y + a x y . Then M ss ϑ α ( Q, I g ) is cut out by x y − ( a + 1) x y − a x y − a x y − a x y − a x y − a x y x y − x y x y − x y .Denote by X the projection of M ss ϑ α ( Q, I g ) onto the first factor P of P × P . Then X is given by the following equationdet − a x − a x − a x + x − a x − ( a + 1) x − a x x − x x − x = 0 . which is exactly x f ( x , x , x ) = 0, the degeneracy locus of η . Note that the abovematrix is of rank 2 for any point ( x , x , x ) in X . This shows that M ss ϑ α ( Q, I g ) isa graph of X and thus they are isomorphic as varieties. (cid:3) EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 92 Remark . In order to relate the moduli space M ss ϑ α ( Q, I g ) to the degeneracylocus, one should work with “actual” rather than formal deformations. However,the Poisson structures are a first-order phenomenon and when the degeneracy locusof a Poisson structure is not irreducible, the corresponding formal deformation doesnot appear to admit an obvious algebraization. We expect this to be the reason whyProposition 10.38 can in this form only be proved for the case of Poisson structureswith reducible degeneracy loci. Example 10.40. Consider again the Poisson structure η with degeneracy locus λ x x x + λ x of Examples 10.31 and 10.33. The representative g of η is given by g x y = λ x y + λ x y and g x y = 0 = g x y . The projection X of M ss ϑ α ( Q, I g )to the first factor P of P × P is given by the equationdet − λ x x ( − λ − x x − x x − x = 0 , which is λ x x x + λ x = 0, the degeneracy locus of η . Note that the abovematrix is of rank 2 for any point ( x , x , x ) in X . Thus, M ss ϑ α ( Q, I g ) is isomorphicto the degeneracy locus λ x x x + λ x = 0 of η .11. Deformations of reconstruction algebras In this section we illustrate deformations of path algebras of quivers with relationsfor another family of examples, namely reconstruction algebras of rational surfacesingularities (see Wemyss [125, 126]). We have already seen deformations of pre-projective algebras of type e A n , which are reconstruction algebras for the n (1 , n − § 10 we shall now revisitreconstruction algebras for the case of the k (1 , 1) singularity with more emphasison the interplay with geometry. The k (1 , 1) singularity is rather simple from ageometric perspective and thus admits a more compact description, but we expectsimilar results to hold for any other rational surface singularity.Let k ≥ X be the k (1 , 1) singularity and let e X = Tot O P ( − k )denote its minimal resolution. This simple case already nicely illustrates manydifferent phenomena: • The reconstruction algebra A of the k (1 , 1) singularity is the endomorphismalgebra of a tilting bundle on e X and thus one has both geometric andalgebraic descriptions of their deformations, both describing deformationsof the Abelian category coh( e X ) (cf. § • From a geometric perspective e X is a smooth quasi-projective surface whichadmits both commutative and noncommutative deformations. • Commutative deformations of e X correspond to (algebraizable) PBW defor-mations of the endomorphism algebra of the tilting bundle. • The surface e X is not affine, but its geometric (commutative) deformationsare smooth affine surfaces, which can be seen directly from the point of viewof quivers after an application of Morita equivalence (see § • Deformations of e X induce (commutative and noncommutative) deforma-tions of these singularities (see § EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 93 • Giving a noncommutative Gr¨obner basis for A one can show that certainnoncommutative deformations arising as quantizations of (degenerate) Pois-son structures on e X also admit an algebraization. • Moduli spaces of semistable representations for the deformed reconstructionalgebra recover both commutative deformations of e X and the degeneracyloci of Poisson structures. The minimal resolution. We work over k = C . Let k ≥ X be the k (1 , 1) surface singularity, i.e. X = C / Γwhere Γ < GL ( C ) is a cyclic group of order k with the generator acting on C by (cid:18) ω ω (cid:19) for ω some primitive k th root of unity. (For k = 2 we have that Γ < SL ( C ), whencewe sometimes need to treat the cases k = 2 and k ≥ e X = Tot O P ( − k ) for k ≥ X , which is asmooth quasi-projective surface.The pullback of the tilting bundle O P ⊕ O P (1) on P along the projection e X π P is O e X ⊕ O e X (1), which is a tilting bundle on e X . We have(11.1) A = End( O e X ⊕ O e X (1)) ≃ k (cid:18) x , x y ,..., y k − ... (cid:19). I where I is the ideal generated by x y j − x y j − y j x − y j − x j = 1 , . . . , k − A is the reconstruction algebra A k, for the (type A) k (1 , 1) singularity (cf.Wemyss [126]). We use the usual notation Q for the quiver in (11.1) and label theleft vertex of Q corresponding to O e X by 0 and the right vertex corresponding to O e X (1) by 1.11.1. Hochschild cohomology. The derived equivalence D b ( A ) ≃ D b ( e X ) givenby the tilting bundle induces isomorphisms HH i ( A ) ≃ HH i ( e X ) (see § § e X obtained in [15, 16, 17].) EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 94 On the geometric side the Hochschild cohomology may be calculated via theHochschild–Kostant–Rosenberg theorem (10.12). Namely, the only non-zero Hoch-schild cohomology groups of e X areHH ( e X ) ≃ H ( O e X )HH ( e X ) ≃ H ( T e X )HH ( e X ) ≃ H (Λ T e X ) ⊕ H ( T e X )HH ( e X ) ≃ H (Λ T e X ) . (Note that H i ( e X, F ) = 0 for any i ≥ F since e X is coveredby two affine open sets and Λ i T e X = 0 for i ≥ e X is a surface.)The surface e X is a toric surface covered by only two affine open sets, and thecohomology groups appearing on the right-hand side can thus easily be calculatedvia ˇCech cohomology — in the case of line bundle coefficients (i.e. O e X or Λ T e X ≃ O e X ( − k + 2)) the cocycles can simply be read off the corresponding toric diagrams.To give an explicit comparison between the algebraic and geometric side, let usrepresent cohomology classes in H i (Λ j T e X ) as ˇCech cocycles for the cover U = { U, V } where U = { ( z, u ) ∈ C } and V = { ( ζ, v ) ∈ C } and for z, ζ = 0 we have ( ζ, v ) =( z − , z k u ) on U ∩ V ≃ C × × C . (Here z and ζ are the local coordinates on P .)On the U -chart, we may give an O e X ( U )-basis of O e X ( U ) , T e X ( U ) , Λ T e X ( U ) bythe constant function 1, the vector fields ∂∂z and ∂∂u , and the bivector field ∂∂z ∧ ∂∂u ,respectively. In the following lemma, we express the cohomology groups appearingin HH i ( e X ) in this basis. Lemma 11.3. The cohomology groups relevant to the deformation theory appearingin the Hochschild–Kostant–Rosenberg decomposition of HH i ( e X ) are the following. H ( O e X ) ≃ k [ u, zu, . . . , z k u ] = k [ z , . . . , z k ] / ( z i z j +1 − z i +1 z j ) ≤ i For A = k Q/I as in (11.1), we can give the followingreduction system(11.4) R = (cid:8) ( x y j , x y j − ) , ( y j − x , y j x ) (cid:9) The reduction system R (11.4) has overlap ambigui-ties S = (cid:8) x y j x (cid:9) 0) 0 < j < kβ := ( x y , x y , . . . , x y k − ) ⊕ (0 , . . . , β := ( x y , x y , . . . , x y k − ) ⊕ (0 , . . . , β := ( x y , x y , . . . , x y k − ) ⊕ (0 , . . . , k = 2 by α := e ⊕ − e β symp := e ⊕ . EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 96 Of course all of these are elements in ker ∂ ⊂ A ⊕ k − ⊕ A ⊕ k − . (Note that for k = 2, β , β , β are also 2-cocycles, but they can be obtained from β symp by multiplyingby the paths x y , x y , x y viewed as elements in HH ( A ) = Z( A ).)Similar to the case of P , one may obtain the following correspondence between“algebraic” 2-cocycles in HH ( A ) and “geometric” 2-cocycles in HH ( e X )(11.7) commutative noncommutative algebraic α j β l β symp geometric z − k + j ∂∂u z l u ∂∂z ∧ ∂∂u ∂∂z ∧ ∂∂u ( k ≥ 2) ( k ≥ 2) ( k = 2) (cf. Lemma 11.3). Note that β symp corresponds to the canonical holomorphic sym-plectic form on the (open) Calabi–Yau surface Tot O P ( − ≃ T ∗ P .11.2. Commutative deformations. The following theorem constructs a familyof deformations of A which correspond to “classical” geometric deformations of e X .Let α , . . . , α k − be as in (11.7). Theorem 11.8. ( i ) The element g = α t + · · · + α k − t k − is a Maurer–Cartan element of the L ∞ algebra p ( Q, R ) b ⊗ ( t , . . . , t k − ) and the associateddeformed algebra b A g = ( A J t , . . . , t k − K , ⋆ g ) is isomorphic to the quotient algebra k Q J t , . . . , t k − K / b I g , where b I g is thecompletion of the two-sided ideal I g generated by x y j − x y j − − e t j y j − x − y j x + e t j < j ≤ k − . ( ii ) The formal deformation b A g in ( i ) admits an algebraization A g = ( A [ t , . . . , t k − ] , ⋆ g ) ≃ k Q [ t , . . . , t k − ] /I g . ( iii ) Evaluating the algebraization in ( ii ) to t i λ i for some λ = ( λ , . . . , λ k − ) ∈ k k − we obtain a Maurer–Cartan element g λ of p ( Q, R ) and the associateddeformation A g λ is Morita equivalent to its center Z( A g λ ) ≃ e A g λ e ≃ e A g λ e precisely when λ = ( λ , . . . , λ k − ) = 0 . In this case we have alge-bra isomorphisms Z( A g ) ≃ e i A g e i ≃ k [ z , . . . , z k ] (cid:14)(cid:0) rank (cid:0) z z + λ z + λ ··· z k − + λ k − z z z ··· z k (cid:1) ≤ (cid:1) . Proof. By Theorem 7.36, it suffices to check that the reduction system R g = { ( s, f s + g s ) | s ∈ S } is reduction unique (see § S = { x y j x } This proves ( i ), and ( ii ) then follows from Theorem 9.3; the proof of ( iii ) is com-pletely analogous to the proof of Corollary 9.33 ( ii ). (cid:3) Deformations of singularities. The results of Theorem 11.8 can be relatedto the deformations of the k (1 , 1) singularity as follows.The algebra e Ae ≃ e Ae ≃ Z( A ) ≃ k [ z , . . . , z k ] / ( z i z j +1 − z i +1 z j ) ≤ i 1) singular-ity. Indeed, considering the Maurer–Cartan element g = α t + · · · + α k − t k − , theevaluation at ( t , . . . , t k − ) = ( λ , . . . , λ k − ) ∈ k k − \ { } yields an algebra Moritaequivalent to the coordinate ring of the affine variety X λ . Remark . The construction of deformations of singularities works for generalnoncommutative resolutions of singularities. Let Z be a Noetherian commutative k -algebra, which is singular. A noncommutative resolution of Z is a k -algebra of theform A = End Z ( Z ⊕ M ), where M is a finitely generated Z -module, such that A has finite global dimension. Let e ∈ A be the idempotent of A corresponding to thedirect summand Z of Z ⊕ M . If A can be written as k Q/I such that e correspondssome vertex of Q , then deformations of A give a family of deformations of Z = eAe by Remark 7.50. EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 98 Noncommutative deformations. Let k ≥ β , β , β be as in(11.7). The following result is the noncommutative analogue to Theorem 11.8. Theorem 11.12. ( i ) The element g = β t + β t + β t is a Maurer–Cartanelement of p ( Q, R ) b ⊗ ( t , t , t ) and the associated formal deformation b A g = ( A J t , t , t K , ⋆ g ) is isomorphic to k Q J t , t , t K / b I g , where b I g is the completion of the two-sidedideal generated by y j − x − y j x − y j − x t − y j x t − y j +1 x t x y j − x y j − < j ≤ k − where we set y k x := y k − x . ( ii ) The formal deformation associated to β t + β t admits an algebraizationand thus g µ = µ β + µ β is a Maurer–Cartan element of p ( Q, R ) for any µ = ( µ , µ ) ∈ k .Proof. As in the proof of Theorem 11.8, we only need to check reduction-uniquenesson elements in S . That is, we have x y j x x y j − x x y j x + x y j − x t + x y j x t + x y j +1 x t . < j < k − g satisfies the Maurer–Cartan equation.Now ( ii ) follows from Proposition 9.16 ( ii ), noting that g = β t + β t satisfiesthe condition ( ≺ ), since x y ≻ x y , x y and x y j ≻ x y j − , x y j . (cid:3) Remark . Denoting by A µ the algebra obtained by evaluating the algebraiza-tion given in Theorem 11.12 ( ii ) at µ = ( µ , µ ) ∈ k , the algebra e A µ e is anoncommutative deformation of the k (1 , 1) singularity e Ae . In fact, for µ = 0this evaluation can be viewed as a “ q -deformation” of the algebra, which turns outto leave the singularity category of the k (1 , 1) singularity invariant. The details forthe general case of cyclic quotient surface singularities will appear in future workjoint with M. Kalck.11.4. The Calabi–Yau case. For k = 2, the (1 , 1) singularity is just the A sin-gularity whose minimal resolution e X ≃ Tot O P ( − ≃ T ∗ P is an “open” Calabi–Yau surface. Theorem 11.14. ( i ) Any -cochain in P defines a Maurer–Cartan elementof p ( Q, R ) b ⊗ m . ( ii ) Let g ∈ Hom( k S, A ) b ⊗ ( t , t ) be the Maurer–Cartan element g = α t + β t = e ( t + t ) ⊕ ( − e ) t . The corresponding deformation ( A J t , t K , ⋆ g ) admits an algebraization ( A [ t , t ] , ⋆ g ) isomorphic to the quotient algebra A g = k Q [ t , t ] /I g , where I g is the two-sided ideal generated by x y − x y − e t − e t y x − y x + e t . EFORMATIONS OF PATH ALGEBRAS OF QUIVERS WITH RELATIONS 99 ( iii ) Evaluating the algebraization of ( ii ) at t = λ and t = µ for some λ, µ ∈ k ,the subalgebras e A λ,µ e and e A λ,µ e are commutative if and only if µ = 0 in which case A λ, for λ = 0 is Morita equivalent to a deformation of the A surface singularity. ( iv ) Evaluating the algebraization of ( ii ) at t = 0 and t = 1 we have that e A , e ≃ U( sl ) / ( C + ) and e A , e ≃ U( sl ) / ( C ) where U( sl ) is the universal enveloping algebra of sl = h X, Y, H i and C = XY + Y X + H is the Casimir element.Proof. Assertion ( i ) follows from the fact that P = 0 (cf. Remark 7.15), ( ii ) followsfrom Theorem 9.3, ( iii ) from Theorem 11.8 ( iii ), and ( iv ) is well known (see forexample Schedler [115, § II.1]). (cid:3) Remark . Note that e A , e ≃ U( sl ) / ( C + ) has infinite global dimensionwhereas e A , e ≃ U( sl ) / ( C ) has finite global dimension [43, Thm. 0.4]. Indeed,Crawford [42, Cor. 1.2.6] showed that there is an equivalence of triangulated cat-egories of singularities D sg ( e A , e ) ≃ D sg ( e Ae ). (The latter is the singularitycategory of the A singularity.)11.5. Moduli spaces of quiver representations. As in the case of noncommu-tative projective planes (see § A = k Q/I can be understood as the endomorphism algebraof a tilting bundle on the minimal resolution e X and it was shown in Karmazyn [75,Cor. 5.4.2] that e X ≃ M ss ϑ α ( Q, I ) for some particular choice of stability function ϑ and dimension vector α .Moreover, let g λ ∈ Hom( k S, A ) be a Maurer–Cartan element corresponding toan element in H ( T e X ) under(11.16) HH ( A ) ≃ HH ( e X ) ≃ H (Λ T e X ) ⊕ H ( T e X ) ⊕ H ( O e X )and let A g λ = k Q/I g λ be the corresponding deformation. Then one can recoveralso the (commutative) deformations of the minimal resolution as moduli spaces,namely X λ ≃ M ss ϑ α ( Q, I g λ ) for the same ϑ and α (see Karmazyn [76]).Now let g µ correspond to an element η µ ∈ H (Λ T e X ) under (11.16) so that thecorresponding deformation A g µ = k Q/I g µ of A corresponds to a quantization of thePoisson structure η µ . Similar to the case of P (cf. Proposition 10.38), the modulispace, which is a commutative algebraic variety, should intuitively speaking recoverthe moduli space of “commutative points” of the noncommutative deformation.Indeed, there is an isomorphism M ss ϑ α ( Q, I g µ ) ≃ dgn( η µ ) ⊂ e X , i.e. the moduli spacerecovers the degeneracy locus dgn( η µ ) of the Poisson structure and dgn( η µ ) can beviewed as the subvariety of e X along which the deformation of the multiplication istrivial and still commutative.Let us illustrate these statements also for the k (1 , 1) singularity, so let A = k Q/I be as in (11.1) and fix the dimension vector α = (1 , 1) and stability function ϑ =( − , g λ = λ α + · · · + λ k − α k − λ = ( λ j ) j ∈ k k − g µ = µ β + µ β µ = ( µ l ) l ∈ k be the Maurer–Cartan elements of p ( Q, R ) obtained by evaluation from the alge-braizations in Theorems 11.8 ( ii ) and 11.12 ( ii ). Here α j , β l are as in (11.7) andlet η µ denote the Poisson structure corresponding to g µ , which is given in local U -coordinates by ( µ + µ z ) u ∂∂z ∧ ∂∂u . Proposition 11.17. We have the following isomorphisms M ss ϑ α ( Q, I ) ≃ X = e X M ss ϑ α ( Q, I g λ ) ≃ X λ M ss ϑ α ( Q, I g µ ) ≃ dgn( η µ ) ⊂ e X. Proof. As in the proof of Lemma 10.35 one sees that a representation is just givenby assigning x , x , y , . . . , y k − arbitrary constants, and such a representation issemistable precisely when x and x are not both 0, so that before imposing therelations I , the moduli space M ss ϑ α ( Q ) is the space k \ { } × k k quotiented by G = k × × k × .It is straight forward to check that by imposing the relations I and I g λ onerecovers exactly e X and its commutative deformations — for example, one mayverify this in the coordinate charts { x = 0 } and { x = 0 } . (See [16, § e X and its deformations in coordinates.)Note that when x i , y j are constants, they commute, in which case half of therelations imposed by I and I g λ are redundant. However, the relations in I g µ do notexhibit this kind of symmetry, and thus impose extra relations in e X cutting outprecisely the degeneracy locus dgn( η µ ). (cid:3) Remark . In order to give a geometric interpretation of the moduli space, oneshould work with an “actual” rather than a formal deformation of A , i.e. thosedeformations obtained by evaluating a formal deformation at some value of the pa-rameter. The only PBW deformations of A which correspond to the quantizationof some Poisson structure are those corresponding to the quantization of a holomor-phic symplectic structure, i.e. the case Γ < SL ( C ) as for example (1 , M ss ϑ α ( Q, I g µ ) and the degeneracy locus dgn( η )are empty . 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