aa r X i v : . [ m a t h . QA ] S e p COUNTING USING HALL ALGEBRAS III. QUIVERS WITHPOTENTIALS
JIARUI FEI
Abstract.
For a quiver with potential, we can associate a vanishing cycleto each representation space. If there is a nice torus action on the potential,the vanishing cycles can be expressed in terms of truncated Jacobian algebras.We study how these vanishing cycles change under the mutation of Derksen-Weyman-Zelevinsky. The wall-crossing formula leads to a categorification ofquantum cluster algebras under some assumption. This is a special case ofA. Efimov’s result, but our approach is more concrete and down-to-earth.We also obtain a counting formula relating the representation Grassmanniansunder sink-source reflections.
Introduction
We continue our development on algorithms to count the points of varieties re-lated to quiver representations. In this note, we focus on the quivers with potentials.A potential W on a quiver Q is just a linear combination of oriented cycles of Q .It can be viewed as a function on a certain noncommutative space attached to Q .When composed with the usual trace function, it becomes a regular function ω on each representation space Rep α ( Q ). This function further descends to variousmoduli spaces. In this paper, all potentials are assumed to be polynomial, i.e., afinite linear combination of oriented cycles.Let f be a regular function on a complex variety X . Consider the scheme theo-retic critical locus { df = 0 } of f . Behrend, Bryan and Szendr¨oi define in [3] a class[ ϕ f ( X )] ∈ K (Var C )[ L − ] associated to each such locus, essentially given by themotivic Milnor fibre of the map f . When X admits a suitable torus action, thisclass can be expressed as [ ϕ f ( X )] = [ f − (0)] − [ f − (1)] . Deligne’s mixed Hodge structure on compactly supported cohomology gives riseto the E -polynomial homomorphism E : K (Var C ) → Z [ x, y ] given by E ([ Y ]; x, y ) = X p,q x p y q X i ( − i dim H p,q ( H ic ( Y, Q )) . The E -polynomial could sometimes be computed using arithmetic method. By a spreading out of a complex variety Y , we mean a separated scheme Y over a finitelygenerated Z -algebra R with an embedding ϕ : R ֒ → C such that the extension of Mathematics Subject Classification.
Key words and phrases.
Quiver Representation, Quiver with Potential, Ringel-Hall Alge-bra, Donaldson-Thomas Invariants, Vanishing Cycle, Virtual Motive, Moduli space, Representa-tion Grassmannian, Quantum Cluster Algebra, Cluster Character, Quantum Dilogarithm, Wall-Crossing, BB-Tilting, Mutation, Jacobian Algebra, Polynomial-count. This definition of [ ϕ f ( X )] differs from the original one by a negative sign. scalars Y ϕ ∼ = Y . Following N. Katz [11, Appendix], we say that Y is polynomial-count if there is a polynomial P Y ( t ) ∈ Z [ t ] and a spreading out Y such that forevery homomorphism ϕ : R → F q to a finite field, the number of F q -points of thescheme Y ϕ is P Y ( q ). Furthermore, the definitions descend to the Grothendieckgroup K (Var C ). It is known ([11, Theorem 6.1.2, 6.1.3]) that Lemma 0.1.
Assume that γ ∈ K (Var C ) is polynomial-count with counting poly-nomial P γ ( t ) ∈ Z [ t ] , then the E -polynomial of γ is given by E ( γ ; x, y ) = P γ ( xy ) . In this note, we directly work over finite fields F q and follow an algebraic approachto compute P γ ( t ) in the above quiver setting. Namely, γ is the class [ ϕ ω ], definedby the regular function ω on representation spaces. In fact, we will work with thegenerating series of the virtual point counts | ϕ ω ( X ) | vir for all dimension vectors: V ( Q, W ) := X α | ϕ ω (Rep α ( Q )) | vir | GL α | vir x α . Here, by definition the virtual point count is related to the ordinary point countby a q -shift: | ϕ ω ( X ) | vir := q − dim X | ϕ ω ( X ) | . Our main results contain two wall-crossing formulas – one for the ordinary V ( Q, W ), the other for the one with a framing stability ν ∞ :T( Q, W ) := X β (cid:12)(cid:12)(cid:12) ϕ ω (Mod ν ∞ (1 ,β ) ( Q )) (cid:12)(cid:12)(cid:12) vir x (1 ,β ) , where Mod να ( Q ) denotes the GIT moduli of α -dimensional ν -stable representations.To state them, let µ k be the mutation of ( Q, W ) in the sense of Derksen, Weyman,and Zelevinsky [7]. Let E k := exp q ( q / q − x k ) , V := V ( Q, W ) , V ′ := V ( µ k ( Q, W )) andT ′ := T( µ k ( Q, W )), then under the technical condition , of Section 4, Theorem 0.2.
We have that E ′ k Φ ∨ k ( VE − k ) = V ′ = Φ k ( E − k V ) E ′ k ;(0.1) ( E ′ k ) − Φ k (T) E ′ k = T ′ = Φ ∨ k ( E k T E − k ) . (0.2) Here, multiplications are performed in appropriate completed quantum Laurent se-ries algebras. Φ ∨ k and Φ k are certain linear monomial change of variables. There are two main ingredients in the proof. One is a construction of S. Moz-govoy, which relates the Hall algebra of a quiver to the quantum Laurent seriesring. The other is a ‘dimension reduction’ technique used by A. Morrison and K.Nagao. The equation (0.1) already appeared in [20, 14], but we put it in a rightassumption .Nagao also suggested in [20] that this theory can be used to study quantumcluster algebra. We follow his suggestion and use (0.2) to categorify quantumcluster algebras under the assumption of the existence of certain potentials. If acluster algebra has such a categorification, then its strong positivity will be impliedby a result of [6] on the purity of the vanishing cycles. In [20] the author only treats a rather special case when k is a strict sink/source . Moreover,the proof contains a gap due to the incorrectness of Lemma 4.3. In [14] the author considered avariation of V and assumed an unproven conjecture (Conjecture 3.2). OUNTING USING HALL ALGEBRAS III. QUIVERS WITH POTENTIALS 3
Let B be an n × m matrix with n ≤ m such that the left n × n submatrix of B is skew-symmetric. Let Λ be another skew-symmetric matrices of size m × m .We assume that Λ and B are unitally compatible , that is, B Λ = ( − I n , B a quiver Q without loops and 2-cycles such that b ij = | arrows j → i | − | arrows i → j | . Such a matrix is called the B -matrix of Q . We endow Q with some potential W having nice properties.Let µ k t be a sequence of mutations , and set ( Q t , W t ) = µ k t ( Q, W ). Let x g ( g ∈ Z m ≥ ) be some initial cluster monomial in the quantum Laurent polynomial ring X Λ (see (6.1)). We extend QP ( Q gt , W t ) from ( Q t , W t ) by adding a new vertex ∞ and g i new arrows from i to ∞ . We apply the inverse of µ k t to ( Q gt , W t ), and obtain aQP ( Q g , W g ) := µ − k t ( Q gt , W t ). Let e B be the B -matrix of Q g . Theorem 0.3.
The mutated cluster monomial X t ( g ) := µ k t ( x g ) is equal to X β (cid:12)(cid:12)(cid:12) ϕ ω g (Mod ν ∞ (1 ,β ) ( Q g )) (cid:12)(cid:12)(cid:12) vir x (1 ,β ) e B . This result may just be a special case of [9], where A. Efimov assumed a muchweaker condition on the potential W . However, our approach and result are moredown-to-earth and computable. It depends only on [3] rather than [16].The second application is on the generating series counting subrepresentationsof representations of quiver (with zero potential). Let s be a sink of Q , and M be arepresentation of Q . We assume that M does not contain the simple representation S s as a direct summand. LetT( M ) := X β q − h M − β,β i Q (cid:12)(cid:12) Gr β ( M ) (cid:12)(cid:12) x (1 ,β ) , where Gr β ( M ) is the variety parameterizing β -dimensional quotient representationsof M . Theorem 0.4. T( M ) and T( µ s M ) are also related via (0.2) . In particular, if M is polynomial-count , that is, all its Grassmannians Gr β ( M ) are polynomial-count,then so are all reflection equivalent classes of M . This note is organized as follows. In Section 1, we recall some basics aboutquiver representations and their Hall algebras. In Section 2, we recall the conceptof quiver with potential and a cut, and its associated algebras. In Section 3, werecall a construction of Mozgovoy and the dimension reduction technique (Lemma3.2). In Section 4, we recall the mutation of QP with a cut, and set up the keyassumption for our main results. When this assumption holds is illustrated inCorollary 4.4, whose proof will be given in the appendix. In Section 5, we prove ourfirst two main results – Theorem 5.2 and Theorem 5.3. In Section 6, we prove ourthird main result, Theorem 6.8, on a categorification of quantum cluster algebras.In Section 7, we prove our last main result, Theorem 7.4, on the representationGrassmannians under reflections.
Notations and Conventions. • All modules are right modules and all vectors are row vectors. • For an arrow a , ta is the tail of a and ha is the head of a . • For any representation M , we use M to denote its dimension vector. JIARUI FEI • S i is the simple module at the vertex i , and P i is its projective cover. • Superscript ∗ is the trivial dual Hom K ( − , K ).1. Basics on Quivers and their Hall algebras
From now on, we assume our base field K = F q to be the finite field with q elements. Let Q be a finite quiver with the set of vertices Q and the set of arrows Q . We write h− , −i Q for the usual Euler form of Q : h α, β i Q = X i ∈ Q α ( i ) β ( i ) − X a ∈ Q α ( ta ) β ( ha ) for α, β ∈ Z Q We also have the antisymmetric form ( − , − ) associated to Q . The matrix of ( − , − )denoted by B is given by(1.1) b ij = | arrows j → i | − | arrows i → j | . Let KQ be the path algebra of Q over K . For any three KQ -modules U, V and W with dimension vector β, γ and α = β + γ , the Hall number F WUV countsthe number of subrepresentations S of W such that S ∼ = V and W/S ∼ = U . Wedenote a W := | Aut Q ( W ) | , where | X | is the number of K -rational points of X . Let H ( Q ) be the vector space of all formal (infinite) linear combinations of isomorphismclasses of KQ -modules. Lemma 1.1 ([22],[23, Proposition 1.1]) . The completed
Hall algebra H ( Q ) is theassociative algebra with multiplication [ U ][ V ] := X [ W ] F WUV [ W ] , and unit [0] . Let mod KQ (resp. mod α KQ ) be the category of all finite dimensional (resp. α -dimensional) KQ -modules. For a subcategory C of mod KQ , we denote χ ( C ) := P M ∈C [ M ]. We use the shorthand χ and χ α for χ (mod KQ ) and χ (mod α KQ ).Let ( T , F ) be a torsion pair ([2, Definition VI.1.1]) in mod KQ , then for any M ∈ mod KQ , there exists a short exact sequence 0 → L → M → N → L uniquein T and N unique in F . In terms of the Hall algebra, this says that Lemma 1.2. χ = χ ( F ) χ ( T ) . A weight σ is an integral linear functional on Z Q . A slope function ν is a quotientof two weights σ/θ with θ ( α ) > α . Definition 1.3.
A representation M is called ν -semi-stable (resp. ν -stable) if ν ( L ) ν ( M ) (resp. ν ( L ) < ν ( M )) for every non-trivial subrepresentation L ⊂ M .We denote by Rep να ( Q ) the variety of α -dimensional ν -semistable representationsof Q . By the standard GIT construction [15], there is a categorical quotient q :Rep να ( Q ) → Mod να ( Q ) and its restriction to the stable representations Rep ν · st α ( Q )is a geometric quotient .A slope function ν is called coprime to α if ν ( γ ) = ν ( α ) for any γ < α . So if ν is coprime to α , then there is no strictly semistable (semistable but not stable)representation of dimension α . In this case, Mod να ( A ) must be a geometric quotient. OUNTING USING HALL ALGEBRAS III. QUIVERS WITH POTENTIALS 5
Lemma 1.4 (Harder-Narasimhan filtration) . Every representation M has a uniquefiltration M ⊂ M ⊂ · · · ⊂ M m − ⊂ M m = M such that N i = M i /M i − is ν -semi-stable and ν ( N i ) > ν ( N i +1 ) . We fix a slope function ν . For a dimension vector α , let χ να := P M ∈ Rep να ( Q ) [ M ].The existence of the Harder-Narasimhan filtration yields the following identity inthe Hall algebra H ( Q ). Lemma 1.5 ([21, Proposition 4.8]) . χ α = X χ να · · · · · χ να s , where the sum runs over all decomposition α + · · · + α s = α of α into non-zerodimension vectors such that ν ( α ) < · · · < ν ( α s ) . In particular, solving recursivelyfor χ να , we get (1.2) χ να = X ∗ ( − s − χ α · · · · χ α s , where the sum runs over all decomposition α + · · · + α s = α of α into non-zerodimension vectors such that ν ( P kl =1 α l ) < ν ( α ) for k < s . For α ∈ Z Q , we write x α for Q i ∈ Q x α ( i ) i . Let X ( Q ) be the quantum Laurentseries algebra Q ( q )[[ x i ]] i ∈ Q with multiplication given by x β x γ = q − ( β,γ ) x β + γ . Lemma 1.6 ([21, Lemma 6.1]) . The map R : [ M ] q h α,α i Q x α a M is an algebrahomomorphism H ( Q ) → X ( Q ) . Since a M = | GL α · M || GL α | , we have that(1.3) Z χ α = q h α,α i Q | Rep α ( Q ) || GL α | x α . QP with a Cut
We fix a quiver Q without loops or oriented 2-cycles and a potential W on Q .Let d KQ be the completion of the path algebra KQ with respect to the m -adictopology, where m is the two-sided ideal generated by arrows of Q . When dealingwith completed path algebras, all its ideals will be assumed to be closed. Recallthat a potential W is a linear combination of oriented cycles of Q .For each arrow a ∈ Q , the cyclic derivative ∂ a on d KQ is defined for each cycle a · · · a d as ∂ a ( a · · · a d ) = X a i = a a i +1 · · · a d a · · · a i − . For each potential W , its Jacobian ideal ∂W is the (closed two-sided) ideal in d KQ generated by all ∂ a W . Let J ( Q, W ) = d KQ/∂W be the
Jacobian algebra . If W ispolynomial and KQ/∂W is finite dimensional, then the completion is unnecessary.This is the assumption we make throughout the paper.Let ω : Rep( Q ) → K be the trace function corresponding to the potential W de-fined by M tr( W ( M )). Note that ω is an additive function, so it in fact descendsto the Grothendieck group: ω : K (mod( Q )) → K . By abuse of notation, we also JIARUI FEI use ω for the same trace function defined on the affine representation varieties andmoduli spaces. It is well-known that a representation M of Q is a representationof J ( Q, W ) if and only if it is in the critical locus of ω (i.e., dω ( M ) = 0).Following [12], we define Definition 2.1. A cut of a QP ( Q, W ) is a subset C ⊂ Q such that the potential W is homogeneous of degree 1 for the degree function deg : Q → N defined bydeg( a ) = 1 for a ∈ C and zero otherwise.This degree function defines a K ∗ -action on Rep α ( Q )(2.1) ( tM )( i ) = M ( i ) for i ∈ Q , ( tM )( a ) = t deg( a ) M ( a ) for a ∈ Q . The homogenicity of W implies that if M is a representation of the Jacobian algebra,then so is tM . Moreover, the trace function is equivariant: ω ( tM ) = tω ( M ). Definition 2.2.
The algebra J ( Q, W ; C ) associated to a QP ( Q, W ) with a cut C is the quotient algebra of the Jacobian algebra J ( Q, W ) by the ideal generated by C .For the degree function given by a cut, ∂W is a homogeneous ideal so the degreefunction induces a grading on J ( Q, W ) as well. Note that J ( Q, W ; C ) is isomorphicto degree zero part of J ( Q, W ). We denote by Q C the subquiver ( Q , Q \ C ) of Q and by h ∂ C W i the ideal h ∂ c W | c ∈ C i . It is clear that J ( Q, W ; C ) can alsobe presented as \ KQ C / h ∂ C W i . Readers may skip to Example 7.5 to see thesedefinitions in action.We put h α, β i C = X c ∈ C α ( tc ) β ( hc ) , h α, β i J C = h α, β i Q + h α, β i C + h β, α i C . Definition 2.3 ([12]) . A cut is called algebraic if(1) J ( Q, W ; C ) is a finite dimensional K -algebra of global dimension 2;(2) { ∂ c W } c ∈ C is a minimal set of generators of the ideal h ∂ C W i in d KQ .It is clear that for algebraic cuts, the form h− , −i J C is exactly the Euler form of J ( Q, W ; C ). From now on, we assume that all cuts are algebraic.Conversely, any finite-dimensional algebra of global dimension 2 arises as antruncated Jacobian algebra (see [12, Proposition 3.3]). Here is the construction.Given any K -algebra A presented by a quiver Q with a minimal set of relations { r , r , . . . , r l } , we can associate with it a QP ( Q A , W A ) with a cut C as fol-lows: Q A, = Q , Q A, = Q ∐ C with C = { c i : h ( r i ) → t ( r i ) } i =1 ...l , and W A = P li =1 c i r i . If A has global dimension 2, it is known [13, Theorem 6.10] that J ( Q A , W A ) is isomorphic to the algebra Π ( A ) := Q i ≥ Ext A ( A ∗ , A ) ⊗ A i . In par-ticular, J ( Q A , W A ) does not depend on the minimal set of relations that we chose.It is now clear that J ( Q A , W A ; C ) ∼ = A . Moreover, let C be an algebraic cut of( Q, W ) and A = J ( Q, W ; C ), then ( Q, W ) and ( Q A , W A ) are right-equivalent ([7,Definition 4.2]). We do not need this construction in this paper though. OUNTING USING HALL ALGEBRAS III. QUIVERS WITH POTENTIALS 7 A Construction of Mozgovoy
To motivate the definition (3.1), we consider a complex (quasiprojective) variety X with a C ∗ -action. Let ω be a regular function on X , which is equivariant withrespect to a primitive character, i.e., not divisible in the character group of C ∗ . Wefurther assume that lim t → tx exist for all x ∈ X , then according to [3, Proposition1.11], [ ω − (1)] ∈ K (Var C ) is the nearby fibre of ω . Its difference with the centralfibre ω − (0) defines a class called the (absolute) vanishing cycle of ω on X :[ ϕ ω ( X )] := [ ω − (0)] − [ ω − (1)] . Now let X be a variety over K = F q with a K ∗ -action, and ω still be a regularfunction on X . We set(3.1) | ϕ ω ( X ) | := | ω − (0) | − | ω − (1) | . Due to the torus action (2.1), we have that( q − | ω − (1) | = | X | − | ω − (0) | , so | ϕ ω ( X ) | = q | ω − (0) | − | X | q − . (3.2)We denote the q -shifted point count q − dim X | ϕ ω ( X ) | by | ϕ ω ( X ) | vir . We also set | GL α | vir := q − dim GL α | GL α | .Let ω : Rep α ( Q ) → K be the trace function corresponding to the potential W with a cut. Recall that the cut induces a torus action on each Rep α ( Q ) such that ω is equivariant. For h = P c M [ M ] ∈ H ( Q ), we define h := P ω ( M )=0 c M [ M ].Such an h is called equivariant if c M = c tM for any t ∈ K ∗ . Let H eq ( Q ) be thesubalgebra of H ( Q ) consisting of equivariant elements. Lemma 3.1 ([19, Proposition 5.2]) . The map R ω : H eq ( Q ) → X ( Q ) defined by h q R h − R hq − is an algebra morphism. Note that if W is zero, then R ω = R . We see from (3.2) and (1.3) that v α := Z ω χ α = | ϕ ω (Rep α ( Q )) | vir | GL α | vir x α . We denote the generating series R ω χ by V ( Q, W ) := P α v α x α . Lemma 3.2 ([20, Theorem 4.1]) . | ϕ ω (Rep α ( Q )) | = q h α,α i C | Rep α ( J ( Q, W ; C )) | .So v α = q h α,α i JC | Rep α ( J ( Q, W ; C )) || GL α | . For any stability ν , the trace function ω restricts to Rep να ( Q ) → K and descendsto the GIT moduli space Mod να ( Q ). Note that for K = C , [ ω − (0)] − [ ω − (1)] is thevanishing cycle of ω on Mod να ( Q ). Indeed, since the C ∗ -action is induced from acut, the primitive character condition is trivially satisfied. To apply [3, Proposition1.11], we can verify as in [20, Lemma 3.2] that lim t → tx exists in Mod να ( Q ) for any x ∈ Mod να ( Q ). JIARUI FEI
Apply the Hall character R ω to the identity (1.2), then we obtain [19, Theorem5.7]: Proposition 3.3. | ϕ ω (Rep να ( Q )) | vir | GL α | vir = X ∗ ( − s − q P i>j ( α i ,α j ) s Y k =1 v α k ( q ) , where the summation ∗ is the same as in Lemma 1.5. We denote by v να ( q ) the above rational function in q . Following [10], we say analgebra A is polynomial-count if each Rep α ( A ) is polynomial-count. Corollary 3.4.
Assume that the GIT quotient
Mod να ( Q ) is geometric. If J ( Q, W ; C ) is polynomial-count, then so is ϕ ω (Mod να ( Q )) . Definition 3.5.
A pair ( α, ν ) is called numb to a cut C on Q if the vector bundle π : Rep α ( Q ) → Rep α ( Q C ) restricts to ν -semistable representations.Later we will need the following generalization of Lemma 3.2. The proof is thesame as that in [20]. For readers’ convenience, we copy the proof here. Lemma 3.6. If ( α, ν ) is numb to C , then | ϕ ω (Rep να ( Q )) | = q h α,α i C | Rep να ( J ( Q, W ; C )) | .So v να = q h α,α i JC | Rep να ( J ( Q, W ; C )) || GL α | . Proof.
By assumption, π : Rep να ( Q ) → Rep να ( Q C ) is a vector bundle of rank d = h α, α i C . The restriction of ω to the fibre π − ( M ) is zero if M ∈ Rep να ( J ( Q, W ; C )),and is a non-zero linear function if x / ∈ Rep να ( J ( Q, W ; C )). Hence | ω − (0) | = q d | Rep να ( J ( Q, W ; C )) | + q d − ( | Rep να ( Q C ) | − | Rep να ( J ( Q, W ; C )) | ) . By (3.2), | ϕ ω (Rep να ( Q )) | = q | ω − (0) | − | Rep να ( Q ) | q − , = q h α,α i C | Rep να ( J ( Q, W ; C )) | . (cid:3) Let { e i } i be the standard basis of Z Q . The k -th (absolute) framing stability ν k is the slope function given by e ∗ k /d , where d ( α ) = P v ∈ Q α ( v ). It is not hard tosee that if all arrows in C end in k , then ( α, ν k ) with α k = 1 is numb to C .4. Mutation of Quivers with Potentials
The key notion in [7] is the definition of mutation µ k of a quiver with potentialsat some vertex k ∈ Q . Let us briefly recall it. The first step is to define thefollowing new quiver with potential e µ k ( Q, W ) = ( e Q, f W ). We put e Q = Q and e Q is the union of three different kinds of arrows • all arrows of Q not incident to k , • a composite arrow [ ab ] from ta to hb for each a and b with ha = tb = k ,and • an opposite arrow a ∗ (resp. b ∗ ) for each incoming arrow a (resp. outgoingarrow b ) at k . OUNTING USING HALL ALGEBRAS III. QUIVERS WITH POTENTIALS 9
The new potential is given by f W := [ W ] + X ha = tb = k b ∗ a ∗ [ ab ] , where [ W ] is obtained by substituting [ ab ] for each words ab as above occurring in(any cyclically equivalent) W .Let A be the algebra J ( Q, W ; C ) and T be the representation A/P k ⊕ τ − S k ,where τ is the classical AR-transformation [2]. Clearly, τ − S k can be presented as P k ( a ) a −−−→ M ha = k P ta → τ − S k →
0, where ( a ) a is the row vector with entries arrowspointing to k . Recall that an A -module T is called tilting if T has finite projectivedimension, Ext iA ( T, T ) = 0 for all i >
0, and there is an exact sequence0 → A → T → T → · · · → T n → , where each T i is a finite direct sum of direct summands of T . Lemma 4.1 ([17, Corollary 2.2.b]) . T is a tilting module if and only if the map P k ( a ) a −−−→ M ha = k P ta is injective. In this case, T is called the BB-tilting module at k . The dual notion of T is theBB-cotilting module T ∨ = A ∗ /I k ⊕ τ S k . What we desire is the following nice situ-ation: , There is an algebraic cut e C on ( e Q, f W ) such that J ( Q, W ; C ) and J ( e Q, f W ; e C )are tilting equivalent via the functor Hom A ( T, − ) or Hom A ( − , T ∨ ).In general, the existence of another (not necessarily algebraic) cut on f W is notguaranteed. However, if we assume that(4.1) all arrows ending at k do not belong to a cut C ,then we can assign a new cut e C containing all • c ∈ C if tc = k , • arrows b ∗ if b / ∈ C , • composite arrows [ ab ] with b ∈ C .This definition is the graded right mutation defined in [1] adapted to our setting.There is a graded version of splitting theorem ([7, Theorem 4.6]). Recall that a(graded) QP ( Q, W ) is trivial if the potential W is in the space KQ spannedby paths of length 2, and if the Jacobian algebra J ( Q, W ) is isomorphic to thesemisimple algebra KQ . A (graded) QP ( Q, W ) is reduced if W ∩ KQ is zero.Applied to QP with a cut, we have Lemma 4.2. ( Q, W, C ) is graded right-equivalent to the direct sum ( Q red , W red , C red ) ⊕ ( Q triv , W triv , C triv ) , where ( Q red , W red , C red ) is reduced and ( Q triv , W triv , C triv ) is trivial , both unique up to graded right-equivalence. We denote the reduced part of ( e Q, f W , e C ) by µ k ( Q, W, C ) := ( Q ′ , W ′ , C ′ ). Theorem 4.3.
Assume that C is a cut satisfying Definition 2.3.(2) and (4.1) ,and that Ext A ( S i , S k ) = 0 for any i = k . Then J ( Q, W ; C ) is tilting equivalent to J ( e Q, f W ; e C ) via Hom A ( T, − ) . Corollary 4.4. If C is an algebraic cut satisfying (4.1) , then C ′ is also algebraic,and J ( Q, W ; C ) is tilting equivalent to J ( e Q, f W ; e C ) via Hom A ( T, − ) . These slightly generalize the main results of [18]. We will prove them in theappendix. By Lemma 4.2, the above J ( e Q, f W ; e C ) can be replaced by J ( Q ′ , W ′ ; C ′ ).If we want to work with the assumption dual to (4.1), that is, all arrows startingwith k do not belong to C , then we should take the functor Hom A ( − , T ∨ ).The equivalence Hom A ( T, − ) induces a map φ k in the corresponding K -group φ k ([ S i ]) = ( [ S ′ i ] i = k, − [ S ′ k ] + P ha = k [ S ′ ta ] i = k ;(4.2)and its dual Hom A ( − , T ∨ ) induces φ ∨ k given by φ ∨ k ([ S i ]) = ( [ S ′ i ] i = k, − [ S ′ k ] + P tb = k [ S ′ hb ] i = k. Since the K -groups of mod( J ( Q, W ; C )) and mod( J ( Q ′ , W ′ ; C ′ )) can be iden-tified with Z Q , by slight abuse of notation, we also write φ k and φ ∨ k for thecorresponding linear isometries on Z Q . Due to the equivalence, we have that h α, β i J C = h φ k α, φ k β i J ′ C . Moreover, it is easy to verify that ( α, β ) = ( φ k α, φ k β ) ′ ,or equivalently, B ′ = φ k Bφ T k , where ( − , − ) ′ is the antisymmetric form of Q ′ .We denote mod( A ) k := { M ∈ mod A | Hom A ( S k , M ) = 0 } , mod( A ) k := { M ∈ mod A | Hom A ( M, S k ) = 0 } . Note that under the assumption , , mod( J ( Q, W ; C )) k (cid:0) resp. mod( J ( Q ′ , W ′ ; C ′ )) k (cid:1) is the torsion (resp. torsion-free) class determined by the tilting module T [2, VI.2].So mod( J ( Q, W ; C )) k ∼ = mod( J ( Q ′ , W ′ ; C ′ )) k . In particular, for α ′ = φ k ( α ) we have that(4.3) | Rep α ( J ( Q, W ; C )) k || GL α | = | Rep α ′ ( J ( Q ′ , W ′ ; C ′ )) k || GL α ′ | . Wall-crossing Formula
Let h S k i be the subcategory of mod KQ generated by the simple S k . Definition 5.1.
We denote E k := R ω χ ( h S k i ) = P n q n / | GL n | x nk = exp q (cid:16) q / q − x k (cid:17) .Recall the generating series V ( Q, W ) defined before Lemma 3.2 and the quantumLaurent series algebra X ( Q ) defined before Lemma 1.6. We set V := V ( Q, W ) ∈ X ( Q ) and V ′ := V ( µ k ( Q, W )) ∈ X ( Q ′ ) . Let Φ k (resp. Φ ∨ k ) be the ring homomor-phism X ( Q ) → X ( Q ′ ) defined by x α ( x ′ ) φ k α (cid:0) resp. ( x ′ ) φ ∨ k α (cid:1) . Theorem 5.2.
Assuming the condition , , we have that E ′ k Φ ∨ k ( VE − k ) = V ′ = Φ k ( E − k V ) E ′ k . OUNTING USING HALL ALGEBRAS III. QUIVERS WITH POTENTIALS 11
Proof.
We apply the character R ω to the torsion-pair identity in H ( Q ) (Lemma 1.2) χ (mod( Q ) k ) χ ( h S k i ) = χ = χ ( h S k i ) χ (mod( Q ) k ) . By Lemma 3.1 we get E − k V = Z ω χ (mod( Q ) k )= X α q h α,α i JC | Rep α ( J ( Q, W ; C )) k || GL α | x α , (similar to Lemma 3.6)= X α q h φ k ( α ) ,φ k ( α ) i J ′ C | Rep φ k ( α ) ( J ( Q ′ , W ′ ; C ′ )) k || GL φ k ( α ) | x α , (4 . V ′ E ′− k = X α q h α,α i J ′ C | Rep α ( J ( Q ′ , W ′ ; C ′ )) k || GL α | ( x ′ ) α , Hence V ′ = Φ k ( E − k V ) E ′ k . The other half is similar. (cid:3)
Framing.
We freeze a vertex ∞ of Q , that is, we do not allow to mutate at ∞ .Let mod ( Q ) be all modules supported outside ∞ . Note that mod ( Q ) is an exactsubcategory of mod( Q ). In particular, it is a torsion-free class, and let T ( Q ) beits corresponding torsion class.Let T := R ω χ (T ( Q )) and V := R ω χ (mod ( Q )). It follows from the torsion pairidentity that T = V − V . We keep the assumption , . According to Theorem 5.2, we have that E ′ k Φ ∨ k ( VE − k ) = V ′ = Φ k ( E − k V ) E ′ k . From the second equality, we get that V ′ − V ′ = E ′ k − Φ k ( V − E k )Φ k ( E − k V ) E ′ k , = E ′ k − Φ k ( V − V ) E ′ k , hence we get a formula for T ′ := R ω ′ χ (T ( Q ′ )) T ′ = E ′ k − Φ k ( T ) E ′ k . Similarly using the first equality, we obtain that T ′ = Φ ∨ k (cid:0) E k TE − k (cid:1) . We can also treat mod ( Q ) as a torsion class, and work with its torsion-free classF . If we set F = R ω F , then F = VV − , and we have the dual formula E ′ k Φ ∨ k ( F ) E ′ k − = F ′ = Φ k ( E − k FE k ) . Consider the subcategory T ( Q ) of T ( Q ), which contains all representationshaving dimension one at the vertex ∞ . It is well-known that the category T ( Q ) contains exactly the ν ∞ -stable representations with dimension 1 at the vertex ∞ .Let(5.1) T( Q, W ) := ( q − q − ) Z ω χ (T ( Q )) . Since the dimension vector (1 , β ) is coprime to the slope function ν ∞ , the modulispace Mod ν ∞ (1 ,β ) ( Q ) is a geometric quotient, and thus we haveT( Q, W ) = ( q − q − ) | ϕ ω (Rep ν ∞ (1 ,β ) ( Q )) | vir | GL (1 ,β ) | vir x (1 ,β ) . = X β (cid:12)(cid:12)(cid:12) ϕ ω (Mod ν ∞ (1 ,β ) ( Q )) (cid:12)(cid:12)(cid:12) vir x (1 ,β ) . (5.2)Now we replace all T ( Q ) by T ( Q ) in the above argument for T ′ , and we caneasily see that Theorem 5.3.
Assuming the condition , , we have that (5.3) E ′ k − Φ k (T) E ′ k = T ′ = Φ ∨ k ( E k T E − k ) . In the next section, by abuse of notation, we will write µ k for the operatorAd − ( E ′ k ) ◦ Φ k , and µ ∨ k for the operator Φ ∨ k ◦ Ad( E k ).6. Application to Cluster Algebras
Let B be an n × m matrix with n ≤ m . The principal part B p of B is bydefinition the left n × n submatrix. We assume that B p is skew-symmetric. Let Λbe another skew-symmetric matrices of size m × m . We assume that Λ and B are unitally compatible , that is, B Λ = ( − I n , B an (ice) quiver Q without loops and 2-cycles satisfying(1.1). The vertices in [ n + 1 , m ] are frozen vertices . We denote by Q p the principalpart of Q , that is, the full subquiver of Q by forgetting all frozen vertices. Thematrix B is called the B -matrix of Q .Let X Λ be the quantum Laurent polynomial ring Z [ q ± ][ x ± , x ± , · · · , x ± m ] withmultiplication given by(6.1) x α x β = q Λ( α,β ) x α + β . Here, we write Λ( − , − ) for the associated bilinear form of Λ. As an Ore domain [5,Appendix], X Λ is contained in its skew-field of fractions F ( X Λ ). Definition 6.1.
A toric frame is a map X : Z m → F ( X Λ ), such that X ( α ) = ρ ( x η ( α ) ) for some automorphism ρ of the skew-field F ( X Λ ), and some automorphism η of the lattice Z m .By abuse of notation we can view X Λ naturally as the toric frame: X Λ ( α ) = x α .Let { e i } ≤ i ≤ m be the standard basis of Z m . We also denote by φ k the matrix ofthe linear isometry (4.2), and by φ p k its restriction on the principal part Q p . Forany integer b , we write [ b ] + for max(0 , b ). Definition 6.2. A seed is a triple (Λ , B, X ) such that X ( g ) X ( h ) = q Λ( α,β ) X ( g + h )for all g, h ∈ Z m . The mutation of (Λ , B, X ) at k is a new triple (Λ ′ , B ′ , X ′ ) = µ k (Λ , B, X ) defined by(Λ ′ , B ′ ) = ( φ T k Λ φ k , φ p k Bφ T k ) , (6.2) OUNTING USING HALL ALGEBRAS III. QUIVERS WITH POTENTIALS 13 and X ′ is detetmined by the following exchange relation X ′ ( e k ) = X (cid:0) X ≤ j ≤ m [ − b kj ] + e j − e k (cid:1) + X (cid:0) X ≤ j ≤ m [ b kj ] + e j − e k (cid:1) , (6.3) X ′ ( e j ) = X ( e j ) 1 ≤ j ≤ m, j = k. (6.4)Since φ k = φ − k , we see that (Λ ′ , B ′ ) is also unitally compatible. The automor-phism ρ for X ′ was constructed explicitly in [5, Proposition 4.2]. One should noticethat the mutation µ k is an involution.Let T n be the n -regular tree with root t . There is a unique way of associatinga seed (Λ t , B t , X t ) for each vertex t ∈ T n such that(1) (Λ t , B t , X t ) = (Λ , B, X Λ ),(2) if t and t ′ are linked by an edge k , then the seed (Λ t ′ , B t ′ , X t ′ ) is obtainedfrom (Λ t , B t , X t ) by the mutation at k . Definition 6.3.
The quantum cluster algebra C (Λ , B ) with initial seeds (Λ , B, X Λ )is the Z [ q ± ]-subalgebra of F ( X Λ ) generated by all cluster variables X t ( e i ) (1 ≤ i ≤ n ), coefficients X t ( e i ) and X t ( e i ) − ( n + 1 ≤ i ≤ m ).Recall the operators µ k = Ad − ( E ′ k ) ◦ Φ k and µ ∨ k = Φ ∨ k ◦ Ad( E k ). Moreover E ( y ) = exp q (cid:16) q / q − y (cid:17) can also be written as the formal product ∞ Y l =0 (1 + q l + y ) − . It satisfies E ( q ± y ) = (1 + q ± y ) ± E ( y ) . Let Y ( Q p ) be the quantum Laurent polynomial (rather than Laurent series) al-gebra in variables { y i } i ∈ Q having the same multiplication rule as X ( Q p ) , that is, y β y γ = q − ( β,γ ) y β + γ . From the fact that y i E ( y k ) = E ( q b ki y k ) y i and E ( y k ) − E ( q b y k ) = b Y l =1 (1 + q l − y k ) , we can easily deduce the following Y -seeds mutation formula Lemma 6.4 ([14, (4.11)]) . µ ∨ k ( y i ) = µ k ( y i ) = y − k ( i = k ) ,y e i +[ − b ik ] + e k | b ik | Y l =1 (1 + q sgn( b ik )( l − ) y k ) sgn( b ik ) otherwise . Let F ( Y ( Q p ) ) be its skew-field of fraction of Y ( Q p ) . From the above formula, wesee that applying a sequence of mutation operators to a Laurent polynomial, weget an element in F ( Y ( Q p ) ) rather than an arbitrary series. We consider the latticemap Z n → Z m , β βB . This map induces an operatorb : Y ( Q p ) → X Λ , y β x βB . By the unital compatibility of Λ and B , we have that αBβ T = Λ( αB, βB ). So weconclude that The inverse of this product is the q -Pochhammer symbol ( − q y ; q ) ∞ . Lemma 6.5.
The operator b is an algebra homomorphism, and thus induces askew-field homomorphism b : F ( Y ( Q p ) ) → F ( X Λ ) . Let k s := ( k , k , . . . , k s ) be a sequence of edges connecting t and t s . We write µ k s for the sequence of mutation µ k s · · · µ k µ k . For simplicity, we write B r for B t r and X r for X t r . The next lemma says that the operator b is compatible withmutations. Lemma 6.6. b ◦ µ − k s ( y β ) = X s ( βB s ) for any β ∈ Z n .Proof. Using the unital compatibility of Λ and B , this is clearly reduced to provefor β = e i . We prove by induction on s . For s = 0, it is trivial. Suppose that it istrue for s , thenb ◦ µ − k s +1 ( y e i ) = b ◦ µ − k s ( µ − k s +1 y e i ) , Lemma 6.4 ======== b ◦ µ − k s (cid:16) y e i +[ b s +1 ki ] + e k | b s +1 ik | Y l =1 (cid:0) q sgn( b s +1 ik )( l − ) y e k (cid:1) sgn( b s +1 ik ) (cid:17) , Lemma 6.6 ======== X s ( e i B s + [ b sik ] + e k B s ) | b ski | Y l =1 (cid:0) q sgn( b ski )( l − ) X s ( e k B s ) (cid:1) sgn( b ski ) . On the other hand, X s +1 ( e i B s +1 ) = X s +1 (cid:0) X j b s +1 ij e j (cid:1) , = q Λ s +1 ( e i B s +1 − b sik e k ,b sik e k ) X s +1 (cid:0) X j = k b s +1 ij e j (cid:1) X s +1 ( b s +1 ik e k ) , = X s (cid:0) X j = k b s +1 ij e j (cid:1) · (cid:16) X s (cid:0) X ≤ j ≤ m [ b s − kj ] + e j − e k (cid:1) + X s (cid:0) X ≤ j ≤ m [ b skj ] + e j − e k (cid:1)(cid:17) b ski , = X s (cid:0) X j = k b s +1 ij e j (cid:1) · (cid:16) X s (cid:0) X ≤ j ≤ m [ b s − kj ] + e j − e k (cid:1)(cid:0) q X s ( e k B s ) (cid:1)(cid:17) b ski , = X s (cid:0) X j = k ( b sij + [ b sik ] + b skj − b ski [ − b skj ] + ) e j (cid:1) · X s (cid:0) b ski (cid:0) X ≤ j ≤ m [ − b skj ] + e j − e k (cid:1)(cid:1) · | b ski | Y l =1 (cid:0) q sgn( b ski )( l − ) X s ( e k B s ) (cid:1) sgn( b ski ) , = X s ( e i B s + [ b sik ] + e k B s ) | b ski | Y l =1 (cid:0) q sgn( b ski )( l − ) X s ( e k B s ) (cid:1) sgn( b ski ) . (cid:3) For any t ∈ T n , there is a unique sequence of edges k t connecting t and t . Let W be some non-degenerate ([7, Definition 7.2, Proposition 7.3]) potential of Q , andset ( Q t , W t ) = µ k t ( Q, W ). We shall assume the following condition for W :(6.5) For any t ∈ T n , and any k ∈ Q t , the assumption , holds for ( Q t , W t ).We do not know if such a potential exists for any quiver without loops or 2-cycles. OUNTING USING HALL ALGEBRAS III. QUIVERS WITH POTENTIALS 15
To give another definition of X t ( g ) for g ∈ Z m ≥ , we consider the extended QP( Q gt , W t ) from ( Q t , W t ) by adding a new vertex ∞ and g i new arrows from i to ∞ .We apply the inverse of µ k t to ( Q gt , W t ), and obtain a QP ( Q g , W g ) := µ − k t ( Q gt , W t ).Let ω g be the trace function corresponding to the potential W g .We freeze the same set of vertices of Q g as that of Q . Although the extendedvertex ∞ is not frozen, we will never perform mutation at ∞ henceforth. Let e B bethe B -matrix of Q g . Note that Q is the full subquiver of Q g without the vertex ∞ so e B is obtained from B by adjointing a row corresponding to the vertex ∞ . Definition 6.7.
For any g ∈ Z m ≥ , we define X t ( g ) = b (cid:0) T( µ − k t ( Q gt , W t )) (cid:1) , whereT( Q, W ) is defined in (5.1).By Theorem 5.3, we have that(6.6) T( µ − k t ( Q gt , W t )) = µ − k t (cid:16) ( q / − q − / ) y ∞ q / − q − / (cid:17) = µ − k t ( y ∞ ) . Theorem 6.8.
Under the assumption (6.5) , definition 6.7 defines the quantumcluster algebra C (Λ , B ) . In particular, we have the following quantum cluster char-acter X t ( g ) = X β (cid:12)(cid:12)(cid:12) ϕ ω g (Mod ν ∞ (1 ,β ) ( Q g )) (cid:12)(cid:12)(cid:12) vir x (1 ,β ) e B . Proof.
We trivially extend Λ to e Λ := ( ), and thus we have the natural embedding X Λ ֒ → X e Λ . Note that e B e Λ = (cid:0) − I (cid:1) , so they are not unitally compatible. Wesay that they are unitally compatible on the principal part. However, it still makesperfect sense if we define ( e B ′ , e Λ ′ ) by (6.2) and e X ′ ( e i ) by the relation (6.3)–(6.4).Clearly, for any k = ∞ , ( e B ′ , e Λ ′ ) is also unitally compatible on the principal part.This is all we need for the following analogue (6.7) of Lemma 6.6 to hold: For each t ∈ T n , we associate as before e X t , then(6.7) b ◦ µ − k t ( y β ) = e X t ( β e B t ) . Moreover, e Λ ′ extends Λ ′ in the same way: e Λ ′ := (cid:0) ′ (cid:1) so that we have the naturalembedding X t ( Z m ) ֒ → e X t ( Z m +1 ) for each t ∈ T n . Hence,b ◦ µ − k t ( y ∞ ) = e X t ( e ∞ e B t ) = X t ( g ) . The last statement on the explicit formula of X t ( g ) follows from (6.6), (5.2), andLemma 6.5. (cid:3) Remark . We can view (1 , β ) e B as βB − g t , where g t is the extended g -vector corresponding to the mutated cluster monomial. Example 6.10.
Consider the quiver 2 b (cid:26) (cid:26) ✹✹✹✹✹✹✹✹✹✹✹✹✹ (cid:15) (cid:15) ∞ a D D ✡✡✡✡✡✡✡✡✡✡✡✡✡ ; ; ✇✇✇✇✇✇✇✇ c o o c c ●●●●●●●● with potential abc . We perform a sequence of mutations { , , , } , and obtain thequiver 2 b (cid:26) (cid:26) ✹✹✹✹✹✹✹✹✹✹✹✹✹ ∞ O O { { ✇✇✇✇✇✇✇✇ ●●●●●●●● a D D ✡✡✡✡✡✡✡✡✡✡✡✡✡ c o o with the same potential. We choose c as the cut. It is easy to count the vanishingcycles for each dimension vector. For example, for β = (1 , , | ϕ ω g (Mod ν ∞ (1 ,β ) ( Q g )) | = q (2 q + 2) . Note that 2 q + 2 counts neither the representation Grassmannian of P ⊕ P ⊕ P of the Jacobian algebra2 (cid:30) (cid:30) ❃❃❃❃❃❃❃❃ @ @ (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) o o nor that of the algebra 2 (cid:30) (cid:30) ❃❃❃❃❃❃❃❃ @ @ (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) . Here, the dotted line between two arrows means a relation given by the vanishing ofcomposition. In particular, the condition ‘numb’ cannot be removed from Lemma3.6.7.
Application: Representation Grassmannians and Reflections
Let s be a sink of Q , and M be a representation of Q . We assume that M doesnot contain the simple representation S s as a direct summand. LetT( M ) := X q − h M − β,β i Q | Gr β ( M ) | x (1 ,β ) . We want to compare T( M ) with T( µ s ( M )).We say an algebra A is extended from Q by M if A = KQ [ M ] := (cid:0) KQ M K (cid:1) . Thisis an algebra of global dimension two, so we can complete it to a QP ( Q A , W A )with a cut C such that J ( Q A , W A ; C ) = A (see Section 2). We freeze the extendedvertex ∞ of A , then Lemma 7.1. T( Q A , W A ) = T( M ) .Proof. Since all arrows in C end in ∞ , by Lemma 3.6,T( Q A , W A ) = Z ω χ (T ( Q A )) = ( q − q − ) X α q h (1 ,β ) , (1 ,β ) i Q [ M ] | Rep ν ∞ (1 ,β ) ( Q [ M ]) || GL (1 ,β ) | x (1 ,β ) , where h− , −i Q [ M ] is the Euler form of KQ [ M ], and ν ∞ is the framing stability.Rep ν ∞ (1 ,β ) ( Q [ M ]) can be identified with { ( N, f ) ∈ Rep β ( Q ) × Hom(
M, K β ) | f ∈ Hom Q ( M, N ) is surjective } . OUNTING USING HALL ALGEBRAS III. QUIVERS WITH POTENTIALS 17
So the quotient Rep ν ∞ (1 ,β ) ( Q [ M ]) / GL β is the representation Grassmannian Gr β ( M ),and thusT( Q A , W A ) = ( q − q − ) X β q q − h M − β,β i Q | Rep ν ∞ α ( Q [ M ]) || GL β || GL | x (1 ,β ) , = X β q − h M − β,β i Q | Gr β ( M ) | x (1 ,β ) . (cid:3) Remark . The analogous statement does not hold for quivers with potentials ingeneral (see Example 6.10).For any s ∈ Q (not necessarily a sink), the cut C of Q A satisfies the conditionsin Corollary 4.4, so we get another algebra A ′ = End A ( T ) = J ( Q ′ A , W ′ A ; C ′ ). Lemma 7.3.
The algebra A ′ is extended from Q ′ := µ s ( Q ) by M ′ := µ s ( M ) .Proof. Let ∞ be the extended vertex of Q [ M ], and P ˆ ∞ := A/P ∞ . ThenEnd A ( P ˆ ∞ ) = KQ and Hom A ( P ˆ ∞ , P ∞ ) = M. Since there are no incoming arrows to ∞ for Q [ M ], we have that P ˆ ∞ ∼ = KQ . Let T = A/P s ⊕ T s be the BB-tilting module of A at s , and T ˆ ∞ := T /P ∞ . We need toshow that End A ( T ˆ ∞ ) = Kµ s ( Q ) and Hom A ( T ˆ ∞ , P ∞ ) = µ s ( M ) . Let T ′ = KQ/P s ⊕ T ′ s be the BB-tilting module of KQ at s . Since M does notcontain S s as a direct summand the quiver of Q [ M ] has no arrow from ∞ to s . So T s = T ′ s , and thus T ˆ ∞ = T ′ . Hence, End A ( T ˆ ∞ ) = Kµ s ( Q ). For the second one, weconsider µ s ( M ) = Hom Q ( T ′ , Hom A ( P ˆ ∞ , P ∞ )) , = Hom A ( T ′ ⊗ KQ P ˆ ∞ , P ∞ ) , = Hom A ( T ˆ ∞ , P ∞ ) . (cid:3) It follows from the previous two lemmas that
Theorem 7.4. T( M ) and T( M ′ ) are related via (5.3) . In particular, if M is polynomial-count , that is, all its Grassmannians Gr β ( M ) are polynomial-count,then so are all reflection equivalent classes of M . Example 7.5.
Consider the following quiver2 b ,b ,b (cid:30) (cid:30) ❃❃❃❃❃❃❃❃ a ,a ,a @ @ (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) c ,c ,c o o with potential W = P I :=( i,j,k ) ∈ S ( − sgn I a i b j c k and an algebraic cut C = { c , c , c } .Then the algebra J ( Q, W ; C ) is Beilinson’s quiver algebra for P . This algebra isextended from the quiver 2 b ,b ,b −−−−−→ M of dimension (3 , (see [10, Example 7.8]). To compute T( M ), we can first compute T( µ ( M )), where µ ( M ) can be presented by the following base diagram • b (cid:127) (cid:127) ⑦⑦⑦⑦ b (cid:31) (cid:31) ❅❅❅❅ ◦ ◦• b / / b ? ? ⑦⑦⑦⑦ ◦ • b o o b _ _ ❅❅❅❅ The black (resp. white) dots are a basis at vertex 3 (resp. vertex 2); The letter onan arrow represents the identity map on the arrow of the same letter.T( µ ( M )) = 1+ x (1 , , + x (1 , , +[3]( x (1 , , + x (1 , , + x (1 , , + x (1 , , )+3 q x (1 , , . Here [ n ] is the quantum number q − n (cid:0) q n − q − (cid:1) . Using Theorem 7.4, we find thatT( M ) = 1 + x (1 , , + x (1 , , + [3]( x (1 , , + x (1 , , + x (1 , , ) + 3 q x (1 , , + [3][3]( x (1 , , + x (1 , , ) + ( q + q − )[3]( x (1 , , + x (1 , , )+ ( q − q − )[3][5]( x (1 , , + x (1 , , ) + ( q − q − )[4][5] x (1 , , . Employing the methods developed in [10], we can compute | ϕ ω (Mod να ( Q )) | and | Mod να ( J ( Q, W ; C )) | for all α with α = 1 and generic ν .8. Appendix: Proof of Theorem 4.3
Theorem 4.3 generalizes the main result of [18] from APR-tilting modules toBB-tilting modules (see after Lemma 4.1). We slightly simplify its proof as well.We follow the matrix notation r ( − ) c in [18], that is, we write the row index r andcolumn index c as left and right subscripts respectively. Lemma 8.1 ([4, Proposition 3.3]) . Let Q be a finite quiver and A be a finitedimensional basic algebra. Let R be a set of relations in Q , and we assume thatany r ∈ R is a formal linear sum of paths in Q with a common start tr and acommon end hr . Then A can be presented as d KQ/ h R i if and only if there is analgebra homomorphism π : d KQ → A such that the sequence M tr = i π ( e hr ) A r (cid:0) π ( a − r ) (cid:1) a −−−−−−−−→ M ta = i π ( e ha ) A a (cid:0) π ( a ) (cid:1) −−−−−→ rad( π ( e i ) A ) → is exact for any i ∈ Q . Here, a − is the formal inverse of a defined by a − ( a a · · · a m ) = ( a · · · a m if a = a, otherwise . We set P in = M ha = k P ta and P out = M tb = k,b/ ∈ C P hb . Recall from Lemma 4.1 that thesummand T k := τ − S k in the BB-tilting module can be presented as(8.1) 0 → P k α −→ P in g −→ T k → , where α := ( a ) a and g := a ( g a ). OUNTING USING HALL ALGEBRAS III. QUIVERS WITH POTENTIALS 19
Using the presentation \ KQ C / h ∂ C W i of J ( Q, W ; C ). We have for i = k (8.2) · · · → P ′ → M hc = i,c ∈ C P tc ∂ cb −−→ M tb = i P hb β −→ P i → S i → , where β := b ( b ) and ∂ cb := c ( ∂ c ∂ b W ) b . Since the cut C satisfies Definition 2.3.(2),the first three terms are part of the minimal projective resolution of S i . We assumethat the projective P ′ is minimal as well.This fits in the following commutative diagram0 / / c ki P k ⊕ α / / (cid:127) _ ι (cid:15) (cid:15) c ki P in ⊕ g / / ∂ acb (cid:15) (cid:15) c ki T k / / f := c ( f c ) (cid:15) (cid:15) P ′ h / / M hc = i,c ∈ C P tc ∂ cb / / M tb = i P hb β / / P i / / S i Here c ki = | C ∩ Q ( k, i ) | , and the first row is a direct sum of c ki copies of (8.1); Themap ι is the natural embedding, the map ∂ acb is given by the matrix a,c ( ∂ a ∂ c ∂ b W ) b ,and f is induced from ∂ acb . We then take the mapping cone of the above diagram,and cancel out the last term c ki P k . We end up with(8.3) P ′ h −→ c ki P in ⊕ M hc = i,tc = k P tc g ′ := (cid:16) ⊕ g ∂ acb ∂ cb (cid:17) −−−−−−−−−−→ c ki T k ⊕ M tb = i P hb f ′ := (cid:16) − fβ (cid:17) −−−−−−−→ P i → S i Recall our setting in Section 4. Let e Q e C be the quiver obtained from e Q by forget-ting all arrows in e C . To apply Lemma 8.1, we construct an algebra homomorphism π : \ K e Q e C → End A ( T ) as follows. For any direct summands T i , T j of T , we willview Hom A ( T i , T j ) under the natural embedding into Hom A ( T, T ). Let id i be theidentity map in Hom A ( T i , T i ). We define(1) π ( e i ) = id i ,(2) π ( a ) = a ∈ Hom A ( P i , P j ) for i, j = k ,(3) π ( a ∗ ) = g a ∈ Hom A ( P ta , T k ), π ( b ∗ ) = − f c ∈ Hom A ( T k , P hc ) for b ∈ C , π ([ ab ]) = ba ∈ Hom A ( P ha , P tb ) for b / ∈ C .Recall that f W := [ W ] + X ha = tb = k b ∗ a ∗ [ ab ] , and e C contains all • c ∈ C if tc = k , • arrows b ∗ if b / ∈ C , • composite arrows [ ab ] with b ∈ C .So the corresponding relations ∂ e C f W are given by • R = { ∂ c [ W ] } tc = k , • R = { a ∗ [ ab ] } b/ ∈ C , • R = { ∂ ab W + b ∗ a ∗ } b ∈ C .We will see that Theorem 4.3 is an immediate consequence of the following twolemmas. Lemma 8.2.
We have the following exact sequence
Hom A ( T, P out ) ◦ r −−→ Hom A ( T, P in ) ◦ g −→ rad(Hom A ( T, T k )) → , where r is the matrix b { ba } a .Proof. We apply Hom A ( T, − ) to the exact sequence (8.1), and get0 → Hom A ( T, P k ) ◦ α −−→ Hom A ( T, P in ) ◦ g −→ Hom A ( T, T k ) → Ext A ( T, P k ) → Ext A ( T, P in ) . The last term Ext A ( T, P in ) vanishes because the first map below is surjectiveHom A ( P in , P in ) −→ Hom A ( P k , P in ) −→ Ext A ( T k , P in ) → . Next, Ext A ( T, P k ) is one-dimensional because of the following exact sequenceHom A ( P in , P k ) −→ Hom A ( P k , P k ) → Ext A ( T k , P k ) → . Finally, we claim the image of ◦ r is exactly the image of ◦ α . By the definition of r , it suffices to show that Hom A ( T, P out ) ◦ β −→ Hom A ( T, P k ) is surjective. But thecokernel of ◦ β is Hom A ( T, S k ) = 0. (cid:3) Applying Hom A ( T, − ) to (8.3), we get the complexHom A ( T, c ki P in ⊕ M hc = i,tc = k P tc ) ◦ g ′ −−→ Hom A ( T, c ki T k ⊕ M tb = i P hb ) ◦ f ′ −−→ Hom A ( T, P i ) → Hom A ( T, S i ) Lemma 8.3. If Ext A ( S i , S k ) = 0 for any i = k , this complex is exact and induces Hom A ( T, c ki P in ⊕ M hc = i,tc = k P tc ) −→ Hom A ( T, c ki T k ⊕ M tb = i P hb ) −→ rad(Hom A ( T, P i )) . Proof.
We first show that the complex is exact at Hom A ( T, c ki T k ⊕ L tb = i P hb ). Weapply Hom A ( T, − ) to the exact sequence0 → Im h → c ki P in ⊕ M hc = i P tc → Im g ′ → , and getHom A ( T, c ki P in ⊕ M hc = i,tc = k P tc ) → Hom A ( T, Im g ′ ) → Ext A ( T, Im h ) . If ϕ ∈ Hom A ( T, c ki T k ⊕ L tb = i P hb ) such that ϕf ′ = 0, then ϕ ( T ) ⊆ Im g ′ . So itsuffices to show that Ext A ( T, Im h ) = 0. The condition Ext A ( S i , S k ) = 0 impliesthat P ′ has no P k as its summands. So Ext A ( T, P ′ ) = 0, and thus0 → Ext A ( T, Im h ) → Ext A ( T, Ker h ) = 0 . Since c ki P in ⊕ M hc = i,tc = k P tc has no P k as its direct summands, for the same reasonthe complex is exact at Hom A ( T, P i ).We remain to show that the cokernel of ◦ f ′ is one-dimensional. Let Ω S i be thefirst syzygy of S i . We apply Hom A ( T, − ) to0 → Ω S i f ′′ −−→ P i → S i → , and obtainHom A ( T, Ω S i ) ◦ f ′′ −−→ Hom A ( T, P i ) → Hom A ( T, S i ) → Ext A ( T, Ω S i ) → Ext A ( T, P i ) = 0 . OUNTING USING HALL ALGEBRAS III. QUIVERS WITH POTENTIALS 21
Since Ext A ( T, c ki P in ⊕ M hc = i,tc = k P tc ) vanishes, the cokernel of ◦ f ′ is the same asthat of ◦ f ′′ . By applying Hom A ( S i , − ) to (8.1), we see that(8.4) Hom A ( T, S i ) = Hom A ( T k , S i ) ⊕ K ∼ = Ext A ( S i , S k ) ∗ ⊕ K. In the meanwhile,Ext A ( T, Ω S i ) = Ext A ( τ − S k , Ω S i ) = Hom A (Ω S i , S k ) ∗ = Hom A (Ω S i , S k ) ∗ , A ( P i , S k ) → Hom A (Ω S i , S k ) → Ext A ( S i , S k ) → Ext A ( P i , S k ) = 0 . So Ext A ( T, Ω S i ) = Ext A ( S i , S k ) ∗ . Together with (8.4), we conclude that the cokernel of ◦ f ′ is K . (cid:3) Proof of Theorem 4.3.
We need to show that the endomorphism algebra End A ( T )of the BB-tilting module T is isomorphic to J ( e Q, f W ; e C ). Recall that the BB-tiltingmodule T is obtained from L i ∈ Q P i by just replacing P k with T k . So accordingto Lemma 8.1, it suffice to check that(1) The map g in Lemma 8.2 and f ′ in Lemma 8.3 agrees with the map π ;(2) The map r in Lemma 8.2 and g ′ in Lemma 8.3 agrees with the desiredrelations R , R and R (defined before Lemma 8.2).(1) is clear from the definition of g, f ′ , and π . For (2), we observe that • The map r agrees with π ( a ∗− r ) for r ∈ R and π ( a ∗ ) = g a ; • The component map ∂ cb in g ′ agrees with π ( b − r ) for r ∈ R and π ( b ) = b ; • Similarly, for b ∈ C the component map ∂ acb (resp. g ) in g ′ is responsiblefor the summand ∂ ab W (resp. b ∗ a ∗ ) in R . (cid:3) Finally, we prove Corollary 4.4.
Proof of Corollary 4.4.
Since the cut satisfies (4.1), there is no relation startingfrom S k . So S k has projective dimension one, and we have that 0 → P out → P k → S k →
0. Hence Hom A ( T, P out ) = Hom A ( T, P k ), and the map ◦ r in Lemma 8.2 isin fact injective. Now J ( Q, W ; C ) has global dimension 2, so P ′ in (8.3) is zero,and thus the map ◦ g ′ of Lemma 8.3 is injective. We conclude that J ( e Q, f W ; e C ) hasglobal dimension 2 as well. The two resolutions of Lemma 8.2 and 8.3 also implythat { ∂ c f W } c ∈ e C is a minimal set of generators in h ∂ e C f W i . (cid:3) Acknowledgement
The author thanks Mathematical Science Research Institute in Berkeley (MSRI)for its hospitality and support during the research program Cluster Algebras ofFall 2012 when most of results are obtained. He also wants to thank ProfessorBernhard Keller for his encouragement. Finally he thanks the anonymous refereefor the careful review and helpful comments.
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Shanghai Jiao Tong University, School of Mathematical Sciences
E-mail address : [email protected]@sjtu.edu.cn