On the Ext algebras of parabolic Verma modules and A infinity-structures
aa r X i v : . [ m a t h . R T ] J un On the Ext algebras of parabolic Verma modules and A ∞ -structures Angela Klamt and Catharina Stroppel a,b a Department of Mathematics, Universitetsparken 5, 2100 Copenhagen (Denmark); email:[email protected] b Department of Mathematics, University of Bonn, Endenicher Allee 60, 53115 Bonn(Germany); email: [email protected]
Abstract
We study the Ext-algebra of the direct sum of all parabolic Verma modules inthe principal block of the Bernstein-Gelfand-Gelfand category O for the her-mitian symmetric pair ( gl n + m , gl n ⊕ gl m ) and present the corresponding quiverwith relations for the cases n = 1 , . The Kazhdan-Lusztig combinatorics isused to deduce a general vanishing result for the higher multiplications in the A ∞ -structure of a minimal model. An explicit calculations of the higher multi-plications with non-vanishing m is included. Keywords:
Extensions, Kazhdan-Lusztig, A-infinity, parabolic Verma modules
Introduction
In 1988 Shelton determined inductively the graded dimension of the spaces ofextensions
Ext k ( M ( λ ) , M ( µ )) = L k ≥ Ext k ( M ( λ ) , M ( µ )) of parabolic Vermamodules M ( λ ) and M ( µ ) in the parabolic category O p for the Hermitian sym-metric cases [Sh]. More recently Biagioli reformulated the result combinatoriallyand obtained a closed dimension formula [Bi]. A nice feature is the fact that(parabolic) Verma modules form an exceptional sequence; i.e. they are labeledby a partially ordered set (Λ , ≤ ) of highest weights such that for all k ≥ thefollowing holds: Hom( M ( λ ) , M ( λ )) = C and Ext k ( M ( λ ) , M ( µ )) = 0 unless λ ≤ µ. A priori the set Λ is infinite, but the category O p decomposes into indecom-posable summands, so-called blocks, each containing only finitely many of theparabolic Verma modules. Taking M to be the direct sum of those which appearin the principal block yields a finite dimensional vector space Ext(
M, M ) whichdecomposes as the direct sum of e µ Ext(
M, M ) e λ = Ext( M ( µ ) , M ( λ )) , where e µ is the projection onto M ( µ ) along the sum of the other direct factors of M . Itcomes along with a natural algebra structure (the Yoneda product) which canbe obtained by viewing Ext(
M, M ) as the homology of the algebra Hom( P • , P • ) Preprint submitted to Arxiv October 5, 2018 ith P • a projective resolution of M ; the multiplication is given by the com-position of maps between complexes. The construction of these projective res-olutions and chain maps requires quite detailed knowledge of the projectivemodules and morphisms between them. Note that already the question aboutnon-vanishing Hom -spaces between parabolic Verma modules is non-trivial (cf.[Bo] or [Hu, Theorem 9.10]). The aim of this paper is to explore this Ext-algebra in more detail for the Hermitian symmetric case of ( gl m + n , gl m ⊕ gl n ) .In [BS3] Brundan and the second author developed a combinatorial descriptionof the category O p for g = gl m + n and p the parabolic subalgebra with Levicomponent gl m ⊕ gl n via a slight generalization of Khovanov’s diagram algebra(cf. Theorem 3.1). Using these combinatorial techniques along with classicalLie theoretical results, provides enough tools to compute projective resolutionsand their morphisms. As a crucial tool and byproduct we obtain a version ofthe Delorme-Schmid theorem (cf. [De], [Sc]) in our situation. The main resultsof the first part of the paper are Theorems 5.3 and 5.4, which give an explicitdescription of the Ext -algebra in terms of a path algebra of a quiver with rela-tions for the cases n = 1 and n = 2 , respectively. The first algebra also occursin the context of knot Floer Homology, [KhSe], see also [AK]. For a connectionto sutured Floer homology we refer to [GW].In the context of Fukaya categories these algebras come along with a natural A ∞ -algebra structure which encodes more information about the object. An A ∞ -algebra, also known in topology as a strongly homotopic associative alge-bra, has higher multiplications satisfying so-called Stasheff relations (cf. [Ke]).As Keller for instance points out, working with minimal models provides thepossibility to recover the algebra of complexes filtered by a family of modules M ( i ) from some A ∞ -structure on Ext( L M ( i ) , L M ( i )) . This A ∞ -structureis constructed in the form of a minimal model, i.e. deduced from an algebrastructure on H ∗ (Hom( L P ( i ) • , L P ( i ) • )) . In particular, there is a natural A ∞ -structure on our space of extensions Ext(
M, M ) . Since the projective objectsare filtered by parabolic Verma modules and therefore parabolic Verma modulesgenerate the bounded derived category D b ( O p ) it is of interest to know moreabout these A ∞ -structures. In the second part of the paper we construct anexplicit minimal model for our Ext -algebra from above. The results from thefirst part, in particular the explicit construction of projective resolutions, allowus to analyse the higher multiplications. For the construction of the minimalmodels we mimic the approach of [LPWZ] and combine formulas obtained byMerkulov [Me] (for the case of superalgebras) and Kontsevich and Soibelman[KoSo] (for the F -case). As for the Ext-algebra structure itself we keep trackof all the signs (which sometimes leads to tedious computations). Using thesetechniques, we achieve the first vanishing theorem (Theorem 6.7) in case n = 1 .In this theorem we get the formality of the Ext -algebra, i.e. we construct aminimal model with vanishing m k for k ≥ . For n = 2 , in the second vanishingtheorem (Theorem 6.9) we have an A ∞ -structure with non-vanishing m butvanishing m k for k ≥ . Thus, we obtain an example of an A ∞ -algebra withnon-trivial higher multiplications. The main result of the paper is presented inthe general vanishing theorem (Theorem 6.6). It says that for arbitrary n we2et a minimal model with vanishing m k for k ≥ n + 2 . A crucial ingredientin the proof is a detailed analysis of the Kazhdan-Lusztig polynomials forcinghigher multiplications to vanish. This article is based on [Kl] and focuses on pre-senting the main results and techniques. Some of the (very) technical detailedcalculations are therefore omitted, but can be found in [Kl]. Acknowledgment
The authors thank Bernhard Keller for helpful discussions and the refereesfor several extremely useful and detailed comments.
1. Preliminaries and Category O p We first recall the definition of the Bernstein-Gelfand-Gelfand category O .For a more detailed treatment see [Hu], [MP].Let g be a finite dimensional reductive Lie algebra over C and h ⊂ b ⊂ g fixedCartan and Borel subalgebras. Denote by Φ ⊂ h ∗ the root system of g relativeto h with the sets ∆ ⊂ Φ + ⊂ Φ of simple and positive roots respectively. For α ∈ Φ we have the root space g α and the coroot α ˇ ∈ h normalized by α ( α ˇ) = 2 .Let g = n − ⊕ h ⊕ n + be the triangular decomposition into negative roots spaces,Cartan subalgebra and positive root spaces. Denote Λ + := { λ ∈ h ∗ |h λ, α ˇ i ≥ for all α ∈ Φ + } , the set of dominant weights.Denote by ρ = P α ∈ Φ + α the half-sum of positive roots and by λ the zeroweight. Let W be the Weyl group with its usual length function w l ( w ) oftaking the length of a reduced expression. We get a natural action of W on h ∗ with fixed point zero. Shifting this fixed point to − ρ defines the dot-action w · λ = w ( λ + ρ ) − ρ . where w ∈ W, λ ∈ h ∗ .For L any Lie algebra we denote by U ( L ) the universal enveloping algebra.For λ ∈ h ∗ and M an arbitrary U ( g ) -module the weight space of weight λ relativeto the action of the Cartan subalgebra h is defined as M λ := { v ∈ M | h · v = λ ( h ) v, ∀ h ∈ h } . We denote by U ( g ) − Mod the category of left U ( g ) -modules.We fix now a standard parabolic subalgebra p containing b . This correspondsto a choice of a subset J ⊂ ∆ with associated root system Φ J ⊂ Φ such that p = l J ⊕ u J with nilradical u J and Levi subalgebra l J = h ⊕ α ∈ Φ J g α .In particular, the choice p = b corresponds to J = ∅ and l J = h , whereas p = g corresponds to J = ∆ and l J = g . Let W p be the Weyl group generatedby all α ∈ J . Denote by W p the set of minimal-length coset representatives for W p \ W , that is W p = { w ∈ W | ∀ α ∈ J : l ( s α w ) > l ( w ) } . Define the set of p -dominant weights as Λ + J := { λ ∈ h ∗ |h λ, α ˇ i ∈ Z + for all α ∈ J } . Denote by E ( λ ) the finite dimensional l J -module with highest weight λ ∈ Λ + J .3 efinition 1.1. The category O p is the full subcategory of U ( g ) − Mod whoseobjects M satisfy the following conditions: O M is a finitely generated U ( g ) -module; O M is h -semisimple, i.e., M = L λ ∈ h ∗ M λ ; O M is locally p -finite, i.e. dim C U ( p ) · v < ∞ for all v ∈ M .We recall a few standard results on O p , see [Hu], [R-C] for details. Definition 1.2.
For λ ∈ Λ + J we define the parabolic Verma module M ( λ ) := U ( g ) ⊗ U ( p J ) E ( λ ) . It has highest weight λ and is the largest quotient contained in O p of theordinary Verma module with highest weight λ . In particular, it has a uniquesimple quotient which is denoted by L ( λ ) . The L ( λ ) , for λ ∈ Λ + J constitutea complete set of non-isomorphic simple objects in O p . The category O p hasenough projective objects; for λ ∈ Λ + J let P ( λ ) be the projective cover of L ( λ ) .The category O p splits into direct summands (so-called ‘blocks’) O p χ , O p = M χ O p χ , indexed by W -orbits under the dot-action. The summand O p χ is the full sub-category of modules containing only composition factors of the form L ( λ ) with λ ∈ χ ∩ Λ + J . In particular M ( λ ) and P ( λ ) are objects of O p χ for λ ∈ χ . Let O p be the principal block of O p corresponding to the orbit through zero which hasprecisely the L ( w · λ ) with w ∈ W p as simple objects. Since we work with leftcosets, for better readability we write P ( x · λ ) =: P ( λ.x ) ; similarly for simplemodules and parabolic Verma modules. Remark 1.3.
To combine later on Lie-theoretical results for O p ( sl m + n ) withcombinatorial results known for O p ′ ( gl m + n ) we will tacitly use the standardequivalence of categories O p ′ ( gl m + n ) ∼ = O p ( sl m + n ) where p ′ is the parabolicsubalgebra with corresponding Levi component gl m ⊕ gl n and p = p ′ ∩ sl m + n .
2. The Ext algebra
We first introduce the homological and internal shift functors, [ i ] and h i i for i ∈ Z , on the category of complexes: Convention 2.1.
For a complex C • = ( C • , d • ) define C [ i ] • by C [ i ] j = C j − i with differential d [ i ] j = ( − i d j . For M a graded A -module define the internalshift M h i i by M h i i j = M j − i . We denote by C • h i i the (internally) shiftedcomplex C • obtained by just shifting each object; the differential maps stayhomogeneous of degree zero. 4et A, B ∈ Ob ( A ) be objects in an abelian category A and assume that A and B have finite projective dimension. Given projective resolutions P • and Q • of A and B , respectively, we define a differential graded structure on Hom( P • , Q • ) with Hom( P • , Q • ) r = Q p Hom( P p , Q p + r ) and differential d p ( f ) = d ◦ f − ( − p f ◦ d (c.f. [GM, III.6.13]). The space of extensions Ext can then becomputed using the derived category,
Ext k ( A, B ) = Hom D ( A ) ( A [0] , B [ k ]) =Hom D ( A ) ( P • [0] , Q • [ k ])= Hom K ( A ) ( P • [0] , Q • [ k ]) =Hom K ( A ) ( P • , Q • )[ k ]= H (Hom( P • , Q • )[ k ]) = H k (Hom( P • , Q • )) , where the third equality holds because P • is a bounded complex of projectives.In other words, Ext k ( A, B ) can be determined by first computing the homomor-phism spaces of the projective resolutions and afterwards taking its cohomology.Cycles in Hom( P • , Q • ) are chain maps (according to the degree commuting oranticommuting) and boundaries are homotopies (up to sign). If considered aschain maps between translated complexes (i.e. in Hom D b ( A ) ( P • [0] , Q • [ k ]) ) withthe sign convention 2.1, the cycles become commuting chain maps and bound-aries stay usual homotopies.We are now interested in the case A = B and the algebra Ext k ( A, A ) = H k (Hom( P • , P • )) . The multiplication in Ext(
A, A ) is induced from the multipli-cation in the algebra Hom( P • , P • ) , where it is given by composing of chain maps.Multiplication will be written from left to right, i.e. for α , β ∈ Hom( P • , P • ) wehave ( α · β )( x ) = β ( α ( x )) .If A = L α ∈ I A α and P α • is a projective resolution of A α with correspondingdecomposition P • = L α ∈ I P α • then Id α = [id] ∈ Ext ( A α , A α ) . The elements Id α form a system of mutual orthogonal idempotents, hence we can write Ext k ( A, A ) = M α,β ∈ I Id α Ext k ( A α , A β ) Id β . It is then enough to determine
Ext k ( A α , A β ) for any k , α , β and the productsof elements x ∈ Ext k ( A α , A β ) and y ∈ Ext l ( A β , A γ ) , interpreting their productas x · y = Id α x Id β Id β y Id γ ∈ Ext k + l ( A, A ) . O p ( gl m + n ( C )) via Khovanov’s diagram algebra We specialize now our setup to g = gl m + n ( C ) with the standard Borel subal-gebra b given by upper triangular matrices containing the Cartan h of diagonalmatrices. Let p be the parabolic subalgebra associated to the Levi subalgebra l = gl m ( C ) ⊕ gl n ( C ) . Then our key tool is the following special case of the maintheorem from [BS3], first observed in [St]:5 ∧ ∨ ∨ ∨ · · · − − · · · Figure 1: the zero weight for n = 2 and m = 3 Theorem 3.1.
There is an equivalence of categories from the principal blockof O p to the category of finite dimensional left modules over the Khovanov dia-gram algebra, K nm − mod , sending the simple module L ( λ ) ∈ O p to the simplemodule L ( λ ) ∈ K nm − mod , the parabolic Verma module M ( λ ) to the cell module M ( λ ) and the indecomposable projectives to the corresponding indecomposableprojectives. Here K nm is the algebra defined diagrammatically in [BS3] with an explicitdistinguished basis given by certain diagrams (see below) and a multiplicationdefined by an explicit “surgery” construction which can be expressed in terms ofan extended 2-dimensional TQFT construction, [St], generalizing a constructionof Khovanov [Kh]. The distinguished basis is in fact a (graded) cellular basis inthe sense of Graham and Lehrer [GL] in the graded version of Hu and Mathas[HM]. The algebra is shown to be quasi-hereditary in [BS1]. Hence we havecell or standard modules M ( λ ) , their projective covers P ( λ ) and irreduciblequotients L ( λ ) . This is meant by the notation used in the theorem. K nm and its basic properties For the construction of K nm , we recall from [BS1] the notions of weights,cup/cap/circle diagrams adapted to our situation. Let λ ∈ Λ + J be the highestweight of a simple module in O p = O p ( gl m + n ( C ) and let ρ = ε m + n − + 2 ε m + n − + · · · + ( m + n − ε ∈ h ∗ . The (diagrammatical) weight associated to λ is obtained by labeling the number i on the real line by ∨ if i belongs to I ∨ ( λ ) and by ∧ if i belongs to I ∧ ( λ ) respectively, where I ∨ ( λ ) := { ( λ + ρ, ε ) , . . . , ( λ + ρ, ε m ) } I ∧ ( λ ) := { ( λ + ρ, ε m +1 ) , . . . , ( λ + ρ, ε m + n ) } . Let Λ nm be the set of diagrammatical weights obtained in this way. Note that thelabels are always on the ( m + n ) places i ∈ I = { , . . . , m + n − } which we call vertices . The diagrammatical weight associated to λ is given by putting all ∧ ’sto the left and all ∨ ’s to the right, see Figure 1. In fact, Λ nm consists precisely ofthe diagrams obtained by permuting the n ∧ ’s and m ∨ ’s establishing a bijectionbetween the highest weights of parabolic Verma modules in O p and elements in Λ nm . The dot-action corresponds then to permuting the labels; swapping ∨ ’s tothe right means getting bigger in the Bruhat order, see [BS3, Section 1].We fix the above bijection and do not distinguish in notation between weightsand diagrammatical weights. For λ = λ .x with x ∈ W p we write l ( λ ) for l ( x ) .6 ∨ ∨ ∧ ∨ ∧ ∨ ∧ ∨ ∨ ∧ ∨ ∧ ∨ Figure 2: An oriented cup diagram and an oriented circle diagram.For each i ∈ I define the relative length l i ( λ, µ ) := { j ∈ I | j ≤ i and vertex j of λ is labeled ∨}− { j ∈ I | j ≤ i and vertex j of µ is labeled ∨} (3.1)and note that l ( λ ) − l ( µ ) = P i ∈ I ℓ i ( λ, µ ) by [BS1, Section 5].A cup diagram is a diagram obtained by attaching rays and finitely manycups (lower semicircles) to the vertices I , so that cups join two vertices i ∈ I ,rays join vertices i ∈ I down to infinity, and rays or cups do not intersect otherrays or cups. A cap diagram is the horizontal mirror image of a cup diagram,so caps (i.e. upper semicircles) instead of cups are used. The mirror image of acup (resp. cap) diagram c is denoted by c ∗ .If c is a cup diagram and λ a weight in Λ nm , we can glue c and λ and obtaina new diagram denoted cλ . It is called an oriented cup diagram if • each cup is oriented, i.e. one of its vertices is labeled ∨ , and one ∧ ; • there are not two rays in c labeled ∨∧ in this order from left to right.An example is given in Figure 2.Similarly we can glue λ to a cap diagram c . The result λc is called orientedcap diagram if c ∗ λ is an oriented cup diagram. A circle diagram is obtained bygluing a cup and a cap diagram at the vertices I . It consists of circles and lines.Gluing an oriented cap diagram with an oriented cup diagram along the sameweight gives an oriented circle diagram . For an example, see Figure 2.The degree of an oriented cup/cap diagram aλ (or λb ) means the total num-ber of oriented cups (caps) that it contains. So in K nm one has deg ( aλ ) ≤ n ,since there are at most n cups. The degree of an oriented circle diagram aλb isdefined as the sum of the degree of aλ and the degree of λb . The cup diagramassociated to a weight λ is the unique cup diagram λ such that λλ is an orientedcup diagram of degree 0. (For an explicit construction: Take any two neighbor-ing vertices labeled by ∨∧ and connect them by a cup. Repeat this procedureas long as possible, ignoring vertices which are already joined to others. Finallydraw rays to all vertices which are left.) The cap diagram associated to a weight λ is defined as λ := ( λ ) ∗ . The vector space underlying K nm has a basis { ( aλb ) | for all oriented circle diagrams with λ ∈ Λ nm } . Each basis vector has a well-defined degree, turning the vector space into agraded vector space equipped with a distinguished homogeneous basis. The7lement e λ is defined to be the diagram λλλ . The product of two circle diagrams aλb and cµd is zero except for b = c ∗ . The multiplication of aλb and b ∗ µd worksby the rules of the generalized surgery procedure defined in [BS1, Section 3and Theorem 6.1.]. The vectors { e α | α ∈ Λ nm } form a complete set of mutuallyorthogonal idempotents in K nm . We get K nm = M α,β ∈ Λ nm e α K nm e β where e α K nm e β has basis (cid:8) ( αλβ ) | λ ∈ Λ nm such that the diagram is oriented } . Theorem 3.1 establishes an equivalence of categories between O p and thecategory of finite dimensional K nm -modules. Following [BS1], we consider thecategory K nm − gmod of finite dimensional graded left K nm -modules which canbe seen as a graded version of O p with the following important objects: • The simple modules L ( λ ) with λ ∈ Λ nm .These are -dimensional modules concentrated in degree zero. The idem-potent e λ ∈ K nm acts by the identity, all other e µ by zero. Shifting theinternal degree gives all simple objects, L ( λ ) h i i , i ∈ Z . • The projective cover P ( λ ) = K nm e λ of the simple module L ( λ ) has homo-geneous basis (cid:8) ( αµλ ) | for all α, µ ∈ Λ nm such that the diagram is oriented } ; with the action induced from the diagrammatical multiplication in thealgebra. By shifting the internal degree one obtains a full set of indecom-posable graded projective modules. • The cell or standard modules M ( µ ) with homogeneous basis (cid:8) ( cµ | (cid:12)(cid:12) for all oriented cup diagrams cµ (cid:9) such that ( aλb )( cµ | ) = ( aµ | ) or depending on the elements.After forgetting the grading, these modules correspond via Corollary 3.1 tosimple modules, projectives and Verma modules in the principal block of O p . We have the following theorems about cell module filtrations of projectivesand Jordan-Hölder filtrations of cell modules, which say that K nm is quasi-hereditary in the sense of Cline, Parshall and Scott [CPS].8 emma 3.2 ([BS1, Theorem 5.1]) . For λ ∈ Λ nm , enumerate the elements ofthe set { µ ∈ Λ nm | λµ is oriented } as µ , µ , . . . , µ n = λ so that µ i > µ j implies i < j . Let M (0) := { } and for i = 1 , . . . , n define M ( i ) to be the subspace of P ( λ ) generated by M ( i − and the vectors (cid:8) ( cµ i λ ) (cid:12)(cid:12) for all oriented cup diagrams cµ i (cid:9) . Then M (0) ⊂ M (1) ⊂ · · · ⊂ M ( n ) = P ( λ ) is a filtration of P ( λ ) as graded K nm -module such that M ( i ) /M ( i − ∼ = M ( µ i ) h deg( µ i λ ) i for ≤ i ≤ n . Lemma 3.3 ([BS1, Theorem 5.2]) . For µ ∈ Λ nm , let N ( j ) be the submodule of M ( µ ) spanned by all graded pieces of degree ≥ j . This defines a finite filtrationof the graded K nm -module M ( µ ) with simple subquotients N ( j ) /N ( j + 1) ∼ = M λ ⊂ µ with deg( λµ )= j L ( λ ) h j i . By the BGG-reciprocity [Hu, Theorem 9.8(f)] the two multiplicities d iλ,µ :=[ M ( µ ) : L ( λ ) h i i ] and [ P ( λ ) : M ( µ ) h i i ] are equal and we get the symmetric q-Cartan matrix C Λ nm ( q ) = ( c λ,µ ( q )) λ,µ ∈ Λ nm , where c λ,µ ( q ) := X j ∈ Z dim Hom K nm ( P ( λ ) , P ( µ )) j q j ∈ Z [ q ] . Set d λ,µ ( q ) = P i d iλ,µ q i . Note that this sum in fact contains at most one non-trivial summand, since d λ,µ = 0 implies λµ is oriented and λ ≤ µ in the Bruhatordering, in which case d λ,µ = q deg( λµ ) holds (cf. [BS1, 5.12]).In a cup (cap) diagram we number the cups (caps) , , . . . according to theirright vertex from left two right. For a cup (cap) diagram a we denote by nes a ( i ) for ≤ i ≤ { cups } the number of cups nested in the i th cup.The following provides then explicit lower and upper bounds for the decom-position numbers and the entries of the q -Cartan matrix: Proposition 3.4. In K nm − gmod we have d λ,µ = 0 unless ≤ l ( λ ) − l ( µ ) ≤ n + 2 X i nes λ ( i ) ≤ n . (3.2) In particular, c λ,µ = 0 unless l ( λ ) − l ( µ ) ≤ n + 2 P i nes λ ( i ) ≤ n . Proof.
Assume d λ,µ ( q ) = 0 . This means that λµ is oriented. By [BS1, Lemma2.3] it follows that λ ≤ µ in the Bruhat ordering, which leads to l ( λ ) ≥ l ( µ ) .Now we find λ and µ such that l ( λ ) − l ( µ ) is maximal and λµ is oriented.Fix such λ and consider weights µ of smallest possible length such that λµ isstill oriented. This is obtained if all ∧ ’s and ∨ ’s on the end of a cup in λ are9nterchanged. Since a ∧ on the i th cup has been moved λ ( i ) positionsto the right, the length is changed by P i (2nes λ ( i ) + 1) . Therefore, we obtain ≤ l ( λ ) − l ( µ ) ≤ n + 2 X i nes λ ( i ) . Since P i nes a ( i ) is maximal if all cups are nested (i.e if the j th cup containsprecisely j − cups). In that case we obtain X i nes a ( i ) = 2 n X i =1 ( i −
1) = ( n − n and therefore (3.2) holds. For c λ,µ = 0 a simple L ( λ ) must occur in P ( µ ) ,especially it must occur in some M ( ν ) , i.e. d λ,ν = 0 and d µ,ν = 0 . Therefore, l ( λ ) − l ( ν ) ≤ n + 2 X i nes λ ( i ) and ≤ l ( µ ) − l ( ν ) which implies l ( λ ) − l ( µ ) ≤ l ( λ ) − l ( ν ) ≤ n + 2 X i nes λ ( i ) , which proves the second inequality. To compute the Ext-algebras of Verma modules it will be useful to constructexplicitly linear projective resolutions of the cell modules M ( λ ) ∈ K nm − gmod .Recall that a projective resolution P • is linear if P i is generated by its homoge-neous component in degree i . From the description of projective modules it isclear that L λ ∈ Λ nm P ( λ ) ∼ = K nm is a minimal projective generator of K nm − mod .Any endomorphism is given by right multiplication with an element of the al-gebra, and Hom K nm ( P ( λ ) , P ( µ )) = Hom K nm ( K nm e λ , K nm e µ ) = e λ K nm e µ as vectorspaces, [BS1, (5.9)].To construct the differentials in linear projective resolutions, we study firstthe degree component of Hom K nm ( P ( λ ) , P ( µ )) , i.e. we search for elements ν s.t. deg( λνµ ) = 1 . Since λνµ ) = deg( λν ) + deg( νµ ) , one summand hasto be and the other .1. deg( λν ) = 0 , i.e. λ = ν , so we look for an oriented cap diagram λµ ofdegree . It exists iff λ > µ and µ = λ.w with w changing the ∧ and ∨ (in this ordering) at the end of a cup into a ∨ and ∧ .2. deg( νµ ) = 0 , i.e. µ = ν , so we look for an oriented cup diagram λµ ofdegree 1. It exists iff µ > λ and λ = µ.w with w changing the ∨ and ∧ atthe end of a cap. 10 ∨ ∧ ∨ ∨ ∧ ∨ l i / λ ∨ ∨ ∨ ∨ ∧ ∧ Figure 3: the labeled cap diagramAltogether we get dim Hom K nm ( P ( λ ) , P ( µ )) ≤ and the diagram calculusdefines a distinguished morphism f λ,µ in case this dimension equals .On the other hand, the modules occurring in a linear projective resolution ofcell modules are determined by polynomials p λ,µ defined diagrammatically andrecursively in [BS2, Lemma 5.2.], namely certain Kazhdan-Lusztig polynomialsgoing back to work of Lascoux and Schützenberger [LS].We recall the construction of these polynomials. Set p λ,µ = 0 if λ µ . A labeled cap diagram C is a cap diagram whose unbounded chambers are labeledby zero and given two chambers separated by a cap, the label in the insidechamber is greater than or equal to the label in the outside chamber.
Definition 3.5.
Denote by D ( λ, µ ) the set of all labeled cap diagrams obtainedby labeling the chambers of µ in such a way that for every inner cap c (a capcontaining no smaller one), the label l inside c satisfies l ≤ l i ( λ, µ ) , where i denotes the vertex of c labeled by ∨ . The polynomials are given by p λ,µ ( q ) := X i p ( i ) λ,µ q i := q l ( λ ) − l ( µ ) X C ∈ D ( λ,µ ) q − | C | . (3.3)where | C | denotes the sum of all labels in C . Example 3.6.
Figure 3 presents the possible labeled cap diagrams from D ( λ, µ ) for the chosen λ and µ . Since l ( λ ) − l ( µ ) = 4 , we get p λ,µ ( q ) = q + q . Theorem 3.7 ([BS2, Theorem 5.3], [Kl, Theorem 3.20]) . For λ ∈ Λ nm the cellmodule M ( λ ) has a linear projective resolution P • ( λ ) of the form · · · d −→ P ( λ ) d −→ P ( λ ) ε −→ M ( λ ) −→ (3.4) with P ( λ ) = P ( λ ) and P i ( λ ) = L µ ∈ Λ nm p ( i ) λ,µ P ( µ ) h i i for i ≥ . Using the above observations and tools from the proof of [BS2, Theorem5.3], [Kl, §3.3.3] gives an explicit method to construct projective resolutions ofcell modules in K nm − gmod by an interesting simultaneous induction varyingthe underlying algebra and the highest weights. For K m and K n we have, upto isomorphism, only one indecomposable module, which is projective, simpleand cell module at once. This provides the starting point of the induction. Inthe following we will fix such a projective resolution P • ( λ ) for each λ . Togetherwith the inequalities obtained before, we can deduce:11 roposition 3.8. If a projective module P ( ν ) occurs as a direct summand in P i ( λ ) with P • ( λ ) being the projective resolution constructed above, one has l ( λ ) − i − n − n − X i nes ν ( i ) ! ≤ l ( ν ) ≤ l ( λ ) − i. Proof.
Let C be a cap connected with the j th ∧ occurring in ν and let itbe the k j th cup in our numbering with starting point i . Recall from (3.1)that l i ( λ, ν ) ≤ { k | k ≤ i and vertex k of ν is labeled ∧} , the latter counting thenumbers of ∧ ’s to the left of the cap. This equals j − − nes ν ( k j ) counting toones the left of the j th ∧ without those lying inside the cap, and thus ≤ | C | ≤ X j ∈{ ,...n } cap ending on j th ∧ ( j − − nes ν ( k j )) ≤ n ( n − − X i nes ν ( i ) . If a module P ( ν ) occurs in the resolution (say at homological degree i ), one has p ( i ) λ,ν > , i.e. there is a diagram C such that i = l ( λ ) − l ( ν ) − | C | . Taking theupper and lower bound for C obtained before, one gets l ( λ ) − i − ( n − n − X i nes ν ( i )) ≤ l ( ν ) ≤ l ( λ ) − i and the claim of the proposition follows.The following is a vanishing result for Ext k ( M ( λ ) , M ( µ )) : Lemma 3.9.
For λ , µ ∈ Λ nm we have Hom k ( P • ( λ ) , P • ( µ )) = 0 unless l ( λ ) ≤ l ( µ ) + n + k. (3.5) Proof.
A map between P • ( λ ) and P • ( µ )[ k ] is in each component a morphismbetween graded projective modules. Including the shift we therefore have toconsider morphisms between projectives P ( ν ) occurring in P i ( λ ) and projectives P ( ν ′ ) in P i − k ( µ ) . By Proposition 3.8 we know l ( λ ) − i − n − n − X i nes ν ( i ) ! ≤ l ( ν ) and l ( ν ′ ) ≤ l ( µ ) − ( i − k ) . Therefore, we have l ( λ ) − i − n − n − X i nes ν ( i ) ! − ( l ( µ ) − ( i − k )) ≤ l ( ν ) − l ( ν ′ ) . (3.6)Since we have a morphism between these projectives we get from Lemma 3.4 l ( ν ) − l ( ν ′ ) ≤ n + 2 X i nes ν ( i ) . (3.7)12ombining the two inequalities (3.6) and (3.7), we obtain l ( λ ) − i − n − n − X i nes ν ( i ) ! − ( l ( µ ) − ( i − k )) ≤ n + 2 X i nes ν ( i ) , which implies l ( λ ) ≤ l ( µ ) + n + k . The claim follows.
4. The
Ext -algebra of L x ∈ W p M ( λ · x ) Assume we are in the setup of Section 3 and denote E nm = M x,y ∈ W p Ext ( M ( x · λ ) , M ( y · λ )) = M λ,µ ∈ Λ nm Ext K nm ( M ( λ ) , M ( µ )) . A very useful tool for describing E nm are Shelton’s recursive dimension formulaswhich he established in [Sh] more generally for all the hermitian symmetriccases. For an arbitrary parabolic subalgebra p , there is no explicit formula, noteven a candidate.Abbreviating E k ( x, y ) = dim Ext k ( M ( λ .x ) , M ( λ .y )) for x, y ∈ W p , [Sh,Theorem 1.3] can be formulated as follows: Theorem 4.1 (Dimension of
Ext -spaces) . With g and p as above, let x, y ∈ W p and let s be a simple reflection with x > xs and xs ∈ W p . The dimensions E k ( x, y ) are then given by the following formulas: . E k ( x, y ) = 0 ∀ k unless y ≤ x ;2 . E k ( x, x ) = ( for k = 00 otherwise.For y < x there are the following recursion formulas: . E k ( x, y ) = E k ( xs, ys ) if y > ys and ys ∈ W p ;4 . E k ( x, y ) = E k − ( xs, y ) if ys / ∈ W p ;5 . E k ( x, y ) = E k − ( xs, y ) + E k ( xs, y ) if ys > y but xs > ys ;6 . E k ( x, y ) = E k − ( xs, y ) − E k +1 ( xs, y )+ E k ( xs, ys ) if x > xs > ys > y. To translate between our setup and Shelton’s note that he denotes N y = M ( λ .ω m yω ) where ω and ω m are the longest elements in W and in W p respectively. Then it only remains to observe that for y, x ∈ W we have ω m yω ∈ W p ⇔ y ∈ W p and ω m yω < ω m xω ⇔ y > x in the Bruhat or-der.Although the previous theorem determines all dimension of Ext-spaces, itis convenient to have explicit vanishing conditions. Therefore, we reprove theDelorme-Schmid Theorem (cf. [De], [Sc]) in our situation:13 emma 4.2. For λ, µ ∈ Λ nm we have Ext k ( M ( λ ) , M ( µ )) = 0 ∀ k > l ( λ ) − l ( µ ) . Proof.
We claim that any chain map f : P • ( λ ) → P • ( µ )[ k ] with k > l ( λ ) − l ( µ ) is homotopic to zero. On the k th component f induces a map f k : P k ( λ ) → P ( µ ) = P ( µ ) . For P ( ν ) occurring as a direct summand in P k ( λ ) we have l ( ν ) ≤ l ( λ ) − k < l ( λ ) − ( l ( λ ) − l ( µ )) = l ( µ ) by Lemma 3.8. By Proposition 3.4 L ( ν ) does not occur in M ( µ ) and so the composition P ( ν ) → P ( µ ) → M ( µ ) is zero. Let P T • ( λ ) be the truncated complex with P Ti ( λ ) = 0 for i < and P Ti ( λ ) = P i + k ( λ ) if i ≥ . This is a projective resolution of im d k , and f • induces a morphism e f • : P T • ( λ ) → P • ( µ ) such that · · · P T ( λ ) e f im d k · · · P ( µ ) M ( µ ) 0 where e f is a lift of the zero map. Since the zero map between the complexes isalso a lift of the zero map and two lifts are equal up to homotopy ([GM, TheoremIII.1.3]) the map e f is nullhomotopic by a homotopy H : P T • ( λ ) → P • ( µ )[ − .This extends to a homotopy H : P • ( λ ) → P • ( µ )[ − by defining it to be zeroon the other terms. The claim follows. Remark 4.3.
The result of Lemma 4.2 could also be deduced from Shelton’sformulas or from the explicit formulas [Bi, Theorem 3.4].
5. Special cases
Now we want to describe the
Ext -algebra in the cases ( m, n ) = (1 , N ) and ( m, n ) = (2 , N − . The first algebra is related to algebras appearing in (knot)Floer homology, see [KhSe], [GW], the second invokes our theory in a moresubstantial way and provides interesting A ∞ -structures.Using knowledge about decomposition numbers, the endomorphism spaces ofprojective modules and the projective resolutions together with the tools workedout above, one can choose explicit maps between the projective resolutions fromTheorem 3.7 and determine their linear dependence up to null homotopies. Inthis way we will construct non-trivial elements in Ext i which, using Shelton’sdimension formulas, can be shown form a basis. Finally we compute the mul-tiplication rules. Especially in the case for n = 2 the computations are longand cumbersome and carried out in [Kl]. We present the crucial computationsfor the n = 1 case here, which suffice in this case to get the results by a feweasy straightforward calculations. For the n = 2 case we present the results andmain idea and refer to [Kl] for the details.14able 1: Filtration of projective module P ( λ ) by simple modules, same colourbelonging to the same Verma module λ = ( j ) P ( j ) j = 0 j = N L ( j ) L ( j + 1) L ( j − L ( j ) j = 0 L (0) L (1) j = N L ( N ) L ( N − L ( N ) n = 1 The elements in W p are precisely s · · · s j , ≤ j ≤ N − and we abbreviate ( j ) = λ .s s . . . s j . The filtrations in Theorems 3.2 and 3.3 combined determinethe filtration of projective modules in terms of simple modules presented inTable 1. To compute the combinatorial Kazhdan-Lusztig polynomials whichdetermine the terms of the resolution of the cell module M ( λ ) we consider ( s ) = µ ≥ λ = ( j ) and obtain µ · · · ∨ ∧ · · · ℓ i λ · · · ∨ · · · ∧ · · · and therefore p λ,µ = q j − s . By Theorem 3.7 there is then a unique summandoccurring in the i th position of the resolution of M ( λ ) , namely the projectivemodule P ( j − i ) , and we have the distinguished morphism f k := f k,k +1 , homo-geneous of degree , from P ( k ) to P ( k + 1) . Set d n − k ( n ) = ( − n + k +1 f k . Lemma 5.1.
The chain complex → P (0) h n i d → P (1) h n − i → · · · d n − → P ( n ) → is a (linear) projective resolution of M ( n ) in K N − gmod . Proposition 5.2.
For j ≥ l the identity maps id : P ( s ) → P ( s ) for all s ≤ l define a chain map Id ( j )( l ) : P • ( j ) → P • ( l )[ j − l ] h j − l i which induces a non-trivial element in Ext j − l ( M ( j ) , M ( l )) . For j > l , the maps f s,s − : P ( s ) → P ( s − for all s ≤ l + 1 define a chain map F ( j )( l ) : P • ( j ) → P • ( l )[ j − l − h j − l − i which induces a non-trivial element in Ext j − l − ( M ( j ) , M ( l )) . roof. We have to check that the maps are not nullhomotopic which is clearin the clear in the first case. For F ( j )( l ) , a homotopy would be a map H ∈ Hom j − l − ( P • ( j ) , P • ( l ) h j − l − i ) which cannot exist by Lemma 3.9 since j (cid:2) l + 1 + ( j − l − .The dimension formula from Theorem 4.1 implies that we constructed a basisof E N . By explicitly composing chain maps we obtain the following relations in Hom( P • , P • ) : Id ( j )( l ) · Id ( l )( m ) = Id ( j )( m ) , F ( j )( l ) · F ( l )( m ) = 0 , Id ( j )( l ) · F ( l )( m ) = F ( j )( m ) , F ( j )( l ) · Id ( l )( m ) = F ( j )( m ) Reformulating the above result in terms of quivers, we obtain:
Theorem 5.3.
The algebra E N is isomorphic to the path algebra of the quiver ( N ) · · · ( j + 1) ( j ) ( j − · · · (0) with relations • • • = 0 , • • • = • • • . The vertex • labeled i corresponds to the idempotent e λ where λ = λ .s · . . . s i .5.2. The result for n = 2 Now consider ( n, m ) = (2 , N − . The elements in W p are precisely theelements s · . . . s k · s · · · · · s l with ≤ l < k ≤ N . We denote the weight λ = λ .s · . . . · s k · s · . . . · s l by ( k | l ) ; the associated diagrammatical weight has ∧ ’s at the l th and k th position (starting to count with position zero). Theorem 5.4.
The algebra E N is isomorphic to the path algebra of the quiverwhich looks as · · · · · · · · ·· · · ( k + 1 | l + 1) ( k | l + 1) ( k − | l + 1) · · ·· · · ( k + 1 | l ) ( k | l ) ( k − | l ) · · ·· · · ( k + 1 | l −
1) ( k | l −
1) ( k − | l − · · ·· · · · · · · · · or k > l + 2 and in the other cases: . . . ( l | l − · · · ( l | l −
2) ( l − | l − · · · ( l | l −
3) ( l − | l −
3) ( l − | l − · · · . . . · · · with relations as follows (in case that both sides of the relation exist):1. • = − • •• • • • = • •• • • • = • •• • • • = • •• • • • = •• •• • • = • •• • • • = • • • = • • These are all cases occurring in the middle of the quiver, i.e. in the upperdiagram. We also have to look for those at the corner part. Those can be foundin [Kl].
6. The A ∞ -structure on E nm A ∞ -algebras are a generalization of associative algebras, see [Ke] for anoverview, including historical and topological motivation. A very detailed ex-position with most of the proofs is provided in [L-H].17 efinition 6.1. An A ∞ -algebra over a field k is a Z -graded k -vector space A = L p ∈ Z A p endowed with a family of graded k -linear maps m n : A ⊗ n → A, n ≥ of degree − n satisfying the following Stasheff identities: P ( − r + st m r + t +1 (Id ⊗ r ⊗ m s ⊗ Id ⊗ t ) = 0 where for fixed n the sum runs over all decompositions n = r + s + t with s ≥ ,and r, t ≥ .We use the Koszul sign convention ( f ⊗ g )( x ⊗ y ) = ( − | g || x | f ( x ) ⊗ g ( y ) , for tensor products, where x , y , f , g are homogeneous elements of degree | x | , | y | , | f | , | g | respectively. Definition 6.2.
Let A and B be two A ∞ -algebras. A morphism of A ∞ -algebras f : A → B is a family f n : A ⊗ n → B of graded k -linear maps of degree − n such that X ( − r + st f r + t +1 (Id ⊗ r ⊗ m s ⊗ Id ⊗ t ) = X ( − w m q ( f i ⊗ · · · ⊗ f i q ) for all n ≥ . Here, the sum run over all decompositions n = r + s + t and overall decompositions n = i + · · · + i q with ≤ q ≤ n and all i s ≥ respectively.The sign on the right-hand side is given by w = P q − j =1 ( q − j )( i j − .A morphism f is a quasi-isomorphism if f is a quasi-isomorphism. It is strict if f i = 0 for all i = 1 .Our goal is to put an A ∞ -structure on the Ext -algebras E nm . The first stepis to introduce an A ∞ -structure on the cohomology of an A ∞ -algebra (the so-called minimal model) and then realize our Ext-algebra as the cohomology ofan A ∞ -algebra, namely the Hom -algebra introduced earlier.
Theorem 6.3 ([Ka1]) . Let A be an A ∞ -algebra and H ∗ ( A ) its cohomology.Then there is an A ∞ -structure on H ∗ ( A ) such that m = 0 and m is inducedby the multiplication on A , and there is a quasi-isomorphism of A ∞ -algebras H ∗ ( A ) → A lifting the identity of H ∗ ( A ) . Moreover, this structure is unique upto isomorphism of A ∞ -algebras. All known (at least to us) proofs inductively construct the model, but theapproaches are slightly different. We follow here Merkulov’s more general con-struction [Me] in the special situation of a differential graded algebra:
Proposition 6.4 ([Me]) . Take ( A, d ) a differential graded algebra with gradingshift [ ] . Let B ⊂ A be a vector subspace of A and Π : A → B a projectioncommuting with d . Assume that we are given a homotopy Q : A → A [ − suchthat − Π = dQ + Qd. (6.1)18 efine λ n : A ⊗ n → A for n ≥ by λ ( a , a ) := a · a and recursively, λ n ( a , . . . , a n )= − X k + l = nk,l ≥ ( − k +( l − | a | + ··· + | a k | ) Q ( λ k ( a , . . . , a k )) · Q ( λ l ( a k +1 , . . . , a n )) . (6.2) for n ≥ , setting formally Qλ = − Id . Then the maps m = d and m n =Π( λ n ) define an A ∞ -structure for a minimal model on B . Choosing Q in a clever way simplifies computations, but our result willdepend on this choice. We make our choices following [LPWZ]. To define Q ,we first divide the degree n part A n of A into three subspaces, for this, denoteby Z n the cocycles of A and by B n the coboundaries. As we work over a field,we can find subspaces H n and L n such that Z n = B n ⊕ H n and A n = B n ⊕ H n ⊕ L n . (6.3)We identify the n th cohomology group H n ( A ) via (6.3) with H n . We want toapply Proposition 6.4 with the choice of a subspace B = H ∗ ( A ) , the projection Π being the projection on the direct summand H ∗ and the map Q defined asfollows:1. When restricted to Z n by equation (6.1) and the condition that d | Z n equalsto zero, the map Q has to satisfy the relation − Π = dQ.
In particular, dQ | H has to be zero. We choose Q | H = 0 .2. On B n the map Π is zero, and therefore the map Q | B has to satisfy dQ ,i.e. Q has to be a preimage of d . We want to choose this preimage as smallas possible i.e. with no non-trivial terms from Z n (they would anyway beannihilated by d ). Since d is injective on L , we can choose Q | B = ( d | L ) − .3. We briefly outline how to determine Q restricted to L (although it won’tplay any role in our computations later on). From (6.1) we get the restric-tion Qd + dQ. As d ( a ) ∈ B for all a ∈ A we see that Qd | L = ( d | L ) − d | L = 1 , so we candefine Q | L = 0 .Now the construction of a minimal model applies to our situation if we choose A := A nm := Hom( P • , P • ) , where P • is the direct sum of all linear projectiveresolutions of M ( λ ) , λ ∈ Λ nm from 3.7, and E = Ext nm = H ∗ ( A ) .In the following we give an upper bound for the l with m l = 0 . Alreadyin the case n = 2 we can show that not all m l for l > vanish and therefore19ur specific model provides interesting examples of A ∞ -algebras with non-trivialhigher multiplications. We start by stating the following Lemma generalizingthe fact that the multiplication of two morphisms can only be non zero if theylie in appropriate Hom -spaces.
Lemma 6.5.
Let a i , ≤ i ≤ l be homogeneous elements of degree k i in E nm ofthe form a i ∈ Ext k i ( M ( µ i ) , M ( ν i )) 1 ≤ i ≤ l. Then we have λ l ( a , ..., a l ) = 0 unless ν i = µ i +1 for all ≤ i ≤ l − ; and if λ l ( a , ..., a l ) = 0 we have λ l ( a , ..., a l ) ∈ Hom Σ k i +2 − l ( P • ( µ ) , P • ( ν l )) . Proof.
The proof goes by induction on l , using Theorem 6.4, see [Kl]. Theorem 6.6 (General Vanishing Theorem) . The A ∞ -structure on E nm satisfies m l = 0 for all l > n + 2 .Proof. We claim that λ l = 0 if l > n + 2 . Since λ l is linear, it is enoughto show the assertion on nonzero homogeneous basis elements and thereforeby Lemma 6.5 we can take a i ∈ Ext k i ( M ( µ i ) , M ( µ i +1 )) for ≤ i ≤ l. ByLemma 4.2 there are d i ≥ such that k i = l ( µ i ) − l ( µ i +1 ) − d i and there-fore P li =1 k i = l ( µ ) − l ( µ l +1 ) − P li =1 d i . >From Lemma 6.5 we know that λ l ( a , ..., a l ) ∈ Hom Σ k i +2 − l ( P • ( µ ) , P • ( ν l )) . Assume λ l = 0 , so, by Lemma 3.9about the morphisms between our chosen projective resolutions, we know that l ( µ ) ≤ l ( µ l +1 ) + n + P k i + 2 − l, thus l ( µ ) ≤ l ( µ l +1 ) + l ( µ ) − l ( µ l +1 ) − l X i =1 d i + 2 − l + n , which is equivalent to P li =1 d i ≤ n + 2 − l . Since P li =1 d i ≥ , we get ≤ n + 2 − l , equivalently l ≤ n + 2 ; providing the asserted upper bound. E N and E N − In the previous section we established general vanishing results for the highermultiplications; in this section we describe explicit models for our small examples n = 1 and n = 2 . The first result in this situation is the following: Theorem 6.7 (1st vanishing Theorem) . The algebra E N is formal, i.e. thereis a minimal model such that m n = 0 for all n ≥ .Proof. Recall that all multiplication rules in the algebra E N are already deter-mined in A N = Hom( P • , P • ) . Therefore, for all elements a , a ∈ Ext( ⊕ M ( λ ) , ⊕ M ( λ )) = H ∗ (Hom( P • , P • )) identified with the subspace H ∗ via the decomposition from(6.3), the product a · a also lies in the subspace H ∗ and has no boundarycomponent in B ∗ . Since we have chosen Q | H = 0 , we obtain Q ( a · a ) = 0 . Using the construction of the higher multiplications in Proposition 6.4 one gets m n = 0 for all n ≥ . 20 .1.1. The case E N − The case of n = 2 turns out to be more interesting than the case n = 1 studied before, since we have non-vanishing higher multiplications. In contrastto the previous example this phenomenon is possible, since some multiplicationsin A N − = Hom( P • , P • ) are only homotopic to their product in the Ext -algebra.This yields the following theorem:
Theorem 6.8.
In the minimal model above, there are non-vanishing m . A complete list of all higher multiplications m is given in [Kl]. Detailed knowledge about the structure of projective resolutions provides astronger vanishing result than in the general case (see [Kl]):
Theorem 6.9 (2nd Vanishing Theorem) . The A ∞ -structure on E N − given bythe construction above satisfies m n = 0 ∀ n ≥ . In the previous section we proved that there is a minimal model with non-vanishing higher multiplications but this does not answer the question whetherthe algebra is formal. To show that the algebra is not formal, we have to provethat no model exists such that m n = 0 for all n ≥ . As a tool one could useHochschild cohomology. Given a dg-Algebra A one can compute its Hochschildcohomology by using the A ∞ -structure on a minimal model of A (cf. [L-H,Lemma B.4.1] and [Ka2]). Assume that we have found a minimal model on H ∗ ( A ) with m n = 0 for ≤ n ≤ p − . Then the multiplication m p defines acocycle for the Hochschild cohomology of A by the construction in [L-H, LemmaB.4.1]. If we can prove that this class is not trivial, we are done and have shownthat the algebra is not formal. If we cannot, we have to modify our model suchthat m p = 0 and then analyze if m p +1 vanishes. A detailed discussion of thistopic would go beyond the scope of this article. Therefore we only state thefollowing conjecture: Conjecture 6.10.
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