MASSEY PRODUCTS IN GRADED LIE ALGEBRA COHOMOLOGY

arXiv:math/0608389v2 [math.AT] 16 Aug 2006

MASSEY PRODUCTS IN GRADED LIE ALGEBRA COHOMOLOGY DMITRI MILLIONSCHIKOV Abstract. We discuss Massey products in a N-graded Lie algebra cohomology. One of the main examples is so-called ”positive part” L1 of the Witt algebra W . Buchstaber conjectured that H ∗ (L1 ) is generated with respect to non-trivial Massey products by H 1 (L1 ). Feigin, Fuchs and Retakh represented H ∗ (L1 ) by trivial Massey products and the second part of the Buchstaber conjecture is still open. We consider an associated graded algebra m0 of L1 with respect to the filtration by the its descending central series and prove that H ∗ (m0 ) is generated with respect to non-trivial Massey products by H 1 (m0 ).

Introduction In the last thirty years Massey products in cohomology have found a lot of interesting applications in topology and geometry. The existence of non-trivial Massey products in H ∗ (M, R) is an obstruction for a manifold M to be K¨ahler K¨ahler manifolds are formal [8], i.e. their real homotopy types are completely determined by their real cohomology algebras. In their turn formal spaces have trivial Massey products. An important feature of Massey products is the following: a blow-up of a symplectic manifold M along its submanifold N inherits non-trivial Massey products from M to N [3]. This idea was used by McDuff [18] in her construction of simply connected symplectic manifold with no K¨ahler structures. The Massey products and examples of symplectic manifolds with no K¨ahler structures was the subject in [7]. Babenko and Taimanov considered an interesting family of symplectic nilmanifolds related to graded finite-dimensional quotients of a ”positive part” L1 of the Witt algebra W . Applying the symplectic blow-up procedure they constructed examples of simply connected non-formal symplectic 1991 Mathematics Subject Classification. 55S30, 17B56, 17B70, 17B10. Key words and phrases. Massey products, graded Lie algebras, formal connection, Maurer-Cartan equation, representation, cohomology. The research of the author was partially supported by grants RFBR 05-01-01032 and ”Russian Scientific Schools”. 1

2

DMITRI MILLIONSCHIKOV

manifolds in dimensions ≥ 10 [2]. In fact it is possible to use another graded Lie algebra m0 instead of L1 [20]. A few time ago an 8-dimensional example was constructed by Fernandez and Mu˜ noz [11] using another techniques. An initial data of almost all examples of non-formal symplectic manifolds are nilmanifolds related to positively graded Lie algebras and some non-trivial triple Massey products in their cohomology. The present article is devoted to the study of n-fold calssical Massey products in the cohomology of N-graded Lie algebras. Although we started our introduction with finite-dimensional examples related to some nilmanifolds we will focuss our attention to two infinite dimensional N-graded Lie algebras L1 and its associated graded (with respect to the filtration by the ideals C k L1 of the descending central series) Lie algebra grC L1 ∼ = m0 . The reason of this interest comes from the relation of the algebra L1 to the Landweber-Novikov algebra in the complex cobordisms theory that was discovered by Buchstaber and Shokurov in 70-th [5]. In that time the algebra L1 attracted a lot of researchers [13] and the computation of its cohomology by Goncharova [14] is one of the most technicaly complicated results in homology algebra. The cohomology algebra H ∗ (L1 ) has a trivial multiplication. Buchstaber conjectured that the algebra H ∗ (L1 ) is generated with respect to the Massey products by H 1 (L1 ), moreover all corresponding Massey products can be chosen non-trivial. The first part of Buchstaber’s conjecture was proved by Feigin, Fuchs and Retakh [10]. But they represented the cohomology classes from H ∗ (L1 ) by means of trivial Massey products. Twelve years later Artel’nykh represented some of Goncharova’s cocycles by non-trivial Massey products, but his brief article does not contain the proofs. Thus one may conclude that the original Buchstaber conjecture is still open. We recall the necessary information on the cohomology of graded Lie algebras in the first Section and study two main examples H ∗ (L1 ) [14] and H ∗ (m0 ) [12] in the Section 2. In the Section 3 we present May’s approach to the definition of Massey products, his notion of formal connection developped by Babenko and Taimanov in [3] for Lie algebras, we introduce also the notion of equivalent Massey products. The analogy with the classical Maurer-Cartan equation is especially transparent in the case of Massey products of 1-dimensional cohomology classes hω1 , . . . , ωn i. The relation of this special case to the representations theory was discovered in [10], [9]. We discuss it in the Section 4. Following [10] we consider Massey products hω1 , . . . , ωn , Ωi, where ω1 , . . . , ωn are closed 1-forms and Ω is a closed q-form. These Massey

MASSEY PRODUCTS IN LIE ALGEBRA COHOMOLOGY

3

products are related to the differentials of Feigin-Fuchs spectral sequence [10]. The main result of the present article is the Theorem 5.7 stating that the cohomology algebra H ∗ (m0 ) is generated with respect to the non-trivial Massey products by H 1 (m0 ). Another important result is the Theorem 5.5 that contains a list of equivalency classes of trival Massey products of 1-cohomology classes. We show that it is related to Benoist’s classification [4] of indecomposable finite-dimensional thread modules over the Lie algebra m0 . 1. Cohomology of N-graded Lie algebras Let g be a Lie algebra over K and ρ : g → gl(V ) its linear representation (or in other words V is a g-module). We denote by C q (g, V ) the space of q-linear skew-symmetric mappings of g into V . Then one can consider an algebraic complex: d

d

d

dq−1

dq

0 1 2 V −−− → C 1 (g, V ) −−− → C 2 (g, V ) −−− → . . . −−−→ C q (g, V ) −−−→ . . .

where the differential dq is defined by: (1) (dq f )(X1 , . . . , Xq+1 ) =

q+1 X

ˆ i , . . . , Xq+1 ))+ (−1)i+1 ρ(Xi )(f (X1 , . . . , X

i=1

+

X

(−1)

i+j−1

ˆ i, . . . , X ˆ j , . . . , Xq+1 ). f ([Xi, Xj ], X1 , . . . , X

1≤i0 gα be a N-graded Lie algebra and V = ⊕β Vβ is a Z-graded g-module. One can define a decreasing filtration F of (C ∗ (g, V ), d): F 0C ∗ (g, V ) ⊃ · · · ⊃ F q C ∗ (g, V ) ⊃ F q+1 C ∗ (g, V ) ⊃ . . . where the subspace F q C p+q (g, V ) is spanned by p+q-forms c in C p+q (g, V ) such that M c(X1 , . . . , Xp+q ) ∈ Vα , ∀X1 , . . . , Xp+q ∈ g. α=q

The filtration F is compatible with d. Let us consider the corresponding spectral sequence Erp,q : Proposition 1.5 ([13],[10]). E1p,q = Vq ⊗ H p+q (g). Proof. We have the following natural isomorphisms: (2)

C p+q (g, V ) = V ⊗ Λp+q (g∗ ) E0p,q = F q C p+q (g, V )/F q+1 C p+q (g, V ) = Vq ⊗ Λp+q (g∗ ).

p,q p+1,q Now the proof follows from the formula for the dp,q : 0 : E0 → E0

d0 (v ⊗ f ) = v ⊗ df, where v ∈ V, f ∈ Λp+q (g∗ ) and df is the standard differential of the cochain complex of g with trivial coefficients.  2. Cohomology H ∗ (L1 ) and H ∗ (m0 ) Theorem 2.1 (Goncharova,[14]). The Betti numbers dimH q (L1 ) = 2, for every q ≥ 1, more precisely  2 1, if k = 3q 2±q , q dimHk (L1 ) = 0, otherwise.

6

DMITRI MILLIONSCHIKOV

q We will denote in the sequel by g± the generators of the spaces 2 q 3q ±q H 3q2 ±q (L1 ). The numbers 2 are so called Euler pentagonal num2

bers. A sum of two arbitrary pentagonal numbers is not a pentagonal number, hence the algebra H ∗ (L1 ) has a trivial multiplication. 1 1 = [e1 ] and g+ = [e2 ]; Example 2.2. 1) H 1 (L1 ) is generated by g− 2 2 2) the basis of H 2 (L1 ) consists of two classes g− = [e1 ∧e4 ] and g+ = 2 5 3 4 [e ∧e − 3e ∧e ] of weights 5 and 7 respectively.

The cohomology algebra H ∗ (m0 ) was calculated by Fialowski and Millionschikov in [12]. It were introduced two operators in [12]: 1) D1 : Λ∗ (e2 , e3 , . . . ) → Λ∗ (e2 , e3 , . . . ), (3)

D1 (e2 ) = 0, D1 (ei ) = ei−1 , ∀i ≥ 3, D1 (ξ ∧ η) = D1 (ξ) ∧ η + ξ ∧ D1 (η), ∀ξ, η ∈ Λ∗ (e2 , e3 , . . . ).

2) and its right inverse D−1 : Λ∗ (e2 , e3 , . . . ) → Λ∗ (e2 , e3 , . . . ), X i i+1 i (−1)l D1l (ξ)∧ei+1+l , D e = e , D (ξ∧e ) = −1 −1 (4) l≥0

where i ≥ 2 and ξ is an arbitrary form in Λ∗ (e2 , . . . , ei−1 ). The sum in the definition (4) of D−1 is always finite because D1l decreases the second grading by l. For instance, i

k

D−1 (e ∧ e ) =

i−2 X

(−1)l ei−l ∧ek+l+1.

l=0

Proposition 2.3. The operators D1 and D−1 have the following properties: dξ = e1 ∧ D1 ξ, e1 ∧ ξ = dD−1 ξ, D1 D−1 ξ = ξ,

ξ ∈ Λ∗ (e2 , e3 , . . . ).

Theorem 2.4 ([12]). The infinite dimensional bigraded cohomology H ∗ (m0 ) = ⊕k,q Hkq (m0 ) is spanned by the cohomology classes of e1 , e2 and of the following homogeneous cocycles: X (5) ω(ei1 ∧ . . . ∧eiq ∧eiq +1 ) = (−1)l D1l (ei1 ∧ · · · ∧ eiq ) ∧ eiq +1+l , l≥0

where q ≥ 1, 2 ≤ i1