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Massey Products in Graded Lie Algebra Cohomology ∗ MASSEY PRODUCTS IN GRADED LIE ALGEBRA COHOMOLOGY ¤ DMITRY MILLIONSCHIKOV y Department of Mechanics and Mathematics, Moscow State University, Leninskie Gory, 119992 Moscow, Russia E-mail: [email protected] 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 1 ¤ by H (L1). Feigin, Fuchs and Retakh represented H (L1) by trivial Massey prod- ucts and the second part of the Buchstaber conjecture is still open. We consider the associated graded algebra m0 of L1 with respect to the ¯ltration by its descend- ¤ ing central series and prove that H (m0) is generated with respect to non-trivial 1 Massey products by H (m0). 1. Introduction In the last thirty years cohomological Massey products 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 KÄahler.A KÄahlermanifold M is formal [8], i.e. its real homotopy type is completely determined by the cohomology algebra H¤(M; R). 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 along its submanifold inherits non-trivial Massey products [3]. This idea was used by McDu® [20] in her construction of a simply connected symplectic manifold with no KÄahlerstructure. The Massey products in cohomology of symplectic manifolds was the subject in [7]. Babenko and Taimanov considered an interesting family of symplectic ¤ MSC 2000: 55S30, 17B56, 17B70, 17B10. Keywords: Massey products, graded Lie algebras, formal connection, Maurer-Cartan equation, representation, cohomology. y Work supported by the grant RFBR 05-01-01032. 353 354 nilmanifolds related to ¯nite-dimensional quotients of a "positive part" L1 of the Witt algebra W . Applying to them the symplectic blow-up proce- dure they constructed examples of simply connected non-formal symplectic manifolds in dimensions ¸ 10 [2]. A few time ago an 8-dimensional ex- ample was constructed by Fernandez and Mu~noz[11] by means of another techniques. The present article is devoted to the study of n-fold classical Massey prod- ucts in the cohomology of N-graded Lie algebras. We will focuss our atten- tion to two main examples: L1 and m0. The in¯nite dimensional N-graded Lie algebra L1 is ¯ltered by the ideals of the descending central series and one can consider its associated graded Lie algebra m0 = grC L1. The reason of this interest comes from the relation of L1 to the Landweber-Novikov algebra in the complex cobordisms theory discovered by Buchstaber and Shokurov [5]. It follows from the Goncharova theorem [14] that the cohomology algebra ¤ H (L1) has a trivial multiplication. Buchstaber conjectured that the alge- ¤ 1 bra H (L1) is generated with respect to the Massey products by H (L1), moreover the corresponding Massey products can be chosen as non-trivial ones. The ¯rst part of Buchstaber's conjecture was proved by Feigin, Fuchs and Retakh [10]. However they represented Goncharova's basic cocycles of ¤ H (L1) using by trivial Massey products. Twelve years later Artel'nykh represented some of Goncharova's cocycles by non-trivial Massey products and unfortunately his brief article does not contain the proofs. Hence the original Buchstaber's conjecture is still open. It was pointed out to the author by May that it follows from the Corol- ¤ 1 lary 5.17 in [19] that the cohomology H (L1) is generated by H (L1) with respect to matric Massey products (generalized Massey products) and this property holds for some class of graded Lie algebras. The question of triv- iality or non triviality of corresponding matric Massey products have not been studied. We recall the necessary information on the cohomology of graded Lie al- ¤ gebras in the ¯rst Section and study two main examples H (L1) [14] and ¤ H (m0) [12] in the Section 3. In the Section 4 we present May's approach to the de¯nition of Massey products, his notion of formal connection de- veloped by Babenko and Taimanov [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 h!1;:::;!ni. The relation of 355 this special case to the representations theory was discovered in [10, 9]. We discuss it in the Section 5. The main result of the present article is the Theorem 6.4 stating that the ¤ cohomology algebra H (m0) is generated with respect to the non-trivial 1 Massey products by H (m0). Another important result is the Theorem 6.3 that presents a list of equivalency classes of trivial Massey products of 1- 1 cohomology classes from H (m0). We show that the Theorem 6.3 is related to Benoist's classi¯cation [4] of indecomposable ¯nite-dimensional thread modules over m0. 2. Cohomology of N-graded Lie algebras Let g be a Lie algebra over K and ½ : g ¡! gl(V ) its linear representation (or in other words V is a g-module). We denote by Cq(g;V ) the space of q-linear skew-symmetric mappings of g into V . Then one can consider an algebraic complex: V ¡¡¡¡!d0 C1(g;V ) ¡¡¡¡!d1 C2(g;V ) ¡¡¡¡!d2 d d ::: ¡¡¡¡!q¡1 Cq(g;V ) ¡¡¡¡!q ::: where the di®erential dq is de¯ned by: Xq+1 i+1 (dqf)(X1;:::;Xq+1) = (¡1) ½(Xi)(f(X1;:::; X^i;:::;Xq+1))+ i=1 X (1) i+j¡1 + (¡1) f([Xi;Xj];X1;:::; X^i;:::; X^j;:::;Xq+1): 1·i<j·q+1 The cohomology of the complex (C¤(g;V ); d) is called the cohomology of the Lie algebra g with coe±cients in the representation ½ : g ¡! V . The cohomology of (C¤(g; K); d)(V = K and ½ : g ¡! K is trivial) is called the cohomology with trivial coe±cients of the Lie algebra g and is denoted by H¤(g). 1 2 ¤ One can remark that d1 : C (g; K) ¡! C (g; K) of the (C (g; K); d) is the dual mapping to the Lie bracket [ ; ] : ¤2g ¡! g. Moreover the condition d2 = 0 is equivalent to the Jacobi identity for [; ]. De¯nition 2.1 A Lie algebra g is called N-graded, if it is decomposed to a direct sum of subspaces such that g = ©igi; i 2 N; [gi; gj] ½ gi+j; 8 i; j 2 N: 356 Example 2.1 The Lie algebra m0 is de¯ned by its in¯nite basis e1; e2; : : : ; en;::: with commutator relations: [e1; ei] = ei+1; 8 i ¸ 2: The algebra m0 has an in¯nite number of N-gradings: m0 = ©i2Nm0i; m01 = Span(e1; e2; : : : ; ek); m0i = Span(ei+k¡1); i ¸ 2: Remark 2.1 We always omit the trivial commutator relations [ei; ej] = 0 in the de¯nitions of Lie algebras. Example 2.2 Let us recall that the Witt algebra W is spanned by di®er- ential operators on the real line R1 with a ¯xed coordinate x d e = xi+1 ; i 2 Z; [e ; e ] = (j ¡ i)e ; 8 i; j 2 Z: i dx i j i+j We denote by L1 the positive part of the Witt algebra, i.e. L1 is a subal- gebra of W spanned by all ei; i ¸ 1. Obviously W is a Z-graded Lie algebra with one-dimensional homogeneous components: W = ©i2ZWi;Wi = Span(ei): Thus L1 is a N-graded Lie algebra. Let g be a Lie algebra. The ideals Ckg of the descending central sequence determine a decreasing ¯ltration C of the Lie algebra g C1g = g ⊃ C2g ⊃ ¢ ¢ ¢ ⊃ Ckg ⊃ ::: ;[Ckg;Clg] ½ Ck+lg: One can consider the associated graded Lie algebra M k k+1 grC g = (grC g)k; (grC g)k = C g=C g: k¸1 Proposition 2.1 We have the following isomorphisms: » » grC L1 = grC m0 = m0: Remark 2.2 (grC m0)1 = Span(e1; e2); (grC m0)i = Span(ei+1); i ¸ 2: Let g = ©®g® be a Z-graded Lie algebra and V = ©¯V¯ is a Z-graded g- ¤ module, i.e., g®V¯ ½ V®+¯: Then the complex (C (g;V ); d) can be equipped 357 q L q with the Z-grading C (g;V ) = ¹ C(¹)(g;V ), where a V -valued q-form c q belongs to C(¹)(g;V ) if and only if for X1 2 g®1 ;:::;Xq 2 g®q we have c(X1;:::;Xq) 2 V®1+®2+:::+®q +¹: This grading is compatible with the di®erential d and hence we have Z- grading in the cohomology: M q q H (g;V ) = H(¹)(g;V ): ¹2Z Remark 2.3 The trivial g-module K has only one non-trivial homogeneous component K = K0. The exterior product in ¤¤(g) induces a structure of a bigraded algebra in the cohomology H¤(g): q p q+p Hk (g) ^ Hl (g) ¡! Hk+l (g): Let g = ©®g® be a N-graded Lie algebra and V = ©¯V¯ is a Z-graded g-module. One can de¯ne a decreasing ¯ltration F of (C¤(g;V ); d): F 0C¤(g;V ) ⊃ ¢ ¢ ¢ ⊃ F qC¤(g;V ) ⊃ F q+1C¤(g;V ) ⊃ ::: where the subspace F qCp+q(g;V ) is spanned by p+q-forms c in Cp+q(g;V ) such that M c(X1;:::;Xp+q) 2 V®; 8X1;:::;Xp+q 2 g: ®¸q The ¯ltration F is compatible with d.
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