Massey Products in Graded Lie Algebra Cohomology (original) (raw)

Cohomology of graded Lie algebras of maximal class

Journal of Algebra, 2006

It was shown by A. Fialowski that an arbitrary infinite-dimensional N-graded "filiform type" Lie algebra g= ∞ i=1 g i with one-dimensional homogeneous components g i such that [g 1 , g i ] = g i+1 , ∀i ≥ 2 over a field of zero characteristic is isomorphic to one (and only one) Lie algebra from three given ones: m 0 , m 2 , L 1 , where the Lie algebras m 0 and m 2 are defined by their structure relations: m 0 : [e 1 , e i ] = e i+1 , ∀i ≥ 2 and m 2 : [e 1 , e i ] = e i+1 , ∀i ≥ 2, [e 2 , e j ] = e j+2 , ∀j ≥ 3 and L 1 is the "positive" part of the Witt algebra. In the present article we compute the cohomology H * (m 0) and H * (m 2) with trivial coefficients, give explicit formulas for their representative cocycles and describe the multiplicative structure in the cohomology. Also we discuss the relations with combinatorics and representation theory. The cohomology H * (L 1

Massey brackets and deformations

Journal of Pure and Applied Algebra, 2001

A generalisation of Massey products in the cohomology of di erential graded Lie algebras is constructed. An application to formal deformations of Lie algebras is given. A similar construction for the associative case is considered.

Cohomology of nilpotent subalgebras of affine Lie algebras

Proceedings of the American Mathematical Society, 1994

We compute the cohomology of the maximal nilpotent Lie algebra of an affine Lie algebra g with coefficients in modules of functions on the circle with values in a representation space of g. These modules are not highest weight modules and are somewhat similar to the adjoint representation. Let g be a finite-dimensional semisimple Lie algebra, b a Bore1 subalgebra of g ,and n+ c b the maximal nilpotent ideal of b. The Bott-Kostant Theorem for Lie algebra cohomology is the following. Theorem [K]. Let V be an irreducible representation of g with dominant highest weight and n a maximal nilpotent subalgebra of g. Then dim Hi(n ; V) is equal to the number of elements of length i in the Weyl group of g. Consider the affine infinite-dimensional graded Lie algebra 6 = g @ @[t , t-'1 corresponding to g , with di = g @ ti. There are at least two analogues of the above Theorem for affine algebras. The most direct analogue is the following: if V is an irreducible representation of the current algebra 6 with dominant highest weight and A+ is a maximal nilpotent subalgebra of 6 , that is, then dimHi(ii+; V) is equal to the number of elements of length i in the Weyl group. This Theorem was proved by Garland in 1975 [GI and Garland and Lepowsky in 1976 (see [GL]). The proof is similar to that of the finitedimensional case. In this paper we present the proof of a different analogue of the Bott-Kostant Theorem obtained jointly with Feigin and announced in [FF]. Namely, we compute the cohomology of A+ with coefficients in modules of functions on the

Lie-Massey brackets and n-homotopically multiplicative maps of differential graded Lie algebras

Journal of Pure and Applied Algebra, 1993

For Alex Heller on his 65th birthday Retakh, V.S.. Lie-Massey brackets and n-homotopically multiplicative maps of differential graded Lie algebras, Journal of Pure and Applied Algebra 89 (1993) 217-229. The definition of rr-homotopically multiplicative maps of differential graded Lie algebras is given. It is shown that such maps conserve n-Lie-Massey brackets.

The Vanishing of the Low-Dimensional Cohomology of the Witt and the Virasoro algebra

arXiv: Rings and Algebras, 2017

A proof of the vanishing of the third cohomology group of the Witt algebra with values in the adjoint module is given. Moreover, we provide a sketch of the proof of the one-dimensionality of the third cohomology group of the Virasoro algebra with values in the adjoint module. The proofs given in the present article are completely algebraic and independent of any underlying topology. They are a generalization of the ones provided by Schlichenmaier, who proved the vanishing of the second cohomology group of the Witt and the Virasoro algebra by using purely algebraic methods. In the case of the third cohomology group though, extra difficulties arise and the involved proofs are distinctly more complicated. The first cohomology group can easily be computed; we will give an explicit proof of its vanishing in the appendix, in order to illustrate our techniques.

The Vanishing of Low-Dimensional Cohomology Groups of the Witt and the Virasoro algebra

2017

A proof of the vanishing of the first and the third cohomology groups of the Witt algebra with values in the adjoint module is given. The proofs given in the present article are completely algebraic and independent of any underlying topology. They are a generalization of the ones provided by Schlichenmaier, who proved the vanishing of the second cohomology group using purely algebraic methods. In the case of the third cohomology group though, extra difficulties arise and the involved proofs are distinctly more complicated.