Lie algebra cohomology of infinite-dimensional modules

Contemporary Mathematics, 1996

We present some recent developments in the application of homological methods to the represention theory of finite dimensional restricted Lie algebras.

2021

A morphism Lie algebra is a triple (g, h, φ) consisting of two Lie algebras g, h and a Lie algebra homomorphism φ : g → h. We define representations and cohomology of morphism Lie algebras. As applications of our cohomology, we study some aspects of deformations, abelian extensions of morphism Lie algebras and classify skeletal morphism sh Lie algebras. Finally, we consider the cohomology of morphism Lie groups and find a relation with the cohomology of morphism Lie algebras. 2010 MSC classification: 17B55, 17B56.

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

arXiv: Differential Geometry, 2018

In this thesis, we introduce a new cohomology theory associated to a Lie 2-algebras and a new cohomology theory associated to a Lie 2-group. These cohomology theories are shown to extend the classical cohomology theories of Lie algebras and Lie groups in that their second groups classify extensions. We use this fact together with an adapted van Est map to prove the integrability of Lie 2-algebras anew.

Bulletin of the American Mathematical Society, 1987

Arxiv preprint arXiv:1108.2806, 2011

In this paper we aim to understand the category of stable-Yetter-Drinfeld modules over enveloping algebra of Lie algebras. To do so, we need to define such modules over Lie algebras. These two categories are shown to be isomorphic. A mixed complex is defined for a given Lie algebra and a stable-Yetter-Drinfeld module over it. This complex is quasi-isomorphic to the Hopf cyclic complex of the enveloping algebra of the Lie algebra with coefficients in the corresponding module. It is shown that the (truncated) Weil algebra, the Weil algebra with generalized coefficients defined by Alekseev-Meinrenken, and the perturbed Koszul complex introduced by Kumar-Vergne are examples of such a mixed complex.

2020

Lie algebra is one of the important types of algebras. Here, we studied lie algebra and discussed its properties. Moreover, we define the Dihedral homology of lie algebra and prove some relations among Hochschild, Cyclic and Dihedral homology of lie algebra. Finally, we prove the Mayer-vetories sequence on Dihedral homology of lie algebra and proved that HC(g,M) ∼=− HD(g,M) ⊕+ HD(g,M).

Journal of the Mathematical Society of Japan, 1994

Mathematics of the USSR-Sbornik, 1984

In this paper we establish a connection between the cohomology of a finite-dimensional modular Lie algebra and its finite-dimensional p-envelopes. We also compute the cohomology of Zassenhaus algebras and their minimal p-envelopes with coefficients in generalized baby Verma modules and in simple modules over fields of characteristic p > 2.

Advances in Mathematics, 1993

In the paper one-and two-dimensional cohomology is compared for finite and infinite nilpotent Lie algebras, with coefficients in the adjoint representation. It turns out that, because the adjoint representation is not a highest weight representation in infinite dimension, the considered cohomology shows basic differences. On my visit to M.I.T., B. Kostant asked the following question: What is the main difference between the cohomology of finite and infinite dimensional nilpotent Lie algebras with coefficients in the adjoint representation, at what points does the generalization of the finite dimensional situation fail? Understanding this difference is especially important as the nilpotent Lie algebra cohomology is very hard to compute and in both finite and infinite dimensional cases only the one-and two-dimensional cohomology is known so far.