# Questions tagged [lie-algebra-cohomology]

Lie algebra cohomology is a cohomology theory for Lie algebras. It can be used to study the topology of Lie groups and homogeneous spaces by relating cohomological methods of Georges de Rham to properties of the Lie algebra.

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### Explicit calculations of Lie algebra homology and cohomology

Are there any textbooks/articles etc. which go through some explicit examples calculating Lie algebra homology and cohomology with coefficients in some representation (for example, the defining ...
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### Do Chevalley-Eilenberg homology functor and taking the cohomology commute?

I've stumbled upon some ideas from homological algebra that I'm trying to piece together from a talk I heard. I don't have much background in this area, so I'm not sure if this is a reasonable thing ... 1 vote
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### Isomorphism of D-modules implied by vanishing of first Lie algebra cohomology

I'm reading the paper Invariant differential operators on a reductive Lie algebra and Weyl group representations by Nolan Wallach. First I recall the notation used in the paper: $V$ is a finite ...
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### Virasoro algebra question: Is there a two-surface in Diff($S^1$) with a non-zero integral over the cocycle in $H^2(\mathfrak{g}, \mathbb{C})$?

I am a physicist so forgive me if this question doesn't make sense. You can start off by defining the Witt algebra, which I'll call $\mathfrak{g}$, as the complexified Lie algebra of vector fields on ...
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Is there a Kunneth Formula for Lie algebra cohomology? Specifically, I want the second cohomology of the Lie algebra $\mathbb{R}\oplus \mathfrak{sl}(2,\mathbb{C})$ to vanish. I would like to say $H^... 4 votes 0 answers 181 views ### Complexes of left-invariant and bi-invariant forms and their cohomology It is well known that for compact and connected Lie groups, one can calculate the de Rham cohomology$H^{k}(G)$using either the subcomplex of left-invariant forms, which leads to the calculation of ... 4 votes 1 answer 196 views ### de Rham counterpart of Lie algebra cohomology valued in arbitrary module Disclaimer: Google search produces a lot of references closely related to my question so I think it should be easy for me to find out the answer on my own, unfortunately due to a complete lack of ... 0 votes 1 answer 99 views ### How does$\mathfrak g$act on$\mathbb C$in the context of Lie algebra cohomology? In this answer, it is pointed out that in the context of Lie algebra cohomology, the Lie algebra$\mathfrak g$acts trivially on$\mathbb C$, i.e. $$a\cdot c=0$$ for all$a\in\mathfrak g,c\in\mathbb C$... 6 votes 1 answer 464 views ### Difference between Koszul and Chevalley-Eilenberg complexes Please have a look at these two definitions: Chevalley-Eilenberg complex Koszul complex (German Wikipedia) Both are from Wikipedia pages on Lie algebra cohomology, and they look rather similar. ... 2 votes 2 answers 808 views ### Central extension of a Lie algebra, why is the bilinear form a 2-cocycle? My professor talked about a central extension of a Lie algebra$\mathfrak g$, which he defined as$\tilde{\mathfrak{g}}=\mathfrak g\oplus\mathbb C$. The Lie bracket on$\tilde{\mathfrak{g}}$is $$\... 0 votes 0 answers 45 views ### How to show that (\Lambda^2(g))^g = H^2(g)? Let g be a semisimple Lie algebra and \Lambda^2(g) = g \wedge g \subset g \otimes g the exterior square of g. Consider the adjoint action of g on g \wedge g and let$$(\Lambda^2(g))^g = \{x \... 3 votes 0 answers 254 views ### Does the adjoint action induce a trivial action on Lie algebra cohomology? Let$k$be an algebraically closed field and$\Gamma$a finite group.$\Gamma$acts on itself via conjugation, and it is true that the induced action on the cohomology algebra$H^{*}(\Gamma,k)$is ... 3 votes 0 answers 118 views ### coboundary operators in relative lie algebra cohomology I am starting to read relative lie algebra cohomology. We define the coboundary operator$d$from$Hom_K(\wedge^q\mathcal{g}/\mathcal{k}, V)$to$Hom_K(\wedge^{q+1}\mathcal{g}/\mathcal{k}, V)$as ... 5 votes 0 answers 206 views ### duality for (co)homology of Lie algebras Let$R$be a commutative ring and$\mathfrak{g}$a Lie$R$-algebra that has an$R$-module basis with$n$elements. What is the relationship between$H_k(\mathfrak{g};R)$,$H_{n-k}(\mathfrak{g};R)$,$H^...
Given a lie algebra $\mathfrak{g}$ over a field $k$, we can define the cohomology groups of $\mathfrak{g}$ as follows: $$H^n(\mathfrak{g},k):=\mathrm{Ext}_{U(\mathfrak{g})}^n(k,k)$$ where \$U(\...