# 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 ...
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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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### Corestriction map in lie algebra cohomology

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(\...
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