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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Proof of $H^2(\mathfrak g_1,\mathfrak g_1)=0$
I want to prove $H^2(\mathfrak g_1,\mathfrak g_1)=0$ where $\mathfrak g_1=\mathbb R$, the $1$-dimensional (abelian) Lie algebra.
I think I need to show two things:
All central extensions $\mathbb R \...
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Does $\mathbb R \overset{\iota}{\hookrightarrow} \mathfrak{aff}(1,\mathbb R) \overset{\pi}{\twoheadrightarrow} \mathbb R$ Lie algebra extension exist?
I think I have an example for a $\mathbb R \overset{\iota}{\hookrightarrow} \mathfrak{aff}(1,\mathbb R) \overset{\pi}{\twoheadrightarrow} \mathbb R$ Lie algebra extension.
$$\iota:\mathbb R\to \...
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What is the second Chevalley-Eilenberg cohomology group of the Lie algebra of $U(1)$?
I am trying to understand a comment on one of my questions in Physics Stackexchange.
The comment itself is this:
$U(1)$ has the second Chevalley-Eilenberg cohomology group trivial, hence by Bargmann'...
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Adjoint representation of a Lie group in homology of its lie algebra
It is known that a (Lie) group acts trivially (by identity) in its homology. It is also known that Lie algebra acts trivially (by zero endomorphisms) in its homology. Can we say that a Lie group acts ...
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Finite dimensional Lie algebras with trivial homology
Are there any known examples of Lie algebras $\mathfrak{g}$ such $H_*(\mathfrak{g})=0$ for all $*>0?$ Better still are there such algebras that this condition holds for arbitrary coefficient ...
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2-cocycles in Lie group vs Lie algebra cohomology (context of projective reps)
I'm confused by the relationship between the cocycle condition in Lie algebras vs Lie groups.
For Lie groups, a 2-cocycle is defined (e.g. here) as a map $\Phi : G \times G \rightarrow \mathbb{F}$ ...
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What's the advantage of defining Lie algebra cohomology using derived functors?
The way I learned Lie algebra cohomology in the context of Lie groups was a direct construction: one defines the Chevalley-Eilenberg complex as
$$C^p(\mathfrak g; V) := \operatorname{Hom}(\bigwedge^p \...
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What is the (co)homology of a free (graded) Lie algebra?
In characteristic $0$, what is the Chevalley-Eilenberg (co)homology of a free (graded) Lie algebra?
Not the definition, but $H^i =$ ??
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Lie algebra cohomology and De Rham cohomology of a compact Lie group
I well know that another question with the same statement has already been published, but I want to ask somethings else.
In particular, here Background for Lie Algebra cohomology and de Rham ...
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Casimir operator and Weibel
On page 245 of Weibel's An introduction to homological algebra, he assigns the following as an exercise:
The image of $c_M$ in the matrix ring $\operatorname{End}_k(M)$ is $r/m$ times the identity ...
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Quillen cohomology of Lie algebras
Fix a base ring $k$ and $k$-Lie algebras $\mathfrak{s}$ and $\mathfrak{t}$, and consider the slice category $\mathfrak{s}/\mathrm{Lie}_k/\mathfrak{t}$. This is a category of universal algebras, so ...
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A small sign error?
I'm trying to do exercise 3.12 of the first chapter in Bourbaki's Lie algebras/groups book, but apparently there is a sign error... somewhere. Let me introduce the objects in question:
First, lets ...
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By what can we extend a Lie algebra?
This Wikipedia article talks about Lie algebra extension by a Lie-algebra, while this other artilce talks about extension by a module. This nLab article mensions central extensions by a ground field.
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Reference Request - Cohomology of the Lie algebra $\mathfrak{u}(n)$ over a finite field?
I'd like to find a reference for the following result: $H^*(\mathfrak{u}(n); \mathbb{F}_p) \cong E_{\mathbb{F}_p}(x_1, x_3, ... , x_{2n-1})$, i.e., that the cohomology of the Lie algebra $\mathfrak{u}(...
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Virasoro algebra question: Is there a two-surface in Diff($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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Kunneth Formula for Lie Algebra Cohomology
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^...
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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 ...
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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 ...
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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 $a\in\mathfrak g,c\in\mathbb C$...
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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. ...
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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
$$\...
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How to show that $(\Lambda^2(g))^g = H^2(g)$?
Let $g$ be a semisimple Lie algebra and $\Lambda^2(g) = g \wedge g \subset g \otimes g$ the exterior square of $g$. Consider the adjoint action of g on $g \wedge g$ and let
$$(\Lambda^2(g))^g = \{x \...
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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 ...
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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 ...
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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^...
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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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A Isomorphism between the extension group and cohomology group of Lie algebras
Within the book An introduction to homological algebra by Weibel, I am trying to prove the following isomorphism, but I am not sure this is true. But I really want to know how to prove or disprove it....
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