# Questions tagged [lie-algebra-cohomology]

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119
questions

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votes

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4k views

### Cohomology of Lie groups and Lie algebras

The length of this question has got a little bit out of hand. I apologize.
Basically, this is a question about the relationship between the cohomology of Lie groups and Lie algebras, and maybe ...

**24**

votes

**1**answer

2k views

### History of Koszul complex

This is a question about the history of commutative algebra. I'm curious why the Koszul complex from commutative algebra is called the Koszul complex? All of Koszul's early papers are about Lie ...

**19**

votes

**3**answers

5k views

### Chevalley Eilenberg complex definitions?

In Weibel's An Introduction to Homological Algebra, the Chevalley-Eilenberg complex of a Lie algebra $g$ is defined as $\Lambda^*(g) \otimes Ug$ where $Ug$ is the universal enveloping algebra of $g$. ...

**17**

votes

**2**answers

661 views

### Generators of the cohomology of a Lie algebra

Fix a characteristic zero ground field. One can easily check that if $\mathfrak g$ is a simple Lie algebra, then the trilinear map map $\omega$ given by $$\omega(x,y,z)=B([x,y],z),$$ with $B$ the ...

**14**

votes

**2**answers

2k views

### Compute Lie algebra cohomology

Is there a computer algebra system that is able to compute the Lie algebra cohomology in a given representation? What if the Lie algebra is finite dimensional?
In my case I would like to be able to ...

**14**

votes

**2**answers

709 views

### Projective modules over quantum groups

My question is short:
How can one calculate $\operatorname{Tor}_{U_q(\mathfrak g)}(k,k)$?
($k$ is the ground field of characteristic zero).
If we had a regular universal enveloping algebra $U(\...

**14**

votes

**0**answers

253 views

### Gel'fand and Fuks' “Globalizing” of cohomology of formal vector fields

I apologize for the length of this question. If anybody already spent some time with cohomology of (formal) vector fields and the results of Gel'fand and Fuks, I imagine a lot can be skipped. I do ...

**13**

votes

**2**answers

1k views

### Torsion for Lie algebras and Lie groups

This question is about the relationship (rather, whether there is or ought to be a relationship) between torsion for the cohomology of certain Lie algebras over the integers, and torsion for ...

**12**

votes

**2**answers

791 views

### What's the most simple proof of Kostant's version of Borel-Weil-Bott for Lie Algebra cohomology?

Besides Kostant's original proof (in http://www.math.tamu.edu/~jml/kostant61.pdf) of the above mentioned theorem (using the Lie Algebra Laplacian), there are a few other approaches:
Casselman-Osborne,...

**10**

votes

**0**answers

78 views

### Non-linear version of the Chevalley–Eilenberg complex

Let $\mathfrak{g}$ be a finite-dimensional Lie algebra. In small degrees, the differentials of the Chevalley–Eilenberg complex $C^\bullet(\mathfrak{g}, \mathfrak{g})$ with values in the adjoint ...

**9**

votes

**3**answers

512 views

### Invariants of exterior powers

Let $\mathfrak{g}$ be the Lie algebra of $GL(n,\mathbb{R})$. Let $\theta(X) = - X^T$ be the Cartan involution on $\mathfrak{g}$; it induces decomposition as $\mathfrak{g} = \mathfrak{k} \oplus \...

**9**

votes

**3**answers

1k views

### Algebraic/Categorical motivation for the Chevalley Eilenberg Complex

Is there a purely algebraic or categorical way to introduce the Chevalley-Eilenberg complex in the definition of Lie algebra cohomology?
In group cohomology, for example, the bar resolution of a ...

**9**

votes

**2**answers

637 views

### Lyndon–Hochschild–Serre spectral sequence for not normal subgroup

Is there analog of Lyndon–Hochschild–Serre spectral sequence for not normal subgroup?
What can you say about it? Can you describe $E^{p, q}_1$ ? What is about $E^{p, q}_2$?
What is the best technique ...

**9**

votes

**1**answer

649 views

### invariant symmetric bilinear forms and Lie algebra cohomology

What are the most general conditions on a Lie algebra $\mathfrak{g}$ over a field $\mathbb{k}$ such that the space of invariant symmetric bilinear forms is isomorphic to $H^3(\mathfrak{g},\mathbb{k})$?...

**9**

votes

**3**answers

905 views

### What is a Homotopy between $L_\infty$-algebra morphisms II

I would like to continue on question I asking, what is a homotopy between
Lie infinity algebras, since I'm not satisfied in two directions:
1.) The naive approach to define a homotopy would be ('...

**9**

votes

**1**answer

229 views

### Matsushima-Murakami Isomorphism for $L^2$-cohomology

Let $\mathbf{G}$ be a reductive connected linear algebraic group over a totally real global number field, say $\mathbb{Q}$. Let $\mathbb{A}=\mathbb{R}\times\mathbb{A}_f$ be the ring of rational adele.
...

**8**

votes

**3**answers

754 views

### Poincaré 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.
In Algebra, Geometry, and Software Systems by Joswig & Takayama on p.200, it ...

**8**

votes

**2**answers

933 views

### Relative Lie Algebra cohomology and sheaf cohomology

(I apologize in advance for the vagueness of my question). Let $G$ be a reductive algebraic group over $\mathbb C$ with Lie algebra $\frak g$ and Borel $B$. I have seen casual references to the fact ...

**8**

votes

**2**answers

760 views

### Can Lie algebra cohomology prove Cartan's Semisimplicity Criterion?

Here is what I mean by "Cartan's semisimplicity criterion":
Let $\mathfrak g$ be a finite-dimensional Lie algebra over a field of characteristic $0$. Assume that the center of $\mathfrak g$ is ...

**8**

votes

**2**answers

358 views

### Relation between Lie Algebra Cohomology and Number of Relations of a Cyclic Module?

Let $\mathfrak{g}$ be a finite dimensional Lie algebra over $k$, let $U$ be its enveloping algebra, and let $M$ be a $\mathfrak{g}$-module (not necessarily finite dimensional). Call the invariant ...

**8**

votes

**1**answer

276 views

### Is this sequence of Lie algebra cohomology a part of spectral sequence?

There is an exact sequence
$$0 \to H^2(\mathfrak{g}, k) \to H^1(\mathfrak{g}, \mathfrak{g}^*) \to H^0(\mathfrak{g}, S^2\mathfrak{g}) \xrightarrow{d} H^3(\mathfrak{g}, k) \to H^2(\mathfrak{g}, \...

**8**

votes

**1**answer

180 views

### Simple identity on Lie algebras in a note of Koszul

In a 1947 Comptes Rendus note (T224, p. 448), Koszul makes the following claim (paraphrased, hopefully correctly), which seems like it should have a simple proof I am missing.
Given a compact, ...

**8**

votes

**0**answers

138 views

### Chevalley-Eilenberg cohomology of polynomial vector fields on $\mathbb{A}^2$

I have a question similar to one given here.
What is the cohomology of the Lie algebra of polynomial vector fields on an affine space $\mathbb{A}^2$ over a field of characteristic $0?$ (I'm interested ...

**8**

votes

**0**answers

222 views

### Homology of an interesting Lie algebra

Let $U$ and $V$ be finite dimensional complex vector spaces (or perhaps graded vector spaces).
Let $E(U)$ be the "square zero extension" $\mathbb C \oplus U$, made into a commutative ring in such a ...

**7**

votes

**4**answers

888 views

### lists of computed cohomologies?

Is there any comprehensive list of examples for computed
1) de-Rham cohomology-groups
2) Lie-algebra-cohomology groups $H^i(\mathfrak{g},\mathbb{R})$
3) equivariant de-Rham cohomology groups ?
...

**7**

votes

**4**answers

633 views

### Is a quasi-iso in Lie algebra cohomology necessarily an iso?

Let $\mathfrak g$ be a Lie algebra (if it matters, right now I only care about finite-dimensional Lie algebras in characteristic $0$, although I'm never opposed to hearing about more general cases). ...

**7**

votes

**2**answers

1k views

### Whitehead lemmas in Lie algebra cohomology for non-algebraically closed fields

I read in Weibel's homological algebra that Whitehead's first and second lemmas are true for any characteristic 0 field. I mean the following:
Whitehead Lemma(s): Let g be a semisimple Lie algebra ...

**7**

votes

**2**answers

1k views

### What is the (Koszul? derived?) interpretation of a pair of Lie algebras with the same cohomology?

There are many words and sentences in mathematics that I basically completely don't understand, including the words "Koszul" and "derived". But rather than ask for a complete description of such ...

**7**

votes

**1**answer

186 views

### Is the Chevalley-Eilenberg cohomology the only interesting cohomology for Lie algebra?

When talking about the cohomology space of a Lie algebras, it comes naturally to refer to the Chevalley-Eilenberg cohomology, is there other interesting type of cohomology for Lie algebra?

**7**

votes

**1**answer

356 views

### Is Nijenhuis–Richardson bracket a BV bracket?

Let $g$ be a finite dimensional Lie algebra, and let me denote $A=(\bigwedge g^* \otimes g, d)$ the Chevalley-Eilenberg complex that calculates cohomology of the Lie algebra with coefficients in the ...

**7**

votes

**1**answer

817 views

### Hochschild (co)homology and representation theory

Dear members of Mathoverflow,
I just discovered the notion of Hochschild (co)homology. I understand well the formalism however I am wondering about the meaning of this (co)homology for representation ...

**7**

votes

**0**answers

210 views

### Induction from the Borel subalgebra to BGG category $\mathcal{O}$

Let $\mathfrak{g}$ be a finite dimensional complex semisimple Lie algebra with a choice of Cartan subalgebra $\mathfrak{h}$, Borel $\mathfrak{b}$, and nilpotent radical $\mathfrak{n}$. Let $\mathcal{O}...

**6**

votes

**1**answer

190 views

### Derivations of universal enveloping algebra of Lie algebras

We know a lot about derivations of Lie algebra. However, for the universal enveloping algebra of Lie algebra, we have few references about it.
My question: describing the derivations of enveloping ...

**6**

votes

**0**answers

107 views

### Cyclic version of Lie algebra cohomology

Lie algebra cochains have a differential $d$ where $d^2 =0$ because of the Jacobi identity, which can be written in the cyclic form or the Leibniz form. $L_\infty$ algebra cochains have a ...

**6**

votes

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105 views

### Are two Lie algebra deformations with cohomologous tangents isomorphic?

Let $(\mathfrak{g},[-,-])$ be a finite-dimensional Lie algebra. I'm interested in real Lie algebras, but feel free to complexify if this helps.
Say I'm interested in classifying isomorphism ...

**6**

votes

**0**answers

179 views

### vanishing of Lie algebra cohomology with coefficients in an infinite-dimensional module

Let $G$ be a real semisimple Lie group, $K$ its maximal compact subgroup, $\mathfrak g, \mathfrak k$ the corresponding Lie algebras. Let $V$ be a locally convex, Hausdorff vector space, which is a $G$-...

**6**

votes

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372 views

### Lagrangian (classical) BRST cohomology groups

I'm trying to understand BRST complex in its Lagrangian incarnation i.e. in the form mostly closed to original Faddeev-Popov formulation. It looks like the most important part of that construction (...

**6**

votes

**0**answers

198 views

### The meaning of a “subcomplex” of the Cartan-Eilenberg of a Lie algebra

Let $\mathfrak{g}$ be a finite dimensional real Lie algebra, and $\mathfrak{g}^* $ be the dual vector space. We have the standard Cartan-Eilenberg complex
$(\wedge^{\cdot} \mathfrak{g}^* ,\text{d}_{...

**5**

votes

**2**answers

944 views

### Hochschild and cyclic cohomology of commutative algebra?

I'm wondering how to compute the Hochschild cohomology ${\rm HH}^n(A,A)$ and the cyclic cohomology ${\rm HC}^n(A)$ where $A = \mathbb{C}[t_i]$ is the algebra of polynomials in a finite set of ...

**5**

votes

**2**answers

388 views

### computing second cohomology $H^2(O_a,\mathbb{Z})\cong \mathbb{Z}^n$ of a generic coadjoint orbit

Let $G$ be a compact , connected and simply connected Lie group and $a\in\mathfrak{g}^*$ (dual of Lie algebra of Lie group $G$). Then let $O_a$ be a generic coadjoint orbit then can we say $H^2(O_a,\...

**5**

votes

**1**answer

151 views

### Equivariant cohomology of a semisimple Lie algebra

Suppose $\mathfrak{g}$ is a real Lie algebra integrating to the connected Lie group $G$. One may consider the $G$-equivariant cohomology of $\mathfrak{g}$ ($\mathfrak{g}^*$) where the $G$-action is ...

**5**

votes

**0**answers

220 views

### Relation of BRST model of equivariant cohomology and BRST cohomology?

I'm now reading Kalkman's paper "BRST model for equivariant cohomology and representation for equivariant Thom class". And I've seen his definition for BRST model is
$B=W(\mathfrak{g})\otimes \...

**5**

votes

**0**answers

239 views

### Hochschild cohomology of SU(2)

I have a question about the computation of an Hochschild Cohomology. Or at least about a space which really looks like a cohomology space. All the functions i consider are assumed to be smooth.
Let's ...

**5**

votes

**1**answer

449 views

### What is the Schouten bracket for the Chevalley-Eilenberg complex with coefficients in a nontrivial module?

Let $\mathfrak g$ be a Lie algebra. The Chevalley-Eilenberg complex is defined to be $\wedge^* \mathfrak g$ with differential $d\colon \wedge^* \mathfrak g\to \wedge^{*-1}\mathfrak g$ defined by $$d(...

**4**

votes

**4**answers

394 views

### Homology of solvable (nilpotent) Lie algebras

Let $\mathfrak{g}$ be a solvable Lie algebra over $\mathbb{C}$ and $\lambda\in(\mathfrak{g}/[\mathfrak{g},\mathfrak{g}])^*$ be a character of $\mathfrak{g}$. I'm interested in calculating homology for ...

**4**

votes

**2**answers

216 views

### Reference request: Projective representations of a simply connected real semisimple Lie group lift to unitary representations

I recently got interested in representation theory in quantum mechanics and I read the following theorem:
Let $G$ be a simply-connected Lie group with $H^2(\mathfrak{g},\mathbb{R})=0$ and let $\...

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votes

**2**answers

304 views

### Lie algebra cohomology over non-fields

This is probably a very elementary question. I'm trying to get an explicit description of the cochain complex and coboundary maps for Lie algebra cohomology over $\mathbb{Z}$, and more generally, over ...

**4**

votes

**1**answer

196 views

### cohomological representations of GL(N)

I am trying to understand cohomology of $G := GL(N)$. For this I need to understand representations of $G(\mathbb{R})$ with nontrivial $(\mathfrak{g},K_\infty)$-cohomology, where $\mathfrak{g}$ is the ...

**4**

votes

**1**answer

164 views

### What is known about the morphism $H^*_{Lie}(L,L)\to H^*_{Lie}(L,UL)$ induced by $L\hookrightarrow UL$

Let $L$ be a (differential) graded Lie algebra over a field $k$ of characteristic 0, and let $UL$ be the universal enveloping algebra of $L$.
The inclusion $L\hookrightarrow UL$ induces a morphism of ...

**4**

votes

**1**answer

659 views

### Restriction map for Lie algebra/Lie group cohomology associated to a complex semisimple Lie algebra and a semisimple Lie-subalgebra

Let $\mathfrak{g}$ be a finite-dimensional complex semisimple Lie algebra (or the corresponding Lie group). For definiteness, I'll take $\mathfrak{g}$ to be of type $A_n$, that is, $\mathfrak{g} = \...