Questions tagged [lie-algebra-cohomology]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
4
votes
0answers
52 views

Lie algebra “semi” coinvariants

In the process of my research, I've come across the need to understand the following construction: Let $\mathfrak{g}$ be a (finite-dimensional) complex Lie algebra, $\beta\in \mathfrak{g}^*$ a Lie ...
7
votes
1answer
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?
2
votes
0answers
85 views

Rational cohomology of p-adic general linear groups

I wanted to compute the cohomology ring $H^*(GL_n(\mathbb{Z}_p); \mathbb{Q}_p)$ (with $p$ fixed prime as usual). I found some incomplete notes stating that the computation should go as follows. First ...
8
votes
0answers
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 ...
10
votes
0answers
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 ...
5
votes
1answer
152 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 ...
3
votes
0answers
56 views

Family of Lie algebras parametrized by a discrete valuation ring

I have a family of Lie algebras parametrized by a discrete valuation ring, whose generic fiber is reductive and whose special fiber is nilpotent. I'd like to learn about the relationship between the ...
4
votes
0answers
169 views

Cohomology and higher structures

Classically, cohomologies of Lie groups/algebras parametrize extensions. To be precise, given an linear $G$-action on $M$, there is an bijection between $H^2(G;M)$ and the set of extension $E$ of $G$ ...
1
vote
1answer
77 views

Lie algebra cohomology: $H^i(R,V)=H^i(R,V^R)$ with $R$ reductive and $V$ an $R$-module

Let $R$ be a reductive, finite-dimensional Lie algebra over a field of characteristic 0, and let $V$ be a semisimple $R$-module (also finite dimensional). I have seen a reference to the fact that $H^i(...
2
votes
0answers
71 views

Abelian lie algebra homology

Let $\mathfrak g$ be an abelian Lie algebra over $\mathbb Z.$ We can consider its Lie-algebra homology, say as $\mathrm{Tor}^{U(\mathfrak g)}_*(\mathbb Z,\mathbb Z)$ and its group homology as $\mathrm{...
1
vote
0answers
35 views

Regarding linear splitting of lie algebra morphism and their CE complexes

The main question here is to ask if anyone has ever seen/researched the map $\alpha$ below and get a reference regarding it. Also, if anyone tells me the equivalent conditions to $\alpha=0$, I would ...
1
vote
0answers
26 views

How is the product structure induced on Lie algebra homology of matrices?

I have been looking at the Chevalley Eilenberg complex $CE_*(\mathfrak g)$ of a Lie algebra $\mathfrak g$ over a field $k$. $$ \wedge^3\mathfrak g\longrightarrow \wedge^2\mathfrak g\longrightarrow \...
6
votes
1answer
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 ...
0
votes
1answer
89 views

Lie algebra cohomology with values in injective module

I am looking for a proof of the following result: Let $\mathfrak{g}$ be a Lie algebra and $I$ an injective $\mathfrak{g}$-module. Then $\mathrm{H}^q(\mathfrak{g},I)=0$ $\forall q>0$. More precisely,...
3
votes
2answers
207 views

Universal central extension of Lie algebras

In the literature, there is the notion of the universal central extension of a Lie algebra. My question is: is there also a notion of universal extensions that are not central? If yes, can you provide ...
14
votes
0answers
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 ...
4
votes
0answers
53 views

Constructing $\delta$ to show Chevalley-Eilenberg complex $\Lambda \mathfrak g \otimes \mathcal U\mathfrak g \rightarrow k$ is null homotopic

Let $\mathfrak g$ be a finite dimensional Lie algebra over a field $k$ and $(\Lambda^\bullet \mathfrak g \otimes \mathcal U\mathfrak g, d)$ be the Chevalley-Eilenberg chain complex. I would like to ...
6
votes
0answers
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 ...
4
votes
2answers
217 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 $\...
4
votes
1answer
113 views

First adjoint cohomology space of simple Lie algebras

Let $L$ be a central extension of a simple Lie algebra $\mathfrak{g}$ such that $L=[L,L]$. It is not difficult to see that if $H^1(\mathfrak{g}, \mathfrak{g})=0$ then $H^1(L,L)=0$. In other words, if ...
4
votes
4answers
396 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 ...
2
votes
1answer
112 views

Calculation of Dynkin operator on free Lie rings

Let $A$ be a free associative ${\mathbb Z}$-algebra on generators $I = \{ x_i ~:~ I \in {\mathbb N} \}$. Setting the usual bracket $[a,b] = ab-ba$, it forms a free Lie ring $A^{(-)}$. The Dynkin ...
4
votes
0answers
143 views

Deformation of the Hochschild-Kostant-Rosenberg isomorphism for universal enveloping algebra

Let $\mathfrak{g}$ be a Lie algebra over a char. $0$ field and let $\iota: U\mathfrak{g}\rightarrow S\mathfrak{g}$ be the Poincaré-Birkhoff-Witt (PBW) isomorphism, inverse to that natural map from the ...
2
votes
1answer
109 views

Extensions of modules over universal enveloping algebra with fixed central action

Let $\mathfrak{g}$ be a Lie algebra over $\mathbb{C}$, $\mathfrak{z}$ be the center of $\text{U}(\mathfrak{g})$, and $M_1$, $M_2$ be $\text{U}(\mathfrak{g})$-modules on which $\mathfrak{z}$ acts by a ...
3
votes
0answers
69 views

Deformations of nilpotent parts of superalgebras

I have two questions concerning some results in the article "Deformations of nilpotent parts of superalgebras" of N. van den Hijligenberg, J.Math.Phys. 35, 1427 (1994); doi:10.1063/1.530598 After ...
3
votes
0answers
101 views

Third cohomology of Lie algebras and obstructions

In general, the third cohomology of a Lie algebra $\mathfrak{g}$ with values in the Lie algebra itself, $H^3(\mathfrak{g},\mathfrak{g})$, contains obstructions to deformations of the Lie algebra. ...
3
votes
0answers
134 views

Gel'fand's and Fuks' calculation of cohomology of formal vector fields - isomorphic spectral sequences yield isomorphic cohomology?

In this book (and, in what seems to be an equivalent fashion, in this article), Gel'fand and Fuks calculate the Lie algebra cohomology of the formal vector fields $W_n$ on $\mathbb{R}^n$ with trivial ...
2
votes
0answers
145 views

Why Lie algebra (Chevalley–Eilenberg) cohomology are graded Lie algebras but not G-algebras?

I was reading a paper related to Gerstenhaber algebra structure and came across to this- "Lie algebra (Chevalley–Eilenberg) cohomology are graded Lie algebras but not G-algebras(Gerstenhaber algebra)"....
3
votes
1answer
390 views

Lie algebras : Deformations and Rigidity

I am studying deformation (as it is introduced in https://arxiv.org/pdf/math/0611793.pdf or http://web.cs.elte.hu/~fialowsk/pubs-af/condefnew2.pdf) and rigidity of some infinite dimensional Lie ...
2
votes
0answers
112 views

Chevalley-Eilenberg cohomology of polynomial vector fields

Let be $A$ the Lie algebra of polynomial vector fields. A p-cochain $C$ of $A$ is a p-linear alternate map from $A^p$ to $A$. For $p=0$, $C$ is an element of $A$. The coboundary operator $\eth$ is ...
1
vote
0answers
200 views

Adjoint cohomology of Lie algebra commutes with direct sum?

The Witt algebra (denoted by $W$) is an infinite dimensional Lie algebra as: $[L_{m},L_{n}]=(m-n)L_{m+n}; \,\,\,\ m, n\in \mathbb{Z}$. I am looking for second adjoint cohomology $H^{2}(W_{1}\oplus ...
1
vote
1answer
276 views

Cohomology of Infinite Dimensional Lie Algebra

I am looking for a explicit relation for adjoint cohomology of infinite dimensional Lie algebras (specifically smooth vector fields on a manifold), is it possible to apply Hochschild-Serre spectral ...
9
votes
3answers
513 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 \...
6
votes
0answers
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 ...
4
votes
1answer
197 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 ...
8
votes
0answers
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 ...
3
votes
1answer
177 views

Show the Cartan 3-form transgresses to the Killing form in the Weil algebra

Let $G$ be a connected, reductive Lie group, and $W\mathfrak g = (S[\mathfrak g^\vee] \otimes \Lambda[\mathfrak g^\vee],\delta)$ the associated Weil algebra. This is a CDGA equipped with an action of $...
8
votes
1answer
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, ...
1
vote
0answers
157 views

Lie algebras over $\mathbb{C}(t)$

Are there naturally-arising Lie algebras(or superalgebras) over $\mathbf{C}(t)$? I have such an algebra, and I want to compute its cohomology–but I'm out of ideas. Are there known methods or theory ...
8
votes
1answer
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}, \...
2
votes
0answers
76 views

If Lie algebra cohomology $H^2(g, M)=Ext^2_{U(g)}(k, M)$ classify $M$-extensions of $g$, are they $Ext^1_?(g, M)$ for some category?

If $\mathfrak{g}$ is a Lie algebra and $M$ is an abelian $\mathfrak{g}$-module, then Lie algebra cohomology $H^2(\mathfrak{g}, M)=Ext^2_{U(\mathfrak{g})}(k, M)$ classify (abelian) extensions of $\...
7
votes
0answers
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}...
4
votes
1answer
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 ...
3
votes
1answer
171 views

Isomorphisms between extension group and $\mathfrak{u}$-cohomology

Let $\mathcal{O}^{\mathfrak{p}}$ be a parabolic subcategory (see, Humphrey's BGG's Ch.9) of the BGG category $\mathcal{O}$ (w.r.t the Cartan $\mathfrak{h}$ and Borel $\mathfrak{b}$) over a finite-...
5
votes
0answers
221 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 \...
2
votes
1answer
167 views

Lie algebra (co)homology of the Lie algebra of differential operators

Let $X$ be an algebraic variety (or a complex manifold) over $\mathbb C$. Let $D(X)$ be its algebra of differential operators. Mostly I am interested in algebraic differential operators, but the case ...
1
vote
2answers
288 views

Nilpotency of Lie Algebra from Structure Constants

Suppose we have a Lie algebra with structure constants $$\mathrm{d}e^i=\sum_{j<k}a_{ijk}e^j\wedge e^k$$ for some coefficients $a_{ijk}$. In this setting, how may be checked (perhaps ...
5
votes
0answers
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 ...
3
votes
1answer
171 views

A converse to Whitehead's Second Lemma (and more)

Let $k$ be an algebraically closed field of characteristic $0$ and let $\mathfrak{h}$ be a finite dimensional $k$-Lie algebra. I'm interested in knowing which (Lie algebraic) properties $\mathfrak{h}$ ...
9
votes
1answer
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. ...