Questions tagged [lie-algebra-cohomology]

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Lie algebra cohomology of formal non-commutative vector fields

Let $k$ be a field of characteristic $0$ and $A=k\langle\langle x_1,\dotsc,x_n\rangle\rangle$ be a free completed associative algebra. The space of continuous derivations $\mathrm{Der}(A)$ is ...
Qwert Otto's user avatar
9 votes
0 answers
340 views

How large must the characteristic of $k$ be, for the cohomology of the Lie algebra $\mathfrak{sl}_n(k)$ to be exterior as in characteristic zero?

$\DeclareMathOperator\SU{SU}$In this question, all Lie algebra cohomology is of the form $H^*(\mathfrak{g}; k)$, with $k$ the trivial one-dimensional representation of $\mathfrak{g}$. All Lie algebra ...
user509184's user avatar
5 votes
0 answers
327 views

Have you seen this Lie algebra?

Computing something I have come across a Lie algebra $\def\L{\mathfrak L}\def\CC{\mathbb C}\L_N$ that I would like to identify. Fix an integer $N$ such that $N\geq2$, let $\L_N$ be the free complex ...
Mariano Suárez-Álvarez's user avatar
6 votes
0 answers
249 views

Lie algebra cohomology of the space of vector fields

For a (closed and oriented) manifold $M$, the first Lie algebra cohomology $H^1(\mathrm{Vect}(M),C^\infty(M))$ of the space of vector fields with coefficients in smooth functions is isomorphic to $H^1(...
Qwert Otto's user avatar
5 votes
0 answers
141 views

Reference request: Whitehead's lemma for semisimple Lie algebras

Let $\mathfrak{g}$ be a semisimple Lie algebra and let $V$ be a representation. One formulation of Whitehead's lemma is that the Lie algebra cohomology of $V$ is given by $$H^{\bullet}(\mathfrak{g}, V)...
unknownymous's user avatar
3 votes
0 answers
509 views

Particular Lie bialgebra structure

Let $\mathfrak{g}$ be a real finite-dimensional Lie algebra, and let $r\in\bigwedge^2\mathfrak{g}$ be a solution of the Yang-Baxter equation. The Yang–Baxter equation states that for all $x\in\...
user56980's user avatar
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131 views

Lie algebra cohomology: Künneth formula for semidirect product?

In trying to understand an example of Lie algebra for which every one-dimensional extension splits (see this question), but not every extension splits, I found it necessary to use the following ...
Nikhil Sahoo's user avatar
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4 votes
1 answer
254 views

Complexification of a Lie subalgebra of a compact real form

I'm currently reading the paper Lie algebra Cohomology and the Generalized Borel–Weil theorem written by B. Kostant, and I have a question about Remark 3.9 he made. In this paper, $\mathfrak{g}$ is a ...
Ji Woong Park's user avatar
2 votes
0 answers
68 views

Subrepresentations and the induced map on Lie algebra cohomology

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SO{SO}$Setup: Let $G$ be the group $\GL(4, \mathbb{R})$, $B$ denotes the Borel subgroup consisting of upper triangular matrices and $P_{(2,2)}$ be the ...
random123's user avatar
  • 411
3 votes
1 answer
71 views

Non-invariant forms on loop Lie algebra of semisimple Lie group

Let us consider a Lie group $G$ with Lie algebra $\mathfrak{g}$ and let $L\mathfrak{g} = C^\infty(S^1, \mathfrak{g})$ the Lie algebra of the loop group $LG$. My question is about continuous Lie ...
Matthias Ludewig's user avatar
4 votes
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97 views

Modern reference for a theorem by Bott on the Dolbeault cohomology of compact homogeneous manifolds

I am looking for a modern, maybe shorter or even easier, reference for Theorem II of Homogeneous vector bundles (R. Bott, Annals of mathematics, 1957). This is a theorem where the Dolbeault cohomology ...
Max Reinhold Jahnke's user avatar
2 votes
1 answer
254 views

Complete $2$-step solvable Lie algebras

A Lie algebra is complete if its center is zero and all its derivations are inner. I would like to study a class of Lie algebras, in particular Let $C$ be the class of finite dimensional $2$-step ...
user56980's user avatar
  • 432
1 vote
0 answers
28 views

Group action in the vicinity of an orbit where the stabilizer jumps

Consider a manifold $M$ with the action of a Lie algebra $\mathfrak g$. Suppose that the action is free, except for one orbit $O\subset M$ where the stabilizer is a nonzero Lie subalgebra ${\mathfrak ...
amkhlv's user avatar
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1 answer
193 views

Spectral sequence from standard/Verma filtration/flag to compute Lie algebra cohomology of tensor product with respect to $\mathfrak{n}$

I'm not sure this question fully qualifies as a research-level math question, but from my (limited) past experience on stackexchanged I feared this question might not get an answer there. Setting: the ...
Fan Zhou's user avatar
  • 181
4 votes
1 answer
169 views

Divisibility by 2 of invariants forms on reductive Lie algebras and anomaly cancellation for gauge theories

Let $G$ be a connected reductive group over $\mathbb C$ and let $\rho:G\to \operatorname{Sp}(2n,\mathbb C)$ be a homomorphism. You can think about $\rho$ as a linear symplectic representation of $G$ ...
Alexander Braverman's user avatar
5 votes
1 answer
263 views

Infinite dimensional Lie algebras with trivial homology

The basic question is: Does vanishing of homology with trivial coefficients imply triviality of an infinite-dimensional Lie algebra? My question is motivated by acylic groups in group theory. In ...
Grisha Taroyan's user avatar
6 votes
1 answer
358 views

2-shifted Poisson bracket on Lie algebra cohomology

Let $\frak{g}$ be a semisimple Lie algebra, and let $({-},{-})$ be an invariant inner product on $\frak{g}$. The Chevalley–Eilenberg complex $C^*(\frak{g})$ has a natural Poisson bracket of degree $-2$...
Ezra Getzler's user avatar
7 votes
0 answers
197 views

On the center of Koszul Lie algebras

The short question is the following: If a positively graded Lie algebra $\mathfrak g$ over a field $F$ is Koszul, is the center of $\mathfrak g$ concentrated in degree $1$? Let us be more precise. A ...
Simone Blumer's user avatar
2 votes
1 answer
380 views

Chevalley complex and $\text{BG}$

For a long time I've been under the impression that the Chevalley complex $\text{CE}(\mathfrak{g})$ of a semisimple (maybe can weaken this) Lie algebra $\mathfrak{g}$ can be extracted from the ...
Pulcinella's user avatar
  • 5,507
8 votes
0 answers
186 views

Cohomology algebra of the maximal nilpotent subalgebra of a semisimple Lie algebra

The answer to this question may be well-known, but I failed to locate it in any obvious source. From the results of Bott (Ann. Math. 66 (1957), 203-248) and Kostant (Ann. Math. 74 (1961), 329-387), it ...
Vladimir Dotsenko's user avatar
9 votes
1 answer
761 views

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 with coefficients in a vector space $V$ (we assume the real ...
marmistrz's user avatar
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172 views

$G$-equivariant modules and Lie algebra cohomology

$\DeclareMathOperator\Id{Id}\DeclareMathOperator\Ad{Ad}$Is there a link between $G$-equivariant modules and Lie algebra cohomology? Tell me if I'm mistaken: On one side, if $p:E\longrightarrow M$ is ...
AG14's user avatar
  • 59
3 votes
1 answer
104 views

Rigidity of Borel Lie algebras

Let $\mathfrak b$ be a Borel subalgebra of dimension $n$ in a real semisimple Lie algebra $\mathfrak g$. I am trying to reconcile two facts about $\mathfrak b$: $\mathfrak b$ is rigid, that is, the ...
user102887's user avatar
4 votes
0 answers
186 views

Rational cohomology cohomology of $p$-adic analytic groups

It is a result of Lazard that given $G$ a compact $p$-adic analytic group then we have an isomorphism \begin{equation} H^*(G; \mathbb{Q}_p) \cong H^*(T_eG; \mathbb{Q}_p) \end{equation} where $T_eG$ is ...
N.B.'s user avatar
  • 757
3 votes
0 answers
82 views

Degeneration of spectral sequence computing Hochschild cohomology of enveloping algebra of Lie algebroid

Let $L$ be a Lie algebroid on a smooth affine $k$-scheme $X=spec(R)$. Recall that by definition $L$ is a locally free sheaf with the structure of a sheaf of $k$ Lie algebras, so that there exists a ...
user avatar
4 votes
0 answers
75 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 ...
Avi Steiner's user avatar
  • 3,031
8 votes
1 answer
304 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?
Antema Kely's user avatar
5 votes
0 answers
156 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 ...
N.B.'s user avatar
  • 757
8 votes
0 answers
175 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 ...
Grisha Taroyan's user avatar
10 votes
0 answers
96 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 ...
Tobias Diez's user avatar
  • 5,442
5 votes
1 answer
326 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 ...
user18063's user avatar
  • 461
3 votes
0 answers
66 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 ...
mayflowers46's user avatar
4 votes
0 answers
186 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 $G$ ...
Student's user avatar
  • 5,008
1 vote
1 answer
116 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(...
Tainted G's user avatar
2 votes
0 answers
80 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{...
thrw's user avatar
  • 21
1 vote
0 answers
38 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 ...
sock's user avatar
  • 323
1 vote
0 answers
54 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 \...
Lies's user avatar
  • 11
6 votes
1 answer
467 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 ...
user11090426's user avatar
0 votes
1 answer
124 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,...
Sleipnir's user avatar
3 votes
2 answers
393 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 ...
Sleipnir's user avatar
14 votes
0 answers
353 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 ...
Lukas Miaskiwskyi's user avatar
4 votes
0 answers
62 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 ...
sock's user avatar
  • 323
6 votes
0 answers
120 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 ...
Jim Stasheff's user avatar
  • 3,840
4 votes
2 answers
644 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 $\...
Lucas Smits's user avatar
4 votes
1 answer
140 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 ...
Salvatore Siciliano's user avatar
4 votes
4 answers
553 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 ...
Boris Bilich's user avatar
2 votes
1 answer
138 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 ...
Siddhartha's user avatar
4 votes
0 answers
215 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 ...
user avatar
2 votes
1 answer
179 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 ...
Echeru's user avatar
  • 31
3 votes
0 answers
71 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 ...
Sleipnir's user avatar