Questions tagged [lie-algebra-cohomology]
The lie-algebra-cohomology tag has no usage guidance.
140
questions
5
votes
0
answers
133
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)...
- 481
3
votes
0
answers
426
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\...
- 432
1
vote
0
answers
119
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 ...
- 1,165
4
votes
1
answer
226
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 ...
- 211
2
votes
0
answers
64
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 ...
- 401
3
votes
1
answer
69
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 ...
- 9,297
4
votes
0
answers
86
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 ...
2
votes
1
answer
224
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 ...
- 432
1
vote
0
answers
27
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 ...
- 111
4
votes
1
answer
176
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 ...
- 181
4
votes
1
answer
160
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$ ...
- 6,827
5
votes
1
answer
254
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 ...
- 1,070
6
votes
1
answer
327
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$...
- 819
7
votes
0
answers
185
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 ...
2
votes
1
answer
331
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 ...
- 5,166
8
votes
0
answers
172
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 ...
- 15.9k
9
votes
1
answer
621
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 ...
- 289
1
vote
0
answers
159
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 ...
- 49
3
votes
1
answer
99
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 ...
- 73
4
votes
0
answers
162
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 ...
- 737
3
votes
0
answers
77
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 ...
user108998
4
votes
0
answers
74
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 ...
- 3,001
8
votes
1
answer
279
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?
- 175
5
votes
0
answers
139
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 ...
- 737
8
votes
0
answers
172
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 ...
- 1,070
10
votes
0
answers
94
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 ...
- 5,402
5
votes
1
answer
290
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 ...
- 461
3
votes
0
answers
65
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 ...
- 31
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$ ...
- 4,648
1
vote
1
answer
100
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(...
- 13
2
votes
0
answers
77
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{...
- 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 ...
- 323
1
vote
0
answers
51
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 \...
- 11
6
votes
1
answer
411
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 ...
- 165
0
votes
1
answer
112
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,...
- 95
3
votes
2
answers
361
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 ...
- 95
14
votes
0
answers
338
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
0
answers
61
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 ...
- 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 ...
- 3,822
4
votes
2
answers
556
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 $\...
- 169
4
votes
1
answer
135
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 ...
- 5,248
4
votes
4
answers
542
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 ...
- 332
2
votes
1
answer
133
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 ...
- 405
4
votes
0
answers
206
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 ...
user108998
2
votes
1
answer
154
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 ...
- 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 ...
- 95
3
votes
0
answers
135
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.
...
- 95
3
votes
0
answers
142
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
0
answers
163
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)"....
- 576
3
votes
1
answer
762
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 ...
