# Questions tagged [lie-algebra-cohomology]

The lie-algebra-cohomology tag has no usage guidance.

112
questions

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### 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 $G$ ...

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**1**answer

73 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(...

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69 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{...

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34 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 ...

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22 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 \...

**5**

votes

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140 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 ...

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votes

**1**answer

85 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,...

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192 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 ...

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242 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 ...

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49 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 ...

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106 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 ...

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votes

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180 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

**1**answer

109 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

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385 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 ...

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**1**answer

110 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 ...

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132 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 ...

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106 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 ...

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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 ...

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98 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.
...

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131 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 ...

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141 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)"....

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**1**answer

332 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 ...

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84 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 ...

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195 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 ...

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265 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 ...

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votes

**3**answers

486 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 \...

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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 ...

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**1**answer

192 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 ...

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221 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

**1**answer

162 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 $...

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178 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, ...

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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 ...

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240 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}, \...

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74 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 $\...

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203 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}...

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163 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 ...

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**1**answer

168 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-...

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213 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 \...

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**1**answer

162 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 ...

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**2**answers

280 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 ...

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233 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 ...

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**1**answer

170 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 $\mathfrak{h}$ ...

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**1**answer

223 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.
...

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402 views

### mod p cohomology ring of alternating groups

Let $A_n$ be the alternating group of $\{1,2,\cdots,n\}$.
(1). What is the cohomology ring
$$
H^*(A_4;\mathbb{Z}/3)
$$
and its Steenrod operation $P^i$'s?
(2). Are there general results about the ...

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145 views

### Double loop groups and cohomology

Let $G$ be a connected reductive group over $\mathbb{C}$ of Lie algebra $\mathfrak{g}$.
What is the value of $H^{3}(\mathfrak{g}((t))((s)),\mathbb{C})$?

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**1**answer

575 views

### Invariant polynomials with respect to group actions on matrices

Let $\mathfrak{gl}_n(\mathbb{R})$ be the Lie algebra of matrices with real entries and $GL_n(\mathbb{R})$ its associated Lie group. Recall that a linear subgroup $G \subseteq GL_n(\mathbb{R})$ acts by ...

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145 views

### When the restriction of derived equivalence to a summand is a derived equivalence as well

I have a question about the equivalence of derived categories. Let $\mathcal{A} = \mathcal{A}'\oplus \mathcal{A}''$ and $\mathcal{B} = \mathcal{B}' \oplus \mathcal{B}''$ are direct sum of abelian ...

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159 views

### Lie algebra cohomology with values in the ring of smooth functions of a $G$-manifold

Let $G$ be a connected Lie group with Lie algebra $\mathfrak{g}$ acting faithfully on a smooth, finite-dimensional manifold $M$. Let $C^\infty(M)$ and $\mathcal{X}(M)$ denote the ring of smooth ...

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149 views

### Lie group cohomology with coefficients in Lie algebra

I'm looking for a reference, and basic results, about Lie algebra as modules over a Lie group (with the adjoint representation), from the point of view of cohomology. Links with the Lie algebra ...

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756 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,...