# Tagged Questions

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### Prove the Weyl's Theorem by Kostant's $\mathfrak{n}$-cohomology result [migrated]

I've met an exercise in Kumar's book ("Kac-Moody Groups, their Flag Varieties and Representation Theory", Chapter III, page 89, Ex. 3.2. E, (1) & (2)).
But I have no idea about its proof.
Any ...

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**1**answer

260 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 ...

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134 views

### Lie algebra cohomology

Let $\mathfrak{g}$ be a simple complex Lie algebra of $rank(\mathfrak{g})\geq2$ and dimension $d$. Fix a (non-zero) invariant bilinear form $(\cdot,\cdot)$ on $\mathfrak{g}$ and let $\{x_i\}_{1\leq ...

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237 views

### Computation of restricted Lie algebra (co)homology

My question is the following:
Is there a small complex, perhaps analogous to the Chevalley-Eilenberg complex, computing the (co)homology of a restricted Lie algebra over a field of characteristic ...

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**1**answer

247 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 ...

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95 views

### A Isomorphism between the extension group and cohomology group of Lie algebras [closed]

Within the book An introduction to homological algebra by Weibel, I am trying to prove the following isomorphism, but I am not sure this is true. But I really want to know how to prove or disprove ...

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**1**answer

438 views

### parabolic subalgebras and Cartan decomposition

Let $\mathfrak{g}$ be a complex simple Lie algebra and $\mathfrak{k}$ its complex subalgebra such that $(\mathfrak{g},\mathfrak{k})$ is a Hermitian symmetric pair; $\mathfrak{g}= ...

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159 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}^* ...

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255 views

### How can one find generators of basic differential forms on homogeneous spaces?

Dear all,
In short, my problem is that I would like to have a better control of the 1-forms on a homogeneous space. Contrary to the group case, the module of differential form is not trivialisable. ...

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556 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 ...

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458 views

### Schur `multipliers' for Lie algebras

Schur multipliers for group extensions and for Lie groups also
Where are they written for Lie algebras?

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158 views

### Exotic Chains for Group Homology of a Complex Lie Group

Related Question: Exotic Chains for Group Cohomology of a Complex Lie Group
Let's take the group homology of a affine algebraic group over $\mathbb C$ (with its discrete topology). The natural ...

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149 views

### Exotic Chains for Group Cohomology of a Complex Lie Group

Related Question: Exotic Chains for Group Homology of a Complex Lie Group
Let's take the group cohomology of a affine algebraic group over $\mathbb C$ (with its discrete topology). The natural ...

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**1**answer

371 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 ...

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719 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 ...

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545 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 ...

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684 views

### Is there any relation between deformation and extension of Lie algebras?

In a paper of A. Weinstein on the geometry of Poisson manifolds, he relates the formal linearization around a zero, p, of the Poisson bivector to extensions of the Lie algebra induced by the bivector ...

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156 views

### A problem on 2 Lie (co)homology group and central extension

For a perfect Lie algebra L over C, The kernal of its universal central extension is isomorphic to H_2(L,C), and its central extensions is 1-1 to the H^2(L,C).
question(1) Can we know one of the H_2 ...

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291 views

### Cohomologies associated to residually torsion-free nilpotent groups

This question is related to my previous question: Relationship between the cohomology of a group and the cohomology of its associated Lie algebra.
A group $G$ is ${\it residually \ torsion \ free \ ...

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631 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 ...

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460 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 ...

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238 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 ...

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710 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 ...

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334 views

### A question on the construction of finite W-algebras

In a well known construction of finite W-algebras, one first constructs a certain
nilpotent subalgebra $\mathfrak{m}$ along with a character $\chi:\mathfrak{m}\rightarrow \mathbb{C}$.
Then one defines
...

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479 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). ...

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533 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} = ...

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741 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 ?
...

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271 views

### Why are relations of degree 3 or less enough in a presentation of the polynomial current Lie algebra g[t]?

Let $\mathfrak{g}$ be a finite dimensional simple Lie algebra over $\mathbb{C}$.
The polynomial current Lie algebra $\mathfrak{g}[t] = \mathfrak{g} \otimes \mathbb{C} [t]$
has the bracket
$$[xt^r, ...

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2k 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 ...

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315 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 ...