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2
votes
0answers
121 views

Kernel of the Weil homomorphism for compact symmetric spaces

Let $X = G/K$ be a Riemannian symmetric space of compact type and consider the "Weil homomorphism" $$w^\bullet: H^\bullet(BK; \mathbb R) \to H^\bullet(X; \mathbb R),$$ i.e. the map in cohomology ...
6
votes
0answers
111 views

vanishing of Lie algebra cohomology with coefficients in an infinite-dimensional module

Let $G$ be a real semisimple Lie group, $K$ its maximal compact subgroup, $\mathfrak g, \mathfrak k$ the corresponding Lie algebras. Let $V$ be a locally convex, Hausdorff vector space, which is a ...
2
votes
0answers
129 views

How to write BRST-BV for dg-Lie?

The usual BRST-BV implements a Lie algebra and its module in terms of ghosts, etc. Where is there written a corresponding formula incorporating the differential of a dg Lie algebra and module?
3
votes
2answers
290 views

What are cohomology of Lie algebra with coefficients geometrically?

I want to find analog of following two statements. Let $G$ be a discrete group, $M$ is representation of $G$. Local systems on $BG$ are the same as $G$ representations (because $\pi_1 (BG) =G$). Let ...
3
votes
1answer
129 views

Lyndon–Hochschild–Serre spectral sequence for not normal subgroup

Is there analog of Lyndon–Hochschild–Serre spectral sequence for not normal subgroup? What can you say about it? Can you describe $E^{p, q}_1$ ? What is about $E^{p, q}_2$? What is the best technique ...
1
vote
1answer
81 views

Whitehead's second Lemma and invariants of exterior square

Let $k$ be an algebraically closed field of characteristic $0$ and let $\mathfrak{g}$ be a finite dimensional semisimple $k$-Lie algebra. By Whitehead's second Lemma, we know that $H^{2}(\mathfrak{g}, ...
0
votes
1answer
172 views

Compact Lie groups with only 3 dimensional cohomology generators

Let $M$ be a compact connected semi-simple Lie group. Then by Hopf's Theorem $H^*(M;\mathbb Q)=\Lambda[\omega_1,...,\omega_s]$ where $\omega_i\in H^i(M;\mathbb Q)$ , $i\ge 3$ is odd. For which $M$, ...
1
vote
0answers
64 views

Cohomology of a graded differential algebra with L-infinity action by a Lie algebra relative to a sub algebra

Suppose $A$ is a graded differential algebra, $h\subset g$ is an ideal, and that there is an $L_\infty$ action by $g/h$ on $A$. Is there any theorem that gives a quasi-isomorphism between the ...
3
votes
0answers
80 views

Lifting Lie algebra cohomology class to Hochschild cochain

Let $g$ be a Lie algebra, $h\subset g$ an ideal. The associative algebra $N=U(h)\subset U(g)$ can be viewed as a $g$-module. The Lie algebra cohomology ${\rm H}^*(g,U(g))$ is isomorphic to the ...
4
votes
2answers
393 views

Hochschild and cyclic cohomology of commutative algebra?

I'm wondering how to compute the Hochschild cohomology ${\rm HH}^n(A,A)$ and the cyclic cohomology ${\rm HC}^n(A)$ where $A = \mathbb{C}[t_i]$ is the algebra of polynomials in a finite set of ...
5
votes
2answers
271 views

computing second cohomology $H^2(O_a,\mathbb{Z})\cong \mathbb{Z}^n$ of a generic coadjoint orbit

Let $G$ be a compact , connected and simply connected Lie group and $a\in\mathfrak{g}^*$ (dual of Lie algebra of Lie group $G$). Then let $O_a$ be a generic coadjoint orbit then can we say ...
5
votes
3answers
386 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 ...
1
vote
1answer
128 views

Cohomology with coefficient in a Lie algebra

For a topological space X we can consider the coefficient of singular cohomology in a Lie algebra A. Then we obtain a graded Lie algebra, that is [x,y]=(-1)^i+j-1 [y,x], for homogeneous ...
2
votes
0answers
158 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 ...
5
votes
0answers
196 views

Lagrangian (classical) BRST cohomology groups

I'm trying to understand BRST complex in its Lagrangian incarnation i.e. in the form mostly closed to original Faddeev-Popov formulation. It looks like the most important part of that construction ...
3
votes
3answers
277 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 ...
4
votes
0answers
89 views

Topological obstruction to icosahedral symmetry?

Let $G$ be a compact simple lie group of rank $n$. Then the Poincaré series of $G$ is given by $$P(G,q)=\prod_{i=1}^n (1+q^{2d_i-1}),$$ where the integers $d_1\leq d_2\leq \cdots \leq d_n$ are the ...
14
votes
1answer
490 views

History of Koszul complex

This is a question about history of commutative algebra. I'm curios why Koszul complex from commutative algebra is called Koszul complex? All Koszul's early papers are about Lie algebras and Lie ...
6
votes
1answer
257 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 ...
1
vote
0answers
105 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 ...
1
vote
0answers
62 views

Sub Lie algebra noncohomologous to zero

Let $g$ be a Lie algebra and $h$ a subalgebra of $g$. The embedding $h\subset g$ induces a map on the cohomology groups $H^*(g)\to H^*(h)$. I want to determine whether this map is surjective. What are ...
1
vote
1answer
345 views

Computing relative Lie algebra cohomology (as appears in Borel-Weil-Bott theorem)

Suppose $G$ is a complex Lie group, $P$ a Borel subgroup, $E$ a representation of $P$ that induces a vector bundle ${\cal E}$ over $G/P$. The general version of Borel-Weil-Bott theorem, as stated in ...
4
votes
1answer
242 views

p-adic Lie group vs Lie algebra cohomology with mod p coefficients

My question concerns the cohomology of a compact $p$-adic Lie group $G$ (wich is pro-$p$). Let $M$ be a finite dimensional $\mathbb{Q}_p$-vector space with continuous linear $G$-action. Lazard ...
0
votes
0answers
192 views

sh Lie algebra cohomology

For sh Lie algebra cohomology, is there written anywhere a description of H^1(L;L) as sh derivations mod inner ones?
0
votes
0answers
201 views

Cohomology of Lie groups and Lie algebras

If the second cohomology of a Lie algerba $g$ is $H^2(g,Z)=Z$. Then what is the second cohomology of the direct product of $n$ copies of $g$? Is it $Z^n$? Can I think if this cohomology as an integral ...
1
vote
1answer
465 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}= ...
7
votes
1answer
474 views

Hochschild (co)homology and representation theory

Dear members of Mathoverflow, I just discovered the notion of Hochschild (co)homology. I understand well the formalism however I am wondering about the meaning of this (co)homology for representation ...
6
votes
0answers
164 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}^* ...
4
votes
0answers
289 views

LIE ALGEBRA coboundary

There seems to be a problem in the literature about the definition of the 'standard' coboundary on the 'Cartan-Chevalley-Eilenberg' algebra - the problem is the signs! Where/when did things go wrong? ...
3
votes
1answer
263 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. ...
4
votes
2answers
647 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 ...
1
vote
3answers
483 views

Schur `multipliers' for Lie algebras

Schur multipliers for group extensions and for Lie groups also Where are they written for Lie algebras?
1
vote
0answers
164 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 ...
4
votes
0answers
156 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 ...
12
votes
2answers
576 views

Projective modules over quantum groups

My question is short: How can one calculate $\operatorname{Tor}_{U_q(\mathfrak g)}(k,k)$? ($k$ is the ground field of characteristic zero). If we had a regular universal enveloping algebra ...
1
vote
0answers
126 views

A Weyl invariance constructed from Clebsch-Gordan Coefficients.

Let $V$ and $\tilde{V}$ be irreducible representations of SU(N) with tensor decomposition: \begin{equation} V \otimes \tilde{V} = \bigoplus_i U_i \end{equation} \noindent were $U_i$ are also irreps ...
1
vote
2answers
197 views

How to prove H^2(g,J(g)) is nonzero for a semisimple Lie algebra g, where J(g) is the augmentation ideal of g?

Suppose g is a fiinte dimensional semisimple lie algebra over a field with characteristic 0. This question is related to Whitehead's second lemma, which says for finite dimensional g-module M, ...
7
votes
1answer
402 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 ...
2
votes
1answer
499 views

dg-lie structure on $HH^*$ and Koszul duality

This is shamelessly close to my other question: A Question on Koszul duality and $B(\infty)$ structures on $HH^*$. Maybe this one will get a better response. Rather than rewrite that one, I am going ...
11
votes
2answers
765 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 ...
6
votes
2answers
573 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 ...
3
votes
3answers
698 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 ...
2
votes
0answers
159 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 ...
2
votes
0answers
317 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 \ ...
5
votes
2answers
642 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 ...
16
votes
2answers
473 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 ...
9
votes
2answers
2k views

Compute Lie algebra cohomology

Is there a computer algebra system that is able to compute the Lie algebra cohomology in a given representation? What if the Lie algebra is finite dimensional? In my case I would like to be able to ...
4
votes
2answers
243 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 ...
5
votes
1answer
345 views

What is the Schouten bracket for the Chevalley-Eilenberg complex with coefficients in a nontrivial module?

Let $\mathfrak g$ be a Lie algebra. The Chevalley-Eilenberg complex is defined to be $\wedge^* \mathfrak g$ with differential $d\colon \wedge^* \mathfrak g\to \wedge^{*-1}\mathfrak g$ defined by ...
6
votes
2answers
755 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 ...