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3
votes
1answer
77 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 ...
3
votes
1answer
134 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-...
4
votes
0answers
71 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 \...
2
votes
0answers
85 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 ...
1
vote
2answers
224 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 ...
4
votes
0answers
162 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 ...
3
votes
1answer
133 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 $\mathfrak{h}$ ...
9
votes
1answer
142 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. ...
3
votes
2answers
288 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 ...
3
votes
0answers
89 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 $H^{3}(\mathfrak{g}((t))((s)),\mathbb{C})$?
2
votes
1answer
162 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 ...
1
vote
1answer
128 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 ...
4
votes
0answers
137 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 ...
3
votes
0answers
103 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 ...
9
votes
2answers
297 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-...
2
votes
0answers
136 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
138 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 $G$-...
2
votes
0answers
163 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
357 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
261 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 ...
2
votes
1answer
90 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
177 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
90 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 Lie-...
3
votes
0answers
100 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
588 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
312 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 $H^2(O_a,\...
6
votes
3answers
525 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
139 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
214 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 i\...
6
votes
0answers
238 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
348 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
96 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 ...
15
votes
1answer
935 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
286 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
116 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 it....
1
vote
0answers
65 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
424 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
298 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
205 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
213 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
507 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}= \mathfrak{k}\oplus\...
7
votes
1answer
588 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
180 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}^* ,\text{d}_{...
4
votes
0answers
296 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
289 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. ...
5
votes
2answers
888 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
558 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
172 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 free ...
4
votes
0answers
167 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 free ...
14
votes
2answers
633 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 $U(\...