Questions tagged [lie-algebra-cohomology]

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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. ...
Sleipnir's user avatar
3 votes
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146 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 ...
Lukas Miaskiwskyi's user avatar
2 votes
0 answers
167 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)"....
Ripan Saha's user avatar
3 votes
1 answer
842 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 ...
Hamidreza Safari's user avatar
2 votes
0 answers
141 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 ...
user116759's user avatar
1 vote
0 answers
245 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 ...
Hamidreza Safari's user avatar
1 vote
1 answer
374 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 ...
Hamidreza Safari's user avatar
9 votes
3 answers
727 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 \...
Vanya's user avatar
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6 votes
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118 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 ...
José Figueroa-O'Farrill's user avatar
4 votes
1 answer
261 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 ...
Vanya's user avatar
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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 ...
Dan Petersen's user avatar
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3 votes
1 answer
286 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 $...
jdc's user avatar
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8 votes
1 answer
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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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1 vote
0 answers
162 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 ...
user83406's user avatar
10 votes
1 answer
419 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}, \...
evgeny's user avatar
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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 $\...
evgeny's user avatar
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8 votes
0 answers
243 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}...
Julian Kuelshammer's user avatar
4 votes
1 answer
185 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 ...
Bashar Saleh's user avatar
3 votes
1 answer
187 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-...
Steven's user avatar
  • 159
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0 answers
308 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 \...
Misaka01034's user avatar
2 votes
1 answer
209 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 ...
lks8271's user avatar
  • 165
2 votes
2 answers
338 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 ...
Jjm's user avatar
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5 votes
0 answers
309 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 ...
Nicolas's user avatar
  • 51
3 votes
1 answer
213 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 $\mathfrak{h}$ ...
Paul Gilmartin's user avatar
10 votes
1 answer
364 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. ...
Mirko's user avatar
  • 203
3 votes
2 answers
581 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 ...
QSR's user avatar
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3 votes
0 answers
167 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 $H^{3}(\mathfrak{g}((t))((s)),\mathbb{C})$?
prochet's user avatar
  • 3,432
4 votes
1 answer
1k 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 ...
Mike Cocos's user avatar
1 vote
1 answer
172 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 ...
GuNa's user avatar
  • 55
4 votes
0 answers
189 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 ...
José Figueroa-O'Farrill's user avatar
2 votes
0 answers
172 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 ...
geodude's user avatar
  • 2,129
12 votes
2 answers
1k 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,...
Bizfold's user avatar
  • 343
2 votes
0 answers
157 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 ...
Tobias Hartnick's user avatar
6 votes
0 answers
207 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 $G$-...
Dima's user avatar
  • 335
2 votes
0 answers
193 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?
Jim Stasheff's user avatar
  • 3,840
3 votes
2 answers
638 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 $...
quinque's user avatar
  • 375
11 votes
3 answers
954 views

Lyndon–Hochschild–Serre spectral sequence for a non-normal subgroup

Is there an analogue of the Lyndon–Hochschild–Serre spectral sequence for a non-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 ...
quinque's user avatar
  • 375
2 votes
1 answer
254 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}, ...
Paul's user avatar
  • 21
9 votes
3 answers
1k views

Poincaré duality for (co)homology of Lie algebras?

Let $R$ be a commutative ring and $\mathfrak{g}$ a Lie $R$-algebra that has an $R$-module basis with $n$ elements. In Algebra, Geometry, and Software Systems by Joswig & Takayama on p.200, it ...
Leo's user avatar
  • 1,579
0 votes
1 answer
292 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 $M$, $...
user46227's user avatar
1 vote
0 answers
132 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-...
user36075's user avatar
  • 131
3 votes
0 answers
136 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 ...
user36075's user avatar
  • 131
5 votes
2 answers
1k 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 ...
user36075's user avatar
  • 131
5 votes
2 answers
434 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,\...
user avatar
9 votes
3 answers
1k 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 ('...
Mark.Neuhaus's user avatar
  • 2,004
2 votes
1 answer
321 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 ...
Ali Taghavi's user avatar
4 votes
0 answers
301 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\...
user42897's user avatar
6 votes
0 answers
444 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 (...
Sasha Pavlov's user avatar
  • 1,535
5 votes
3 answers
745 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 $...
Ben Knudsen's user avatar
4 votes
0 answers
116 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 ...
Drew Armstrong's user avatar