Questions tagged [lie-algebra-cohomology]
The lie-algebra-cohomology tag has no usage guidance.
63
questions with no upvoted or accepted answers
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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 ...
10
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96
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Non-linear version of the Chevalley–Eilenberg complex
Let $\mathfrak{g}$ be a finite-dimensional Lie algebra. In small degrees, the differentials of the Chevalley–Eilenberg complex $C^\bullet(\mathfrak{g}, \mathfrak{g})$ with values in the adjoint ...
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347
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How large must the characteristic of $k$ be, for the cohomology of the Lie algebra $\mathfrak{sl}_n(k)$ to be exterior as in characteristic zero?
$\DeclareMathOperator\SU{SU}$In this question, all Lie algebra cohomology is of the form $H^*(\mathfrak{g}; k)$, with $k$ the trivial one-dimensional representation of $\mathfrak{g}$. All Lie algebra ...
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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 ...
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Cohomology algebra of the maximal nilpotent subalgebra of a semisimple Lie algebra
The answer to this question may be well-known, but I failed to locate it in any obvious source. From the results of Bott (Ann. Math. 66 (1957), 203-248) and Kostant (Ann. Math. 74 (1961), 329-387), it ...
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176
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Chevalley-Eilenberg cohomology of polynomial vector fields on $\mathbb{A}^2$
I have a question similar to one given here.
What is the cohomology of the Lie algebra of polynomial vector fields on an affine space $\mathbb{A}^2$ over a field of characteristic $0?$ (I'm interested ...
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8
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244
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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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197
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On the center of Koszul Lie algebras
The short question is the following: If a positively graded Lie algebra $\mathfrak g$ over a field $F$ is Koszul, is the center of $\mathfrak g$ concentrated in degree $1$?
Let us be more precise. A ...
6
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254
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Lie algebra cohomology of the space of vector fields
For a (closed and oriented) manifold $M$, the first Lie algebra cohomology $H^1(\mathrm{Vect}(M),C^\infty(M))$ of the space of vector fields with coefficients in smooth functions is isomorphic to $H^1(...
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votes
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120
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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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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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6
votes
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207
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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 $G$-...
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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 (...
- 1,535
6
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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}_{...
- 8,707
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Have you seen this Lie algebra?
Computing something I have come across a Lie algebra $\def\L{\mathfrak L}\def\CC{\mathbb C}\L_N$ that I would like to identify.
Fix an integer $N$ such that $N\geq2$, let $\L_N$ be the free complex ...
- 46.8k
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Reference request: Whitehead's lemma for semisimple Lie algebras
Let $\mathfrak{g}$ be a semisimple Lie algebra and let $V$ be a representation. One formulation of Whitehead's lemma is that the Lie algebra cohomology of $V$ is given by
$$H^{\bullet}(\mathfrak{g}, V)...
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5
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Rational cohomology of p-adic general linear groups
I wanted to compute the cohomology ring $H^*(GL_n(\mathbb{Z}_p); \mathbb{Q}_p)$ (with $p$ fixed prime as usual). I found some incomplete notes stating that the computation should go as follows.
First ...
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308
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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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5
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311
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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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97
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Modern reference for a theorem by Bott on the Dolbeault cohomology of compact homogeneous manifolds
I am looking for a modern, maybe shorter or even easier, reference for Theorem II of Homogeneous vector bundles (R. Bott, Annals of mathematics, 1957). This is a theorem where the Dolbeault cohomology ...
4
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195
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Rational cohomology cohomology of $p$-adic analytic groups
It is a result of Lazard that given $G$ a compact $p$-adic analytic group then we have an isomorphism
\begin{equation} H^*(G; \mathbb{Q}_p) \cong H^*(T_eG; \mathbb{Q}_p) \end{equation}
where $T_eG$ is ...
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4
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75
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Lie algebra "semi" coinvariants
In the process of my research, I've come across the need to understand the following construction:
Let $\mathfrak{g}$ be a (finite-dimensional) complex Lie algebra, $\beta\in \mathfrak{g}^*$ a Lie ...
- 3,029
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186
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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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4
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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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4
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answers
216
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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 ...
user108998
4
votes
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answers
189
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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 ...
- 32.8k
4
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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\...
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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 ...
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375
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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? ...
- 3,850
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184
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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 ...
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4
votes
1
answer
142
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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 ...
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Lie algebra cohomology of formal non-commutative vector fields
Let $k$ be a field of characteristic $0$ and $A=k\langle\langle x_1,\dotsc,x_n\rangle\rangle$ be a free completed associative algebra. The space of continuous derivations $\mathrm{Der}(A)$ is ...
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0
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509
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Particular Lie bialgebra structure
Let $\mathfrak{g}$ be a real finite-dimensional Lie algebra, and let $r\in\bigwedge^2\mathfrak{g}$ be a solution of the Yang-Baxter equation. The Yang–Baxter equation states that for all $x\in\...
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votes
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Degeneration of spectral sequence computing Hochschild cohomology of enveloping algebra of Lie algebroid
Let $L$ be a Lie algebroid on a smooth affine $k$-scheme $X=spec(R)$. Recall that by definition $L$ is a locally free sheaf with the structure of a sheaf of $k$ Lie algebras, so that there exists a ...
user108998
3
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answers
68
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Family of Lie algebras parametrized by a discrete valuation ring
I have a family of Lie algebras parametrized by a discrete valuation ring, whose generic fiber is reductive and whose special fiber is nilpotent. I'd like to learn about the relationship between the ...
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71
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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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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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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 ...
3
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168
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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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votes
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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 ...
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answers
420
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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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68
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Subrepresentations and the induced map on Lie algebra cohomology
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SO{SO}$Setup: Let $G$ be the group $\GL(4, \mathbb{R})$, $B$ denotes the Borel subgroup consisting of upper triangular matrices and $P_{(2,2)}$ be the ...
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answers
85
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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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167
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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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2
votes
0
answers
141
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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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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 $\...
- 1,990
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votes
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answers
172
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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 ...
- 2,129
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157
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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 ...
2
votes
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193
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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?
- 3,850
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votes
0
answers
213
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A problem on 2 Lie (co)homology group and central extension
For a perfect Lie algebra $L$ over $C,$ the kernel of its universal central extension is isomorphic to $H_2(L,C),$ and its central extensions are in 1-1 correspondence to $H^2(L,C).$
Question (1): ...
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