Questions tagged [lie-algebra-cohomology]
The lie-algebra-cohomology tag has no usage guidance.
144
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History of Koszul complex
This is a question about the history of commutative algebra. I'm curious why the Koszul complex from commutative algebra is called the Koszul complex? All of Koszul's early papers are about Lie ...
7
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1
answer
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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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0
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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....
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0
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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
1
answer
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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
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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
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0
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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
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0
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231
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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 ...
4
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1
answer
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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
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1
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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
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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}_{...
4
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0
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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
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1
answer
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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. ...
7
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2
answers
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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
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3
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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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0
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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
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0
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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 ...
14
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2
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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(\...
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0
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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
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2
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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, H^2(g,M)...
10
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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 $H^3(\mathfrak{g},\mathbb{k})$?...
3
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1
answer
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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 ...
13
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2
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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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2
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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
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3
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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
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0
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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): ...
3
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0
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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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2
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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 ...
17
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2
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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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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
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2
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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
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1
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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 $$d(...
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2
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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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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
...
7
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4
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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). ...
3
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1
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When does a VBLA induce an isomorphism on Lie algebroid cohomology?
This question is geared towards the experts, so I will only briefly gloss the definitions. Everything I say is in the category of finite-dimensional smooth manifolds, and whenever I say "$\mathbb Z$-...
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1
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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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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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3
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What is an obviously coordinate-independent description of the Chevellay-Eilenberg complex for a Lie algebroid?
I've read in many places, including the n-Lab page, that a Lie algebroid (which I think of as in the first definition on the n-Lab page) is the same as a vector bundle $A \to X$ and a (properties?) ...
3
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1
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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, yt^...
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3
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Algebraic/Categorical motivation for the Chevalley Eilenberg Complex
Is there a purely algebraic or categorical way to introduce the Chevalley-Eilenberg complex in the definition of Lie algebra cohomology?
In group cohomology, for example, the bar resolution of a ...
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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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Chevalley Eilenberg complex definitions?
In Weibel's An Introduction to Homological Algebra, the Chevalley-Eilenberg complex of a Lie algebra $g$ is defined as $\Lambda^*(g) \otimes Ug$ where $Ug$ is the universal enveloping algebra of $g$. ...
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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 ...