Questions tagged [lie-algebra-cohomology]

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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 ...
Sasha Pavlov's user avatar
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7 votes
1 answer
408 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 ...
Sasha Pavlov's user avatar
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1 vote
0 answers
144 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....
Kan Park's user avatar
1 vote
0 answers
68 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 ...
user36075's user avatar
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1 answer
663 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 ...
user36075's user avatar
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4 votes
1 answer
615 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 ...
Arkandias's user avatar
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0 answers
225 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?
Jim Stasheff's user avatar
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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 ...
user30765's user avatar
4 votes
1 answer
940 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\...
Vít Tuček's user avatar
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7 votes
1 answer
947 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 ...
Damien S.'s user avatar
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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}_{...
Zhaoting Wei's user avatar
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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? ...
Jim Stasheff's user avatar
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3 votes
1 answer
452 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. ...
Amin's user avatar
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7 votes
2 answers
2k 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 ...
user14211's user avatar
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3 answers
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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?
Jim Stasheff's user avatar
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1 vote
0 answers
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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 ...
John Pardon's user avatar
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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 ...
John Pardon's user avatar
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14 votes
2 answers
746 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(\...
John Pardon's user avatar
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1 vote
0 answers
152 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 ...
Kasper's user avatar
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2 answers
252 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, H^2(g,M)...
Henry Lee's user avatar
10 votes
1 answer
777 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 $H^3(\mathfrak{g},\mathbb{k})$?...
Vít Tuček's user avatar
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3 votes
1 answer
787 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 ...
Daniel Pomerleano's user avatar
13 votes
2 answers
2k 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 ...
Christopher Drupieski's user avatar
9 votes
2 answers
1k 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 ...
Chuck Hague's user avatar
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3 votes
3 answers
857 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 ...
Feri's user avatar
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2 votes
0 answers
213 views

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): ...
ren l's user avatar
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3 votes
0 answers
420 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 \ ...
Peter Goetz's user avatar
9 votes
2 answers
838 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 ...
darij grinberg's user avatar
17 votes
2 answers
807 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 ...
Mariano Suárez-Álvarez's user avatar
17 votes
2 answers
3k 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 ...
Michele Torielli's user avatar
4 votes
2 answers
332 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 ...
Vipul Naik's user avatar
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5 votes
1 answer
551 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 $$d(...
Jim Conant's user avatar
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7 votes
2 answers
1k 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 ...
Theo Johnson-Freyd's user avatar
3 votes
1 answer
401 views

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 ...
Jan Weidner's user avatar
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7 votes
4 answers
718 views

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). ...
Theo Johnson-Freyd's user avatar
3 votes
1 answer
335 views

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 "$\mathbb Z$-...
Theo Johnson-Freyd's user avatar
4 votes
1 answer
734 views

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} = \...
Christopher Drupieski's user avatar
7 votes
4 answers
938 views

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 ? ...
student's user avatar
  • 1,212
3 votes
3 answers
618 views

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?) ...
Theo Johnson-Freyd's user avatar
3 votes
1 answer
327 views

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^...
B.R.J.'s user avatar
  • 138
9 votes
3 answers
1k views

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 ...
Hanno's user avatar
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26 votes
2 answers
4k views

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 ...
algori's user avatar
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22 votes
3 answers
7k views

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 $g$. ...
mpdude's user avatar
  • 367
8 votes
2 answers
377 views

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 ...
Greg Muller's user avatar
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