Talk:Lie algebra cohomology

The link to Weyl's theorem currently leads to a disambiguation page. Someone with more knowledge of the subject should correct the link to lead to a specific article.--Bill 19:59, 3 February 2006 (UTC)[reply]

I am going to add some stuff to this from Kassel's quantum groups text unless someone strenuously objects. Myrkkyhammas 16:42, 17 April 2007 (UTC)[reply]

Invariants[edit]

I fixed a bug in the definition of the invariants module. For a Lie algebra ${\mathfrak {g}}$ , the invariants of a ${\mathfrak {g}}$ -module are the elements $m$ such that $xm=0$ for all $x\in {\mathfrak {g}}$ . Then the action on $M^{\mathfrak {g}}$ is trivial (as an endomorphism). For a Lie group $G$ the invariants are the elements such that $gm=m$ for all $g\in G$ so the action on $V^{G}$ is trivial (as an isomorphism). --Xtquique (talk) 19:20, 14 August 2014 (UTC)xtquique[reply]

Examples Needed[edit]

Elementary examples of lie algebra cohomology are needed. In addition, theorems related to the de-Rham cohomology of Lie groups and their lie algebras needs to be added. — Preceding unsigned comment added by 73.181.114.81 (talk) 01:05, 29 May 2017 (UTC)[reply]

agreed, I tried to develop the de Rham direction a bit Olivier Peltre (talk) 17:48, 2 April 2018 (UTC)[reply]

Great work! Wundzer (talk) 21:13, 14 December 2020 (UTC)[reply]

Some resources[edit]

http://www-math.mit.edu/~dav/cohom.pdf
https://www.mathematik.tu-darmstadt.de/media/algebra/dafra/notizen/2018-11-15_Zorbach_liecohomology.pdf
Compute ${\mathfrak {sl}}_{2}(\mathbb {C} )$ and another simple example (low dim so or sp)

Wundzer (talk) 21:13, 14 December 2020 (UTC)[reply]

Completing and elaborating the article[edit]

I will try to develop the article a bit and add a few more references (Cartan, Koszul...). Also in my opinion, the definition with Ext and Tor would be more suited later on as it is a bit over-kill in this case and requires solid knowledge in category theory. Any comments and suggestions welcome! Olivier Peltre (talk) 17:48, 2 April 2018 (UTC)[reply]

Missing definitions[edit]

In the Cohomology in small dimensions section we see this:

The second cohomology group

$H^{2}({\mathfrak {g}};M)$

is the space of equivalence classes of Lie algebra extensions

$0\rightarrow M\rightarrow {\mathfrak {h}}\rightarrow {\mathfrak {g}}\rightarrow 0$

of the Lie algebra by the module $M$ .

A bit hard to understand this because of some lacking definitions.

The Lie algebra extensions article defines only extensions by a Lie algebra, not by a module.
And I think, that 'module' means here a ${\mathfrak {g}}$ -module, but the definition of ${\mathfrak {g}}$ -module is missing from Wikipedia.
According to the introduction, this is related with a concept of Lie module which is also missing from Wikipedia. — Preceding unsigned comment added by 89.135.79.17 (talk) 05:34, 7 October 2019 (UTC)[reply]