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)

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)

Invariants
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

Examples Needed
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)


 * agreed, I tried to develop the de Rham direction a bit Olivier Peltre (talk) 17:48, 2 April 2018 (UTC)
 * Great work! Wundzer (talk) 21:13, 14 December 2020 (UTC)

Some resources
Wundzer (talk) 21:13, 14 December 2020 (UTC)
 * 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)

Completing and elaborating the article
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)

Missing definitions
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.


 * 1) The Lie algebra extensions article defines only extensions by a Lie algebra, not by a module.
 * 2) And I think, that 'module' means here a $$\mathfrak{g}$$-module, but the definition of $$\mathfrak{g}$$-module is missing from Wikipedia.
 * 3) 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)