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• CommentRowNumber1.
• CommentAuthorzskoda
• CommentTimeDec 1st 2009
• (edited Dec 1st 2009)
I added word "abelian" when the article Lie algebra cohomology talks about Lie algebra cohomology as cohomology of CE complex. While one can combine this abelian cohomology with various actions to get nonabelian information, there is also a genuine nonabelian cohomology of Lie algebra as it is for Lie groups. At least some special cases are understood in low dimensions. For example the problem of extensions of Lie algebras by nonabelian Lie algebras leads to 1,2,3 nonabelian cocycles; 2-cocycles are analogues of factor systems. I will create some entry about that, and put a longer exposition in "private" web following my old notes from Spring/Summer 1997. For start I just started a stub nonabelian Lie algebra cohomology.
• CommentRowNumber2.
• CommentAuthorzskoda
• CommentTimeDec 1st 2009
• (edited Dec 1st 2009)

It would be nice if Danny who worked a lot on this topic joins us writing someting about it. I worked on this in 1997. but my results were not so advanced as his. In 2000 there is a short preprint which got just a bit more than where I got, but less than what Danny had

• Dmitri Alekseevsky, Peter W. Michor, Wolfgang Ruppert, Extensions of Lie algebras math.DG/0005042
• CommentRowNumber3.
• CommentAuthorzskoda
• CommentTimeDec 17th 2009
• (edited Dec 17th 2009)

Hi, this is Urs speaking here, using Zoran"s account. (I am at his appartment and cannot use my own machine to go online here.)

I just added to nonabelian Lie algebra cohomology at Abstract definition the details of the anounced claim and proof about how nonabelian Lie algebra cohomology is an example of the general nonsense of cohomology.

• CommentRowNumber4.
• CommentAuthorzskoda
• CommentTimeDec 17th 2009
• (edited Dec 17th 2009)

I (Zoran this time) created Lie algebra extension, an entry related to nonabelian Lie algebra cohomology entry.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeAug 16th 2010

I removed the query box that used to be at the end of the long proof at nonabelian Lie algebra cohomology and replaced it with a note in the text.

The query box had been worried that the proof had a gap due to missing cofibrancy in $dgAlg$, but in fact what the proof really needs is cofibrancy in $dgAlg^{op}$, which is trivial. So all is fine with the proof and only the query box was confused and had to go.

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeAug 16th 2010

Hm, on the other hand the other object should be fibrant in $dgAlg^{op}$. Hm, still needs attention after all.