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• CommentRowNumber1.
• CommentAuthorjim_stasheff
• CommentTimeJul 7th 2012
Tor and Ext are usually defined for modules over an associative ring.
Is there analogous terminology and machinery and formalism for modules over a Lie algebra
*without* passing to the universal enveloping thus `reducing to the previous case'?
cf. Lie algebra cohomology
• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeJul 8th 2012
• (edited Jul 8th 2012)

It is kind of clea what do do in principle: set up a homotopical category of complexes of Lie modules and consider the derived hom and derived tensor product functor. But you are asking for specific literature, right? I’d need to check…

• CommentRowNumber3.
• CommentAuthorzskoda
• CommentTimeJul 9th 2012

$Ext$ is defined in the context of homological algebra in abelian categories and also, more generally, in triangulated categories (as $Hom(X,Y[i])$); I see no need to interpret the abelian category of modules in a specific way via enveloping algebras to make sense of that.

• CommentRowNumber4.
• CommentAuthorMatanP
• CommentTimeJul 9th 2012

One point to start with is Hinich’s paper HOMOLOGICAL ALGEBRA OF HOMOTOPY ALGEBRAS.

using this, one can get a (projective) model structure on DGL and also modules over an algebra. Then, as said in #3 Ext and Tor may be defined.

Hope what I wrote answers your question and not a different one :)

btw, since this model structure admits functorial factorizations, one can have a definition of derived functors on the infinity-category itself and not just on its homotopy category.

• CommentRowNumber5.
• CommentAuthorjim_stasheff
• CommentTimeJul 9th 2012
Thanks to all, so indeed they are still called Tor and Ext as I would have hoped.
Any specific references in the Lie context would be welcome, that the history not be lost.