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1. • CommentRowNumber1.
   • CommentAuthorjim_stasheff
   • CommentTimeJul 7th 2012
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   Author: jim_stasheff
   Format: TextTor 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

   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
2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJul 8th 2012
   • (edited Jul 8th 2012)
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   Author: Urs
   Format: MarkdownItexIt is kind of clea what do do in principle: set up a homotopical category of complexes of Lie modules and consider the [[Ext|derived hom]] and [[Tor|derived tensor product]] functor. But you are asking for specific literature, right? I'd need to check...

   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…

3. • CommentRowNumber3.
   • CommentAuthorzskoda
   • CommentTimeJul 9th 2012
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   Author: zskoda
   Format: MarkdownItex$Ext$ 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.

   $Ext$ 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.

4. • CommentRowNumber4.
   • CommentAuthorMatanP
   • CommentTimeJul 9th 2012
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   Author: MatanP
   Format: MarkdownItexOne 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.

   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.

5. • CommentRowNumber5.
   • CommentAuthorjim_stasheff
   • CommentTimeJul 9th 2012
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   Author: jim_stasheff
   Format: TextThanks 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.

   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.