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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

    ExtExt is defined in the context of homological algebra in abelian categories and also, more generally, in triangulated categories (as Hom(X,Y[i])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.