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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…
$Ext$ 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.
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.
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