## Cohomology of crossed Lie algebras and additive Milnor’s $$K$$- theory. (Cohomologie des algèbres de Lie croisées et $$K$$-théorie de Milnor additive.)(French)Zbl 0818.17022

Summary: In this paper we define modules of (co)-homology $${\mathfrak H}_ 0({\mathfrak G}, {\mathfrak A})$$, $${\mathfrak H}_ 1({\mathfrak G},{\mathfrak A})$$, $${\mathfrak H}^ \circ({\mathfrak G},{\mathfrak A})$$, $${\mathfrak H}^ 1({\mathfrak G},{\mathfrak A})$$ where $${\mathfrak G}$$ and $${\mathfrak A}$$ are Lie algebras with an extra structure (crossed Lie algebras). These modules satisfy the usual properties of cohomological functors, in particular existence of an exact sequence associated to a short exact sequence of coefficients. For a $$k$$-algebra $$A$$, equipped with the trivial Lie algebra structure, we use these homology modules to compare the cyclic homology group $$HC_ 1(A)$$ with an additive analogue of Milnor’s group $$K_ 2^{\text{Madd}}(A)$$.

### MSC:

 17B56 Cohomology of Lie (super)algebras 17B40 Automorphisms, derivations, other operators for Lie algebras and super algebras 17B99 Lie algebras and Lie superalgebras 19D55 $$K$$-theory and homology; cyclic homology and cohomology 18G50 Nonabelian homological algebra (category-theoretic aspects) 18G60 Other (co)homology theories (MSC2010) 19C99 Steinberg groups and $$K_2$$
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### References:

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