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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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