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The aim of this paper is to introduce and study Lie algebras and Lie groups over noncommutative rings. For any Lie algebra $gg$ sitting inside an associative algebra $A$ and any associative algebra $FF$ we introduce and study the algebra $(gg,A)(FF)$, which is the Lie subalgebra of $FF otimes A$ generated by $FF otimes gg$. In many examples $A$ is the universal enveloping algebra of $gg$. Our description of the algebra $(gg,A)(FF)$ has a striking resemblance to the commutator expansions of $FF$ used by M. Kapranov in his approach to noncommutative geometry. To each algebra $(gg, A)(FF)$ we associate a ``noncommutative algebraic group which naturally acts on $(gg,A)(FF)$ by conjugations and conclude the paper with some examples of such groups.
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A representation of the central extension of the unitary Lie algebra coordinated with a skew Laurent polynomial ring is constructed using vertex operators over an integral Z_2-lattice. The irreducible decomposition of the representation is explicitly computed and described. As a by-product, some fundamental representations of affine Kac-Moody Lie algebra of type $A_n^{(2)}$ are recovered by the new method.
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We study five different types of the homology of a Lie algebra over a commutative ring which are naturally isomorphic over fields. We show that they are not isomorphic over commutative rings, even over $mathbb Z,$ and study connections between them. In particular, we show that they are naturally isomorphic in the case of a Lie algebra which is flat as a module. As an auxiliary result we prove that the Koszul complex of a module $M$ over a principal ideal domain that connects the exterior and the symmetric powers $0to Lambda^n Mto M otimes Lambda^{n-1} M to dots to S^{n-1}M otimes M to S^nMto 0 $ is purely acyclic.
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