Lie algebras and Lie groups over noncommutative rings

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.