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.