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 g sitting inside an associative algebra A and any associative algebra F we introduce and study the algebra (g,A)(F), which is the Lie subalgebra of F⊗A generated by F⊗g. In many examples A is the universal enveloping algebra of g. Our description of the algebra (g,A)(F) has a striking resemblance to the commutator expansions of F used by M. Kapranov in his approach to noncommutative geometry. To each algebra (g,A)(F) we associate a “noncommutative algebraic” group which naturally acts on (g,A)(F) by conjugations and conclude the paper with some examples of such groups.