Lie algebras and Lie groups over noncommutative rings

Abstract

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