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.
All Science Journal Classification (ASJC) codes
- Lie algebra
- Lie group
- Noncommutative ring
- Semisimple Lie algebra