Resumen de Lie algebras and lie groups over noncommutative rings

Arkady Berenstein, Vladimir Retakh

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