Lie algebras and Lie groups over noncommutative rings |
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| Authors: | Arkady Berenstein Vladimir Retakh |
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| Affiliation: | a Department of Mathematics, University of Oregon, Eugene, OR 97403, USA
b Department of Mathematics, Rutgers University, Piscataway, NJ 08854, USA |
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| Abstract: | 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. |
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| Keywords: | Lie algebra Semisimple Lie algebra Lie group Noncommutative ring |
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