Lie algebras and Lie groups over noncommutative rings

Arkady Berenstein, Vladimir Retakh

Research output: Contribution to journalArticlepeer-review

3 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)1723-1758
Number of pages36
JournalAdvances in Mathematics
Issue number6
StatePublished - Aug 20 2008

All Science Journal Classification (ASJC) codes

  • General Mathematics


  • Lie algebra
  • Lie group
  • Noncommutative ring
  • Semisimple Lie algebra


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