2007/01/15 by Arkady Berenstein, Berenstein, Arkady, Vladimir Retakh +1
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA) #Representation Theory (math.RT) #math.QA #math.RT
paper · pdf · doi:10.48550/arxiv.math/0701399
Introduction is improved and some typos corrected. To appear in "Advances"
openalex publication_date 2007/01/15 · arxiv created 2008/02/19 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \FF we introduce and study the algebra (≫,A)(\FF), which is the Lie subalgebra of \FF ⊗ A generated by \FF ⊗ ≫. In many examples A is the universal enveloping algebra of ≫. Our description of the algebra (≫,A)(\FF) has a striking resemblance to the commutator expansions of \FF used by M. Kapranov in his approach to noncommutative geometry. To each algebra (≫, A)(\FF) we associate a ``noncommutative algebraic'' group which naturally acts on (≫,A)(\FF) by conjugations and conclude the paper with some examples of such groups.