Noncommutative Loops over Lie Algebras and Lie Groups
نویسنده
چکیده
The aim of this paper is to introduce and study Lie algebras over noncommutative rings. For any Lie algebra g sitting inside an associative algebra A and any associative algebra F we introduce and study the F -loop algebra (g, A)(F), which is the Lie subalgebra of F ⊗ A generated by F ⊗ g. In most examples A is the universal enveloping algebra of g. Our description of the loop algebra has a striking resemblance to the commutator expansions of F used by M. Kapranov in his approach to noncommutative geometry. We also associate with each F -loops algebras (g, A)(F) a “noncommutative algebraic” group which naturally acts on (g, A)(F) by conjugations and discuss examples of such groups.
منابع مشابه
Lie algebras and Lie groups over noncommutative rings
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)...
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