Rings with involution whose symmetric elements are central

Taw Pin Lim

International Journal of Mathematics and Mathematical Sciences, 1980, vol. 3, 1-7

Abstract:

In a ring R with involution whose symmetric elements S are central, the skew-symmetric elements K form a Lie algebra over the commutative ring S . The classification of such rings which are 2 -torsion free is equivalent to the classification of Lie algebras K over S equipped with a bilinear form f that is symmetric, invariant and satisfies [ [ x , y ] , z ] = f ( y , z ) x − f ( z , x ) y . If S is a field of char ≠ 2 , f ≠ 0 and dim K > 1 then K is a semisimple Lie algebra if and only if f is nondegenerate. Moreover, the derived algebra K ′ is either the pure quaternions over S or a direct sum of mutually orthogonal abelian Lie ideals of dim ≤ 2 .

Date: 1980
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:762359

DOI: 10.1155/S0161171280000178

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