International Journal of Mathematics and Mathematical Sciences
Volume 3 (1980), Issue 2, Pages 247-253
doi:10.1155/S0161171280000178
Rings with involution whose symmetric elements are central
Department of Actuarial Mathematics, University of Manitoba, Winnipeg R3T 2N2, Manitoba, Canada
Received 30 May 1978; Revised 6 February 1979
Copyright © 1980 Taw Pin Lim. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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 dimK>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.