Rings with involution whose symmetric elements are central

• Main Author: Taw Pin Lim
• Format: Article
• Language: English
• Published: Hindawi Limited 1980-01-01
• Series: International Journal of Mathematics and Mathematical Sciences
• Subjects: ring with involution; symmetric and skew-symmetric elements; Lie algebra; symmetric and invariant bilinear form; Cartan's criterion of semisimplicity of Lie algebras; pure quaternions; mutually orthogonal abelian Lie ideals.
• Online Access: http://dx.doi.org/10.1155/S0161171280000178
• ISSN: 0161-1712; 1687-0425

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.