Jordan-Lie Inner Ideals of the Orthogonal Lie Algebras

Abstract

Let  be an associative algebra over a field F of any characteristic with involution *  and let K=skew(A)={a in A|a*=-a} be its corresponding sub-algebra under the Lie product [a,b]=ab-ba for all a,b in A . If  for some finite dimensional vector space over  F and * is an adjoint involution with a symmetric non-alternating bilinear form on V , then * is said to be orthogonal. In this paper, Jordan-Lie inner ideals of the orthogonal Lie algebras were defined, considered, studied, and classified. Some examples and results were provided. It is proved that every Jordan-Lie inner ideals of the orthogonal Lie algebras is either eKe*  or  is a type one point space.