Abelian Non Jordan-Lie Inner Ideals of the Orthogonal Finite Dimensional Lie Algebras

Abstract

Let A associative algebra (finite dimensional over ) of any characteristic with involution * and let K = skew(A) = {a ∈ A|a* = -a} be its corresponding sub-algebra under the Lie product [a, b] = ab - ba for all a, b ∈ A. If A = End V for some finite dimensional vector space over and * is an adjoint involution with a symmetric non-alternating bilinear form on V, then * is said to be orthogonal. In this paper, abelian inner ideals which are non Jordan-Lie of such Lie algebras were defined, considered, studied, and classified. Some examples and results were provided. It is proved that every abelian inner ideal which is non Jordan-Lie B of K can be expressed as B = {v, H ⊥}, where v is an isotropic vector of a hyperbolic plane H ⊆ V and H ⊥ is the orthogonal subspace of H.