Let A be an associative algebra over a field š½ of any characteristic with involution and let K = skew (A) = { a ā A | a* = ā a} be its corresponding sub-algebra under the Lie product [a, b] = ab ā ba for all a, b ā A. In this paper, inner ideals of such Lie algebras were defined, considered, studied, and classified. Some examples and results were provided. It is proved that for every Jordan-Lie inner ideal of K, one can find an idempotent e ā A such that this inner ideal may be written in the form eK e*. It is also proved inner ideals of such Lie algebras are regular.
Let Ā be an associative algebra over a field F of any characteristic with involution *Ā 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 Ā for some finite dimensional vector space overĀ 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*Ā or Ā is a type one point space.
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