Generalization of Jordan-Lie inner ideals of finite dimensional associative algebras

Abstract

We generalize Baranov and Shlaka’s results about bar-minimal Jordan-Lie and regular inner ideals of finite dimensional associative algebras. Let [Formula: see text] be a finite dimensional [Formula: see text]-perfect associative algebras [Formula: see text] over an algebraically closed field [Formula: see text] of arbitrary characteristic [Formula: see text] and let [Formula: see text] be a subalgebra of [Formula: see text]. We prove that for any bar-minimal Jordan-Lie inner ideal [Formula: see text] a [Formula: see text], there is a bar-minimal Jordan-Lie inner ideal of [Formula: see text] that contains [Formula: see text] and if [Formula: see text] is regular, then is a regular inner ideal of [Formula: see text] that contains [Formula: see text]. We also prove that for any strict orthogonal pair [Formula: see text] in [Formula: see text], there is a strict orthogonal idempotent pair [Formula: see text] in [Formula: see text] such that [Formula: see text].