Jordan-Lie inner ideals of finite dimensional associative algebras (Q2301874)

Inner ideals of Lie algebras are the analogues of one-sided ideals in associative algebras and of inner ideals in Jordan algebras. Since their introduction by \textit{G. Benkart} [Trans. Am. Math. Soc. 232, 61--81 (1977; Zbl 0373.17003)], inner ideals have proved to be useful for classifying both finite-dimensional and infinite-dimensional Lie algebras. A detailed account can be found in the book of this reviewer [\textit{A. Fernández López}, Jordan structures in Lie algebras. Providence, RI: American Mathematical Society (AMS) (2019; Zbl 1441.17001)]. Let $A$ be a finite-dimensional associative algebra over an algebraically closed field of characteristic $p$ not 2 or 3. Then $A$ becomes a Lie algebra $A^{(-)}$, with Lie bracket $[a,b]=ab-ba$, and let $L=A^{(k)}$ its $k$-derived algebra, $k$ greater than or equal to zero. A subspace $B$ of $L$ is called an isotropic inner ideal if $[B, [B, L]] \subset B$ and $[B, B]=0$. Any isotropic $B$ inner ideal is abelian, i.e. $[B,B]=0$. Abelian inner ideals of $A^{(-)}$ were determined by Benkart, when $A$ is a simple Artinian ring, and by the reviewer, when $A$ is a centrally closed prime ring. In the paper under review the authors give a further step on the determination of isotropic inner ideals of a Lie algebra $L=A^{(k)}$, where $A$ is not necessarily semisimple, proving among other the following result (Corollary 1.6): Let $A$ be a finite-dimensional associative algebra over an algebraically closed field of characteristic $p$ not 2 or 3, and let $B$ be an isotropic inner ideal of $L$. Then $B=eSf \oplus B_R$, where $S$ is a Levi subalgebra of $A$, $(e, f)$ is a pair of orthogonal idempotents of $A$, and $B_R=B \cap R$, with $R$ being the radical of $L$.