On certain classes of algebras in which centralizers are ideals

Towers, David and Saha, Ripan (2021) On certain classes of algebras in which centralizers are ideals. Journal of Lie Theory, 31 (4). pp. 991-1002. ISSN 0949-5932

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Abstract

This paper is primarily concerned with studying finite-dimensional anti-commutative nonassociative algebras in which every centralizer is an ideal. These are shown to be anti-associative and are classified over a general field $F$; in particular, they are nilpotent of class at most $3$ and metabelian. These results are then applied to show that a Leibniz algebra over a field of charactersitic zero in which all centralizers are ideals is solvable.