Journal of Lie Theory 31 (2021), No. 4, 991--1002
Copyright Heldermann Verlag 2021
On Certain Classes of Algebras in which Centralizers are Ideals
Ripan Saha
Department of Mathematics, Raiganj University, Raiganj 733134, India
ripanjumaths@gmail.com
David A. Towers
Department of Mathematics and Statistics, Lancaster University, Lancaster, England
d.towers@lancaster.ac.uk
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 field F of characteristic different from 2; 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.
Keywords: Anti-commutative algebra, anti-associative algebra, Lie algebra, Leibniz algebra, mock-Lie algebra, centralizer, nilpotent algebra.
MSC: 17A30, 17A32, 17B30.
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