On the commutator in Leibniz algebras


Dzhumadil'daev A. S. , Ismailov N. A. , Sartayev B. K.

INTERNATIONAL JOURNAL OF ALGEBRA AND COMPUTATION, vol.32, no.04, pp.785-805, 2022 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 32 Issue: 04
  • Publication Date: 2022
  • Doi Number: 10.1142/s0218196722500333
  • Title of Journal : INTERNATIONAL JOURNAL OF ALGEBRA AND COMPUTATION
  • Page Numbers: pp.785-805
  • Keywords: Leibniz algebras, commutator, anti-commutator, polynomial identities, computer algebra

Abstract

We prove that the class of algebras embeddable into Leibniz algebras with respect to the commutator product is not a variety. It is shown that every commutative metabelain algebra is embeddable into Leibniz algebras with respect to the anti-commutator. Furthermore, we study polynomial identities satisfied by the commutator in every Leibniz algebra. We extend the result of Dzhumadil'daev in [A. S. Dzhumadil'daev, q-Leibniz algebras, Serdica Math. J. 34(2) (2008) 415-440]. to identities up to degree 7 and give a conjecture on identities of higher degrees. As a consequence, we obtain an example of a non-Spechtian variety of anticommutative algebras.