On the commutator in Leibniz algebras

Askar Serkulovich Dzhumadil'daev, N. A. Ismailov, B. K. Sartayev

2022

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DOI 10.1142/s0218196722500333
Data confidence 70 medium
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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.