On the commutator in Leibniz algebras
INTERNATIONAL JOURNAL OF ALGEBRA AND COMPUTATION, vol.32, no.04, pp.785-805, 2022 (SCI-Expanded, Scopus)
- Publication Type: Article / Article
- Volume: 32 Issue: 04
- Publication Date: 2022
- Doi Number: 10.1142/s0218196722500333
- Journal Name: INTERNATIONAL JOURNAL OF ALGEBRA AND COMPUTATION
- Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, Aerospace Database, Applied Science & Technology Source, Communication Abstracts, Computer & Applied Sciences, MathSciNet, Metadex, zbMATH, Civil Engineering Abstracts
- Page Numbers: pp.785-805
- Keywords: Leibniz algebras, commutator, anti-commutator, polynomial identities, computer algebra
- Süleyman Demirel University Affiliated: No
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.