On algebras embeddable into bicommutative algebras

Abstract

We consider algebras embeddable into free bicommutative algebras with respect to commutator and anti-commutator products over a field of characteristic zero. We show that every metabelian Lie algebra can be embedded into a bicommutative algebra with respect to the commutator product. Furthermore, we prove that the class of commutative algebras embeddable into bicommutative algebras with respect to the anti-commutator product forms a variety. As a consequence, we obtain that every metabelian Lie algebra can be embedded into a Novikov algebra with respect to the commutator product.

KEYWORDS:

• Bicommutative algebra
• metabelian Lie algebra
• speciality problem

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 17A30
• 17A50
• 17D25

Acknowledgments

We are grateful to the anonymous reviewer for the careful reading of our text, valuable comments and Remark 4.2.

Disclosure statement

There is no competing interest.

Data availability statement

No data was used for the research.

Ethical approval

Not applicable.

Additional information

Funding

This work was supported by the Science Committee of the Ministry of Science and Higher Education of the Republic of Kazakhstan (Grant No. AP14870282).