Symmetric Zinbiel superalgebras

Abstract

The notion of symmetric Zinbiel superalgebras is introduced. We prove that the nilpotency index of a symmetric Zinbiel superalgebra is not greater than 4 and describe two-generated symmetric Zinbiel algebras and odd generated superalgebras. We discuss identities of mono and binary symmetric Zinbiel and Leibniz algebras. It is proven that each quadratic Zinbiel algebra is 2-step nilpotent. Also, we study double extensions of symmetric Zinbiel algebras.

Keywords:

• Double extension
• dual Leibniz algebra
• Zinbiel algebra

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 17A30
• 17A32

Acknowledgments

The authors thank Pasha Zusmanovich for some useful discussions. We also thank the referee of the paper for some useful suggestions on how to improve the part of mono and binary Leibniz and Zinbiel algebras.

Additional information

Funding

The work is supported by FCT UIDB/00212/2020 and UIDP/00212/2020; RFBR 20-01-00030.