Symmetric Zinbiel superalgebras

TL;DR

This paper introduces symmetric Zinbiel superalgebras, establishing their nilpotency bounds, classifying certain generated structures, and exploring their identities and extensions within algebraic frameworks.

Contribution

It provides new insights into the structure, nilpotency, and classifications of symmetric Zinbiel superalgebras, including their identities and extension methods.

Findings

Nilpotency index of symmetric Zinbiel superalgebras is at most 4

Two-generated symmetric Zinbiel algebras are characterized

Quadratic Zinbiel algebras are 2-step nilpotent

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.