Jacobi - type identities in algebras and superalgebras

TL;DR

This paper introduces new algebraic identities involving commutators and anticommutators, generalizes them to superalgebras, and explores their implications for associativity and symplectic supermanifolds.

Contribution

It presents novel identities in associative algebras and superalgebras, establishing their fundamental role and extending classical results to supergeometry.

Findings

Derived a fundamental identity from which other identities follow

Proved that the fundamental identity implies associativity in the algebra

Extended identities to superalgebras and symplectic supermanifolds

Abstract

We introduce two remarkable identities written in terms of single commutators and anticommutators for any three elements of arbitrary associative algebra. One is a consequence of other (fundamental identity). From the fundamental identity, we derive a set of four identities (one of which is the Jacobi identity) represented in terms of double commutators and anticommutators. We establish that two of the four identities are independent and show that if the fundamental identity holds for an algebra, then the multiplication operation in that algebra is associative. We find a generalization of the obtained results to the super case and give a generalization of the fundamental identity in the case of arbitrary elements. For nondegenerate even symplectic (super)manifolds, we discuss analogues of the fundamental identity.