Shift associative algebras

TL;DR

This paper introduces and studies shift associative algebras, exploring their properties, classifications, and relations to other algebraic structures, including their basis, duality, and dimensional characteristics.

Contribution

It provides a comprehensive analysis of shift associative algebras, including their basis, classification, and connections to other algebraic systems, with new results on their structure and dimensionality.

Findings

Constructed a basis for free shift associative algebra.

Established an analog of Wedderburn-Artin's theorem.

Classified complex 4-dimensional shift associative algebras.

Abstract

We present a comprehensive study of algebras satisfying the identity $(xy)z=y(zx),$ named as shift associative algebras. Our research shows that these algebras are related to many interesting identities. In particular, they are related to anti-Poisson-Jordan algebras and algebras of associative type $\sigma$. We study algebras of associative type $\sigma$ to be Koszul and self-dual. A basis of the free shift associative algebra generated by a countable set $X$ was constructed. An analog of Wedderburn-Artin's theorem was established. The algebraic and geometric classifications of complex $4$-dimensional shift associative algebras are given. In particular, we proved that the first non-associative shift associative algebra appears only in dimension $5$.