$\delta$-Poisson and transposed $\delta$-Poisson algebras

TL;DR

This paper introduces and studies two new classes of Poisson-type algebras, exploring their identities, classifications, and structural properties, including their relations to other algebraic systems and their Koszul and self-dual characteristics.

Contribution

It defines $ ext{delta}$-Poisson and transposed $ ext{delta}$-Poisson algebras, classifies simple cases, and constructs bases for free algebras, advancing understanding of their structure and relationships.

Findings

Classified simple $ ext{delta}$-Poisson and transposed $ ext{delta}$-Poisson algebras.

Established relations to shift associative, $F$-manifold, and Jordan bracket algebras.

Constructed bases for free $ ext{delta}$-Poisson and mixed-Poisson algebras.

Abstract

We present a comprehensive study of two new Poisson-type algebras. Namely, we are working with $\delta$-Poisson and transposed $\delta$-Poisson algebras. Our research shows that these algebras are related to many interesting identities. In particular, they are related to shift associative algebras, $F$-manifold algebras, algebras of Jordan brackets, etc. We classify simple $\delta$-Poisson and transposed $\delta$-Poisson algebras and found their depolarizations. We study $\delta$-Poisson and mixed-Poisson algebras to be Koszul and self-dual. Bases of the free $\delta$-Poisson and mixed-Poisson algebras generated by a countable set $X$ were constructed.