On the Kantor product, II

TL;DR

This paper extends the classification of Kantor products, describes the Kantor square for low-dimensional algebras, and provides methods to construct and classify various algebraic structures such as transposed Poisson and Poisson-Novikov algebras.

Contribution

It introduces explicit descriptions and constructive methods for Kantor squares and related algebraic structures, advancing the understanding of their classifications.

Findings

Explicit descriptions of Kantor squares for low-dimensional algebras

Constructive methods for new transposed Poisson and Poisson-Novikov algebras

Classification techniques for Poisson and commutative post-Lie structures

Abstract

We describe the Kantor square (and Kantor product) of multiplications, extending the classification proposed in [I. Kaygorodov, On the Kantor product, Journal of Algebra and Its Applications, 16 (2017), 9, 1750167]. Besides, we explicitly describe the Kantor square of some low dimensional algebras and give constructive methods for obtaining new transposed Poisson algebras and Poisson-Novikov algebras; and for classifying Poisson structures and commutative post-Lie structures on a given algebra.