New splittings of operations of Poisson algebras and transposed Poisson algebras and related algebraic structures

TL;DR

This paper introduces new algebraic structures arising from different ways of splitting operations in Poisson and transposed Poisson algebras, revealing novel interlaced algebraic frameworks and their representations.

Contribution

It develops a comprehensive theory of mixed splittings of operations in Poisson and transposed Poisson algebras, identifying eight new algebraic structures for each case and analyzing their representation theory.

Findings

Eight algebraic structures for Poisson algebras from mixed splittings.

Eight algebraic structures for transposed Poisson algebras from mixed splittings.

Distinct differences between Poisson and transposed Poisson algebra structures.

Abstract

There are two kinds of splittings of operations, namely, the classical splitting which is interpreted operadically as taking successors and another splitting which we call the second splitting giving the anti-structures of the successors' algebras. The algebraic structures corresponding to them respectively are characterized in terms of representations. Due to the appearance of the two bilinear operations in Poisson algebras and transposed Poisson algebras, we commence to study new splittings of operations in the ``mixed" sense that the commutative associative products and Lie brackets are splitted in different manners respectively, that is, they are splitted interlacedly in three manners: the classical splitting, the second splitting and the un-splitting. Accordingly the corresponding algebraic structures are given. More explicitly, there are 8 algebraic structures interpreted in terms of representations of Poisson algebras illustrating the mixed splittings of operations of Poisson algebras respectively, including the known pre-Poisson algebras. For illustrating the mixed splittings of operations of transposed Poisson algebras, there are 8 algebraic structures interpreted in terms of representations of transposed Poisson algebras on the spaces themselves and another 8 algebraic structures interpreted in terms of representations of transposed Poisson algebras on the dual spaces. Moreover, such a phenomenon exhibits an obvious difference between Poisson algebras and transposed Poisson algebras.