On the generalized Poisson and transposed Poisson algebras

TL;DR

This work characterizes algebras that are simultaneously generalized Poisson and transposed Poisson with a derivation $D$, providing explicit polynomial identities that capture their overlap and showing how polarization and depolarization relate single-operation and two-operation descriptions. It establishes that $a\cdot [b,c]=a\cdot D(b)\cdot c-a\cdot b\cdot D(c)$ and $[a\cdot b,c]=D(a)\cdot b\cdot c+a\cdot D(b)\cdot c-a\cdot b\cdot D(c)$ are the defining relations for such algebras, and demonstrates productive constructions from commutative associative algebras with derivations and from Zinbiel algebras, including an explicit example on $\mathbb{C}[x]$ with $D=\frac{d}{dx}$ leading to $[f,g]=D(f)\int_0^x g\,dx - D(g)\int_0^x f\,dx$. The paper also provides depolarization criteria via a single product $\star$ with $D$, introducing the identities that link $\star$ to the GP/transposed structure and proving the converse. Finally, Itô’s theorem is extended to generalized Poisson algebras, proving that a non-commutative GP algebra that is the sum of two abelian subalgebras is metatrivial, thus yielding an Itô-type metabelian property for this class. Together, these results advance structural understanding and yield concrete constructions of algebras with dual Poisson-type behaviors.

Abstract

We provide the polynomial identities of algebras that are both generalized Poisson algebras and transposed Poisson algebras. We establish defining identities via single operation for generalized Poisson algebras and prove that Ito's theorem holds for generalized Poisson algebras.

Theorems & Definitions (10)

• Theorem 2.1
• Corollary 2.2
• Proposition 2.3
• Corollary 2.4
• Theorem 3.1
• Lemma 4.1
• Theorem 4.2
• Definition 5.1: BCZ23
• Definition 5.2
• Theorem 5.3