(Transposed) Poisson algebra structures on null-filiform associative algebras

TL;DR

The paper addresses the problem of classifying all $(\text{transposed})$ Poisson algebra structures on the canonical null-filiform associative algebra $\mu_0^n$ over $\mathbb{C}$. It constructs a parameterized family $\mathbf{TP}(\alpha_2,\dots,\alpha_n)$ describing the transposed Poisson brackets and uses automorphisms of $\mu_0^n$ to normalize parameters, yielding a complete list of non-isomorphic algebras: $\mathbf{TP}(1,0,\dots,0)$, $\mathbf{TP}(0,\alpha,0,\dots,0)$, and the families $\mathbf{TP}(0,\dots,0,1_s,0,\dots,0,\alpha_{2s-3},0,\dots,0)$ for $4\le s\le n$ (with $\alpha,\alpha_{2s-3}\in\mathbb{C}$), plus the trivial $\mathbf{TP}(0,\dots,0)$ and low-dimensional cases. It further shows that every Poisson structure on $\mu_0^n$ is trivial. The results provide a complete classification that clarifies the landscape of transposed Poisson structures on null-filiform algebras and informs related derivation and Lie-algebraic constructions.

Abstract

In this paper we investigate classifications of all (transposed) Poisson algebras of the associated associative null-filiform algebra

Theorems & Definitions (29)

• Definition 1
• Definition 2
• Proposition 3: Bai
• Definition 4
• Theorem 5: MO
• Theorem 6: aku
• Theorem 7
• proof
• Theorem 8
• proof