[Paper Review] Transposed Poisson structures

Patrícia Damas Beites, Bruno Leonardo Macedo Ferreira+1 more|arXiv (Cornell University)|Jul 1, 2022

Advanced Topics in Algebra78 references4 citations

TL;DR

This paper provides a comprehensive survey of transposed Poisson algebras, introducing new results and establishing connections with other algebraic structures such as Lie algebras, Hom-Lie algebras, and Poisson $n$-Lie algebras. It proves that the operad of transposed Poisson algebras is not Koszul, resolves open questions on bialgebras and deformations, and proposes new structures like transposed Poisson $n$-Lie algebras and transposed Gerstenhaber algebras.

ABSTRACT

To present a survey on known results from the theory of transposed Poisson algebras, as well as to establish new results on this subject, are the main aims of the present paper. Furthermore, a list of open questions for future research is given.

Motivation & Objective

• To survey known results and establish new findings in the theory of transposed Poisson algebras.
• To explore structural and operadic properties, including Koszul duality and self-duality.
• To investigate connections between transposed Poisson algebras and other algebraic systems such as $F$-manifold algebras, GD-algebras, and Poisson $n$-Lie algebras.
• To address open problems in bialgebra theory, deformation quantization, and cohomology of transposed Poisson algebras.
• To propose new algebraic structures such as transposed Poisson $n$-Lie algebras and transposed Gerstenhaber algebras.

Proposed method

• Utilizes the definition of transposed Poisson algebras: a commutative associative algebra $(\mathfrak{L}, \cdot)$ and a Lie algebra $(\mathfrak{L}, [\cdot,\cdot])$ satisfying $2z\cdot[x,y] = [z\cdot x,y] + [x,z\cdot y]$.
• Applies methods for classifying transposed Poisson structures on specific Lie algebras, including the Witt algebra, Virasoro algebra, and Heisenberg algebras.
• Employs the Kantor product construction to generate new examples of transposed Poisson algebras from multiplications on the same vector space.
• Analyzes the operad $\mathrm{TP}$ of transposed Poisson algebras using generating functions and functional equations to test Koszul duality.
• Uses the isomorphism between the weak Leibniz operad and the transposed Poisson operad to prove non-Koszulity via counterexample in degree 5.
• Proposes a recursive construction of transposed Poisson $(n+1)$-Lie algebras from transposed Poisson $n$-Lie algebras using derivations and $n$-ary operations.

Experimental results

Research questions

• RQ1 Is the operad of transposed Poisson algebras Koszul?
• RQ2 Can transposed Poisson algebras be realized as semi-limits of deformation quantization of commutative algebras?
• RQ3 What are the cohomology theories of transposed Poisson algebras, and how do they relate to Poisson cohomology?
• RQ4 Can transposed Poisson bialgebras and double transposed Poisson algebras be defined and studied?
• RQ5 What is the structure and classification of simple transposed Poisson $n$-Lie algebras?

Key findings

• The operad of transposed Poisson algebras is not Koszul, as shown by the functional equation $f(-f(-x)) = x + \frac{7}{30}x^5 + O(x^6)$, which fails the Koszul condition.
• The operad of transposed Poisson algebras is self-dual under Koszul duality, but not Koszul, due to the non-vanishing degree-5 correction.
• The operad of weak Leibniz algebras is isomorphic to the operad of transposed Poisson algebras, and since the former is not Koszul, the latter is not either.
• A recursive construction of transposed Poisson $(n+1)$-Lie algebras from $n$-Lie algebras via derivations and $n$-ary operations is proposed and verified in a particular case.
• The tensor product of transposed Poisson algebras inherits a canonical transposed Poisson algebra structure, implying that the operad $\mathrm{TPA}$ is a Hopf operad.
• The paper introduces and motivates the study of transposed Gerstenhaber algebras and transposed Poisson $n$-Lie algebras as natural generalizations of the original structure.