Double Transposed Poisson Algebras

Abstract: We introduce double transposed Poisson algebras, a noncommutative analogue of the transposed Poisson algebras of Bai, Bai, Guo and Wu that is compatible with the Kontsevich--Rosenberg principle. We first consider a simplified version which we call id-adapted double transposed Poisson algebras and then explore the general definition. We prove that every such structure on a unital associative algebra $\mathbb{A}$ is governed by a single derivation $$\mathbb{A}\to\mathbb{A}\otimes\operatorname{S}(\mathbb{A}/[\mathbb{A},\mathbb{A}])$$. Furthermore, this induces a $\operatorname{GL}_N$-equivariant transposed Poisson structure on each representation algebra $$\mathbb{A}_N=\Bbbk[\operatorname{Rep}_N(\mathbb{A})]$$. We also define $H_0$-transposed Poisson structures, the transposed counterpart of Crawley-Boevey's $H_0$-Poisson structures, and use the trace map to obtain a transposed Poisson structure on the ring of $\operatorname{GL}_N$-invariants $\mathbb{A}_N<sup>{\operatorname{GL}_N}$.

• The paper presents a classification of double transposed Poisson structures via a single derivation mapping to a symmetric algebra.
• It constructs GL-equivariant transposed Poisson structures on representation algebras through trace pushforward, linking noncommutative and commutative invariants.
• The study extends noncommutative geometry and invariant theory, offering new tools for deformation quantization and moduli space analysis.

Double Transposed Poisson Algebras: Noncommutative Structures and Representation Theory

Introduction

The paper "Double Transposed Poisson Algebras" (2607.01066) presents a significant extension of the theory of Poisson algebras by introducing double transposed Poisson algebras, situated within noncommutative geometry. Building on the framework of transposed Poisson algebras as developed by Bai et al. (Bai et al., 2020) and the double Poisson algebra formalism of Van den Bergh [math/0410528], the authors formalize a noncommutative analogue tailored to satisfy the Kontsevich–Rosenberg principle, ensuring compatibility with noncommutative smooth geometry [math/9812158]. The work addresses both foundational questions and applies the resulting theory to moduli of representations, GL-equivariant structures, and invariant theory.

Core Constructions

The authors define two main structures: the id-adapted double transposed Poisson algebra (a simplified model) and the full double transposed Poisson algebra generalization. Both are designed to capture the noncommutative analogue of transposed Poisson brackets, extending the concepts originating from the commutative setting.

A salient theoretical result is the authors’ proof that any double transposed Poisson algebra structure on a unital associative algebra $A$ is uniquely determined by a single derivation:

$D: A \to A \otimes \operatorname{S}(A/[A,A])$

where $\operatorname{S}(A/[A,A])$ denotes the symmetric algebra on the abelianization of $A$. This provides explicit and practical control over the space of such algebraic structures, reducing the classification problem to the specification of appropriate derivations.

Representation Algebras and Invariant Theory

The study further integrates representation-theoretic and invariant-theoretic aspects. For each $N \in \mathbb{N}$, the construction yields a $GL$-equivariant transposed Poisson structure on the representation algebra $A_N = [\operatorname{Rep}_N(A)]$, aligning with the Kontsevich–Rosenberg philosophy of transferring noncommutative geometry structures to moduli of representations. The $GL$-equivariance reflects the natural symmetry acting on $N$-dimensional representations, facilitating the transfer of algebraic structures through the orbit map and invoking Procesi’s invariants [procesi1976].

The authors also introduce and formalize $H_0$-transposed Poisson structures as a noncommutative counterpart to Crawley-Boevey's $D: A \to A \otimes \operatorname{S}(A/[A,A])$0-Poisson structures [math/0506268]. The trace map, central in the transfer of noncommutative geometric features to commutative invariant rings, is shown to push forward the double transposed Poisson structure to the ring of $D: A \to A \otimes \operatorname{S}(A/[A,A])$1-invariants $D: A \to A \otimes \operatorname{S}(A/[A,A])$2. This enables the passage from noncommutative to commutative Poisson structures in a manner compatible with classical invariant theory.

Structural Implications and Theoretical Significance

The approach taken in this work confirms the utility of the Kontsevich–Rosenberg principle in structuring noncommutative analogues of Poisson geometry. It applies the principle not just in the context of double Poisson brackets but extends it to a transposed setting, thereby unifying various recent directions in noncommutative Poisson geometry. The classification of double transposed Poisson structures via derivations provides an actionable tool for constructing new examples and studying their deformation properties.

Critically, the method yields explicit tools for constructing equivariant Poisson structures on representation spaces, with consequences for the study of moduli of representations in noncommutative geometry. Notably, the trace pushforward offers a systematic pathway to obtain transposed Poisson structures on commutative rings of invariants, a process central to the connection between noncommutative and classical geometry.

Numerical Results and Bold Claims

The primary theoretical claim of this paper is that the space of double transposed Poisson structures on a unital associative algebra $D: A \to A \otimes \operatorname{S}(A/[A,A])$3 is classified by the space of derivations $D: A \to A \otimes \operatorname{S}(A/[A,A])$4. This is a strong assertion that consolidates the algebraic structure into a single derivational object, streamlining the construction and analysis of such algebras.

While the paper is primarily algebraic, the application to representation algebras and equivariant structures implies concrete computational frameworks for explicit examples, especially in low-dimensional cases and for path algebras of quivers.

Implications and Future Developments

These results carry several implications:

• Noncommutative Geometric Deformation Theory: The new structure stands to inform the deformation theory of noncommutative spaces, particularly in contexts where double or transposed brackets play a role.
• Representation Theory: The construction of $D: A \to A \otimes \operatorname{S}(A/[A,A])$5-equivariant (transposed) Poisson structures on representation spaces could yield new invariants and facilitate the study of symplectic and Poisson geometry of moduli spaces.
• Quantum Algebras and Deformation Quantization: Given the relationship between Poisson and quantum algebras, prospects for generalizing the approach to $D: A \to A \otimes \operatorname{S}(A/[A,A])$6-products and noncommutative quantization—particularly double star-products as discussed in related works [(Safonkin, 31 May 2025)v4]—are immediate.
• Higher Structures: The connection to $D: A \to A \otimes \operatorname{S}(A/[A,A])$7-Poisson structures and the compatibility with trace processes foreshadows the systematic study of traces and symmetrized invariants in noncommutative Poisson geometry, impacting both theoretical understanding and concrete computations.

Conclusion

The paper develops a robust framework for double transposed Poisson algebras, bridging noncommutative Poisson geometry and the study of representations. By classifying these structures via derivations and demonstrating their compatibility with $D: A \to A \otimes \operatorname{S}(A/[A,A])$8-equivariant and invariant-theoretic contexts, the authors provide new tools for the exploration of noncommutative algebraic geometry and representation theory. The work sets the stage for further investigation into quantization, moduli space geometry, and the role of transposed structures in theoretical and computational aspects of noncommutative geometry.