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Double Poisson brackets on low dimensional algebras

G. I. Sharygin, A. Hernandez Rodriguez

TL;DR

This work analyzes double Poisson brackets in the sense of Van den Bergh on finite‑dimensional algebras, showing that all brackets on matrix algebras are inner and thereby extend to finite‑dimensional semisimple algebras via Morita invariance; it then examines the non‑semisimple case of $2\times 2$ upper triangular matrices, describing all brackets and the induced Poisson structures on low‑dimensional representation spaces, and finally discusses modified double Poisson brackets in the Arthamonov framework. The authors develop the foundational definitions and constructions, connect them to representation spaces $\mathrm{Rep}_n(\mathcal{A})$ and to Hochschild (co)homology, and provide explicit computations illustrating how inner brackets arise from $r\in \Lambda^2\mathcal{A}$ and how nontrivial Poisson structures appear in non‑semisimple settings. Their results clarify when nontrivial Poisson structures can occur in finite dimensions and illuminate links to integrable systems and noncommutative geometry, with potential implications for deformation quantization and representation‑theoretic approaches. The paper thus clarifies the landscape of noncommutative Poisson structures on small algebras and sets the stage for further exploration of their geometric and dynamical consequences.

Abstract

In this paper, we describe double Poisson brackets in the sense of M. Van den Bergh on certain finite-dimensional algebras. In particular we prove that all possible double Poisson brackets on matrix algebras are "inner", i.e. given by some commutators in bimodules. As a corollary of this result, we see that all possible double Poisson brackets in any finite-dimensional semisimple algebras over algebraically closed fields are also given by inner derivations. We further give a description of all double Poisson brackets on the algebra of 2x2 upper triangular matrices. We further discuss Poisson structures induced from the double Poisson brackets in its representation spaces of rank two and three. In the appendix, we describe modified double Poisson brackets (in the sense of S. Arthamonov) on this algebra.

Double Poisson brackets on low dimensional algebras

TL;DR

This work analyzes double Poisson brackets in the sense of Van den Bergh on finite‑dimensional algebras, showing that all brackets on matrix algebras are inner and thereby extend to finite‑dimensional semisimple algebras via Morita invariance; it then examines the non‑semisimple case of upper triangular matrices, describing all brackets and the induced Poisson structures on low‑dimensional representation spaces, and finally discusses modified double Poisson brackets in the Arthamonov framework. The authors develop the foundational definitions and constructions, connect them to representation spaces and to Hochschild (co)homology, and provide explicit computations illustrating how inner brackets arise from and how nontrivial Poisson structures appear in non‑semisimple settings. Their results clarify when nontrivial Poisson structures can occur in finite dimensions and illuminate links to integrable systems and noncommutative geometry, with potential implications for deformation quantization and representation‑theoretic approaches. The paper thus clarifies the landscape of noncommutative Poisson structures on small algebras and sets the stage for further exploration of their geometric and dynamical consequences.

Abstract

In this paper, we describe double Poisson brackets in the sense of M. Van den Bergh on certain finite-dimensional algebras. In particular we prove that all possible double Poisson brackets on matrix algebras are "inner", i.e. given by some commutators in bimodules. As a corollary of this result, we see that all possible double Poisson brackets in any finite-dimensional semisimple algebras over algebraically closed fields are also given by inner derivations. We further give a description of all double Poisson brackets on the algebra of 2x2 upper triangular matrices. We further discuss Poisson structures induced from the double Poisson brackets in its representation spaces of rank two and three. In the appendix, we describe modified double Poisson brackets (in the sense of S. Arthamonov) on this algebra.
Paper Structure (24 sections, 7 theorems, 72 equations)

This paper contains 24 sections, 7 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Double Poisson brackets
  3. Definitions and basic properties
  4. The Poisson structure on $\mathrm{Rep}_n(\mathcal{A})$
  5. Some relevant results
  6. Inner double brackets
  7. Modified Double Poisson brackets
  8. Double Poisson brackets on matrix algebras
  9. Equations for a double Poisson bracket
  10. Smoothness of the algebra of matrices
  11. Double Poisson brackets of the algebra of matrices are inner
  12. Example of a Double Poisson bracket on a non-semisimple algebra
  13. Computation for double Poisson bracket
  14. Poisson structure on $\mathrm{Rep}_2(\mathcal{A}_2)$
  15. Coordinate ring
  16. ...and 9 more sections

Key Result

Proposition 1

vdb Assume that $(\mathcal{A},\{\!\{ -,-\}\!\})$ is a double Poisson algebra. Let where $\{\!\{ a,b\}\!\}=\{\!\{ a,b \}\!\}'\otimes\{\!\{ a,b\}\!\}"$. Then the following statements are true: Remark that for commutative algebras $\mathcal{A}_\flat=\mathcal{A}/[\mathcal{A},\mathcal{A}]=\mathcal{A}$, so in the commutative case we get a Poisson structure on $\mathcal{A}$ in the usual sense.

Theorems & Definitions (20)

  • Remark 1
  • Definition 1
  • Proposition 1
  • Definition 2
  • Definition 3
  • Proposition 2
  • Proposition 3
  • Remark 2
  • Definition 4
  • Definition 5
  • ...and 10 more