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On the construction of polynomial Poisson algebras: a novel grading approach

Rutwig Campoamor-Stursberg, Danilo Latini, Ian Marquette, Junze Zhang, Yao-Zhong Zhang

TL;DR

This work introduces a grading framework to streamline the explicit construction of polynomial Poisson algebras arising as commutants in symmetric and enveloping algebras of Lie algebras. By grading monomials with respect to subalgebras, the authors significantly reduce the space of admissible Poisson-bracket terms and obtain closed, finitely generated polynomial Poisson algebras for several subalgebra chains in $\mathfrak{sl}(3,\mathbb{C})$ and Cartan-centralizers in $A_n$. The main results include the Elliott chain $\mathfrak{so}(3)\subset\mathfrak{su}(3)$ yielding a cubic Poisson algebra with central elements, the reduction $\mathfrak{o}(3)\subset\mathfrak{sl}(3,\mathbb{C})$ with explicit generators and a center, and the Cartan centralizer $S(\mathfrak{sl}(3,\mathbb{C}))^{\mathfrak h}$ analyzed via root systems. Across sections 4 and 5, root-system and grading techniques are integrated to classify and compute admissible monomials, offering a universal, realizations-independent approach applicable to broader subalgebra chains and Racah-type algebras in mathematical physics.

Abstract

In this work, we refine recent results on the explicit construction of polynomial algebras associated with commutants of subalgebras in enveloping algebras of Lie algebras by considering an additional grading with respect to the subalgebra. It is shown that such an approach simplifies and systematizes the explicit derivation of the Lie--Poisson brackets of elements in the commutant, and several fundamental properties of the grading are given. The procedure is illustrated by revisiting three relevant reduction chains associated with the rank-two complex simple Lie algebra $\mathfrak{sl}(3,\mathbb{C})$. Specifically, we analyze the reduction chains $\mathfrak{so}(3) \subset \mathfrak{su}(3)$, corresponding to the Elliott model in nuclear physics, the chain $\mathfrak{o}(3) \subset \mathfrak{sl}(3,\mathbb{C})$ associated with the decomposition of the enveloping algebra of $\mathfrak{sl}(3,\mathbb{C})$ as a sum of modules, and the reduction chain $\mathfrak{h} \subset \mathfrak{sl}(3,\mathbb{C})$ connected to the Racah algebra $R(3)$. In addition, a description of the classification of the centralizer with respect to the Cartan subalgebra $\mathfrak{h}$ associated with the classical series $A_n$ in connection with its root system is reconsidered. As an illustration of the procedure, the case of $S(A_3)^\mathfrak{h}$ is considered in detail, which is connected with the rank-two Racah algebra for specific realizations of the generators as vector fields. This case has attracted interest with regard to orthogonal polynomials.

On the construction of polynomial Poisson algebras: a novel grading approach

TL;DR

This work introduces a grading framework to streamline the explicit construction of polynomial Poisson algebras arising as commutants in symmetric and enveloping algebras of Lie algebras. By grading monomials with respect to subalgebras, the authors significantly reduce the space of admissible Poisson-bracket terms and obtain closed, finitely generated polynomial Poisson algebras for several subalgebra chains in and Cartan-centralizers in . The main results include the Elliott chain yielding a cubic Poisson algebra with central elements, the reduction with explicit generators and a center, and the Cartan centralizer analyzed via root systems. Across sections 4 and 5, root-system and grading techniques are integrated to classify and compute admissible monomials, offering a universal, realizations-independent approach applicable to broader subalgebra chains and Racah-type algebras in mathematical physics.

Abstract

In this work, we refine recent results on the explicit construction of polynomial algebras associated with commutants of subalgebras in enveloping algebras of Lie algebras by considering an additional grading with respect to the subalgebra. It is shown that such an approach simplifies and systematizes the explicit derivation of the Lie--Poisson brackets of elements in the commutant, and several fundamental properties of the grading are given. The procedure is illustrated by revisiting three relevant reduction chains associated with the rank-two complex simple Lie algebra . Specifically, we analyze the reduction chains , corresponding to the Elliott model in nuclear physics, the chain associated with the decomposition of the enveloping algebra of as a sum of modules, and the reduction chain connected to the Racah algebra . In addition, a description of the classification of the centralizer with respect to the Cartan subalgebra associated with the classical series in connection with its root system is reconsidered. As an illustration of the procedure, the case of is considered in detail, which is connected with the rank-two Racah algebra for specific realizations of the generators as vector fields. This case has attracted interest with regard to orthogonal polynomials.

Paper Structure

This paper contains 17 sections, 12 theorems, 184 equations, 6 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Construction of commutant and related polynomial algebra: The polynomial ansatz
  4. The grading of polynomials in $\mathcal{Q}_\mathfrak{g}(d)$
  5. Lie algebra with decomposition into two subalgebras
  6. Lie algebras with a decomposition into three subalgebras
  7. Constructing polynomial Poisson algebras from subalgebras of $\mathfrak{sl}(3,\mathbb{C})$ via the grading method
  8. The reduction chain $\mathfrak{so}(3) \subset \mathfrak{su}(3)$
  9. Elliott chain and construction of the generators via a weight zero type criteria
  10. The reduction chain $\mathfrak{o}(3) \subset \mathfrak{sl}(3,\mathbb{C})$
  11. Reduction chain $\mathfrak{h} \subset \mathfrak{sl}(3,\mathbb{C})$
  12. Grading of monomials in $\mathcal{Q}_{A_n}(n)$
  13. Grading and root systems in $\mathcal{Q}_{A_n}(n)$
  14. Explicit polynomials in $\{\textbf{q}_2,\textbf{q}_r\}$
  15. Cartan centralizer of $S( A_3)$
  16. ...and 2 more sections

Key Result

Proposition 2.1

The coadjoint action of $\mathfrak{a}$ on $S(\mathfrak{g})$ preserves the Poisson-Lie bracket $\{\cdot,\cdot\}$ defined on $S(\mathfrak{g}).$ That is, for all $X_m \in \mathfrak{a},$ and $p,q \in S(\mathfrak{g}),$

Theorems & Definitions (31)

  • Proposition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Proposition 2.5
  • proof
  • Definition 3.1
  • Remark 3.2
  • Lemma 3.3
  • proof
  • ...and 21 more