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Algebraic structures and Hamiltonians from the equivalence classes of 2D conformal algebras

Ian Marquette, Junze Zhang, Yao-Zhong Zhang

TL;DR

The article develops an entirely algebraic route to 2D superintegrable systems by studying centralisers of subalgebras within the universal enveloping algebra of the 2D conformal algebra $\\mathfrak{c}(2)$, exploiting its $\\mathfrak{so}(3,1)$-type subalgebra structure. By solving commuting-PDEs in the symmetric algebra and inducing Poisson and quantum closed quadratic algebras, the authors construct algebraic Hamiltonians from Casimir invariants and identify a range of (often Abelian) and non-Abelian symmetry algebras, including Racah-type realizations. The framework yields explicit generators, their Poisson or commutator brackets, central elements, and Casimirs, connecting to existing work on embeddings into Darboux spaces and illustrating a metric-free approach to superintegrability. This algebraic classification clarifies how subalgebra reductions encode integrability, provides a catalogue of polynomial algebras arising from $\\mathfrak{c}(2)$, and suggests avenues for extending to spaces of non-constant curvature and to broader representation-theoretic settings.

Abstract

The construction of superintegrable systems based on Lie algebras and their universal enveloping algebras has been widely studied over the past decades. However, most constructions rely on explicit differential operator realisations and Marsden-Weinstein reductions. In this paper, we develop an algebraic approach based on the subalgebras of the 2D conformal algebra $\mathfrak{c}(2)$. This allows us to classify the centralisers of the enveloping algebra of the conformal algebra and construct the corresponding Hamiltonians with integrals in algebraic form. It is found that the symmetry algebras underlying these algebraic Hamiltonians are six-dimensional quadratic algebras. The Berezin brackets and commutation relations of the quadratic algebraic structures are closed without relying on explicit realisations or representations. We also give the Casimir invariants of the symmetry algebras. Our approach provides algebraic perspectives for the recent work by Fordy and Huang on the construction of superintegrable systems in the Darboux spaces.

Algebraic structures and Hamiltonians from the equivalence classes of 2D conformal algebras

TL;DR

The article develops an entirely algebraic route to 2D superintegrable systems by studying centralisers of subalgebras within the universal enveloping algebra of the 2D conformal algebra , exploiting its -type subalgebra structure. By solving commuting-PDEs in the symmetric algebra and inducing Poisson and quantum closed quadratic algebras, the authors construct algebraic Hamiltonians from Casimir invariants and identify a range of (often Abelian) and non-Abelian symmetry algebras, including Racah-type realizations. The framework yields explicit generators, their Poisson or commutator brackets, central elements, and Casimirs, connecting to existing work on embeddings into Darboux spaces and illustrating a metric-free approach to superintegrability. This algebraic classification clarifies how subalgebra reductions encode integrability, provides a catalogue of polynomial algebras arising from , and suggests avenues for extending to spaces of non-constant curvature and to broader representation-theoretic settings.

Abstract

The construction of superintegrable systems based on Lie algebras and their universal enveloping algebras has been widely studied over the past decades. However, most constructions rely on explicit differential operator realisations and Marsden-Weinstein reductions. In this paper, we develop an algebraic approach based on the subalgebras of the 2D conformal algebra . This allows us to classify the centralisers of the enveloping algebra of the conformal algebra and construct the corresponding Hamiltonians with integrals in algebraic form. It is found that the symmetry algebras underlying these algebraic Hamiltonians are six-dimensional quadratic algebras. The Berezin brackets and commutation relations of the quadratic algebraic structures are closed without relying on explicit realisations or representations. We also give the Casimir invariants of the symmetry algebras. Our approach provides algebraic perspectives for the recent work by Fordy and Huang on the construction of superintegrable systems in the Darboux spaces.
Paper Structure (28 sections, 6 theorems, 190 equations)

This paper contains 28 sections, 6 theorems, 190 equations.

Table of Contents

  1. Introduction
  2. Superintegrable systems from the centralizers of $\mathcal{U}(\mathfrak{g})$
  3. Conformal algebra $\mathfrak{c}(2)$ and its conjugacy classes
  4. Construction of polynomial algebras from reduction chains of $\mathfrak{c}(2)$
  5. Symmetry algebras from one-dimensional subalgebras
  6. Symmetry algebra from $\mathfrak{a}_1$
  7. Symmetry algebra from $\mathfrak{a}_2$
  8. Symmetry algebra from $\mathfrak{a}_3$
  9. Symmetry algebra from $\mathfrak{a}_4$
  10. Symmetry algebra from $\mathfrak{a}_5$
  11. Symmetry algebra from $\mathfrak{a}_6$
  12. Reduction chain $\mathfrak{a}_c \subset \mathfrak{c}(2)$
  13. reduction chains $\mathfrak{a}_{(ij)} \subset \mathfrak{c}(2)$
  14. Polynomial algebra generated by MASAs
  15. Algebraic Hamiltonians from 2D non-Abelian subalgebras
  16. ...and 13 more sections

Key Result

Proposition 2.5

Let $\mathcal{C}_{\mathcal{U}^k(\mathfrak{g})}(\mathfrak{a}) := \,\mathcal{C}^k(\mathfrak{a})$ and $\mathcal{C}_{\mathcal{S}^k(\mathfrak{g})}(\mathfrak{a}) := \, \mathcal{C}_k(\mathfrak{a})$. Then $\mathcal{C}_{\mathcal{U} (\mathfrak{g})}(\mathfrak{a}) = \bigoplus_{k \geq 0} \mathcal{C}^k(\mathfrak

Theorems & Definitions (20)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Proposition 2.5
  • Definition 2.6
  • Remark 2.7
  • Definition 3.1
  • Lemma 4.1
  • proof
  • ...and 10 more