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Polynomial Poisson Algebras and Superintegrable Systems from Cartan centralisers of Types $B_3$, $C_3$ and $D_3$

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

Abstract

In this work, we construct explicit formulas for the generators of the Cartan centralisers of complex semisimple Lie algebras $B_n,C_n$ and $D_n$, the case $A_n$ being already known \cite{campoamor2023algebraic}. The precise structures for the cases of rank-three simple Lie algebras ($B_3,C_3$ and $D_3$) are provided, and the inclusion relations between the corresponding polynomial Poisson algebras (finitely generated Poisson algebras over $\mathbb{C}[\mathfrak{h}^*]$) are illustrated. We develop the idea of constructing algebraic superintegrable systems and their integrals from the generators of these polynomial Poisson algebras. In particular, we explicitly present the algebraic superintegrable systems corresponding to the Cartan reduction chains $\mathfrak{h} \subset \mathfrak{so}(6,\mathbb{C})$, $\mathfrak{h} \subset \mathfrak{so}(7,\mathbb{C})$, and $\mathfrak{h} \subset \mathfrak{sp}(6,\mathbb{C})$.

Polynomial Poisson Algebras and Superintegrable Systems from Cartan centralisers of Types $B_3$, $C_3$ and $D_3$

Abstract

In this work, we construct explicit formulas for the generators of the Cartan centralisers of complex semisimple Lie algebras and , the case being already known \cite{campoamor2023algebraic}. The precise structures for the cases of rank-three simple Lie algebras ( and ) are provided, and the inclusion relations between the corresponding polynomial Poisson algebras (finitely generated Poisson algebras over ) are illustrated. We develop the idea of constructing algebraic superintegrable systems and their integrals from the generators of these polynomial Poisson algebras. In particular, we explicitly present the algebraic superintegrable systems corresponding to the Cartan reduction chains , , and .
Paper Structure (14 sections, 8 theorems, 205 equations, 1 table)

This paper contains 14 sections, 8 theorems, 205 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Construction of Cartan centralisers of semisimple Lie algebras
  4. Explicit generators for Bn, Cn and Dn Cartan centralisers
  5. Explicit generators of S(Bn)h
  6. Explicit generators of S(Cn)h and S(Dn)h
  7. Generators of $S(D_n)^\mathfrak{h}$
  8. Generators of $S(C_n)^\mathfrak{h}$
  9. Non-commutative polynomial Poisson algebras of rank 3
  10. Explicit generators for Q extunderscoreB3(d)
  11. Explicit generators for Q extsubscriptD3(d')
  12. Explicit generators for Q extsubscriptC3(d")
  13. Conclusions
  14. Declarations

Key Result

Lemma 4.3

Let $\mathcal{Q}_{B_n}(d)$ be the polynomial Poisson algebra defined above. Following properties hold: (i) For all $1 \leq i,j \leq n$, the monomials $\varepsilon_{ij}^-\varepsilon_{ji}^-$, $\varepsilon_{ij}^+\widehat{\varepsilon}_{ij}^+$ and $\varepsilon_i \widehat{\varepsilon}_i$ are indecomposabl

Theorems & Definitions (25)

  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 3.1
  • Definition 4.1
  • Remark 4.2
  • Lemma 4.3
  • proof
  • ...and 15 more