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

TL;DR

This paper constructs explicit generators for polynomial Poisson algebras from Cartan centralisers of certain Lie algebras and uses them to develop algebraic superintegrable systems, providing new explicit examples for rank-three cases.

Contribution

It provides explicit formulas for generators of Cartan centralisers in types B, C, D Lie algebras and constructs associated superintegrable systems, extending known results for type A.

Findings

Explicit generators for Cartan centralisers in B_3, C_3, D_3.

Illustration of inclusion relations between polynomial Poisson algebras.

Construction of algebraic superintegrable systems from these generators.

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 $\mathfrak{h} \subset \mathfrak{sp}(6,\mathbb{C})$.