Subalgebra chains and nuclear physics: Commutant approach and construction of polynomial algebras
Rutwig Campoamor-Stursberg, Danilo Latini, Ian Marquette, Junze Zhang, Yao-Zhong Zhang
TL;DR
This work develops a realization-independent framework for studying subalgebra chains by exploiting the commutant in $U(\mathfrak{g})$ and $\mathfrak{g}'$-invariant polynomials, using the dual space and Lie-Poisson structure to build finitely generated polynomial algebras. It applies the method to key chains relevant in nuclear physics, including $\mathfrak{su}(3) \supset \mathfrak{so}(3)$ (Elliott), $\mathfrak{so}(5) \supset \mathfrak{su}(2) \times \mathfrak{u}(1)$ (Seniority), and $\mathfrak{su}(4) \supset \mathfrak{su}(2) \times \mathfrak{su}(2)$ (Quasisquare), and provides new insights into $\mathfrak{so}(5) \supset \mathfrak{so}(3)$ (Surfon). The authors classify indecomposable $\mathfrak{g}'$-invariant polynomials, derive the corresponding Poisson or polynomial algebras, and identify missing-label operators as combinations within these algebras, highlighting a universal, realization-free character useful for labeling in nuclear models. They also analyze grading, functional relations, and central elements, and outline plans to extend the framework to additional chains.
Abstract
In this paper, we review a new approach to study subalgebra chains $\mathfrak{g} \supset \mathfrak{g}'$ in the context of nuclear physics. This approach does not rely on explicit realizations as bosons or differential operators. We rely on the enveloping algebra, the notion of commutant $C_{U(\mathfrak{g})}(\mathfrak{g}^{\prime})$ and $\mathfrak{g}^{\prime}$-invariant polynomials. This approach builds on those $\mathfrak{g}^{\prime}$-invariant polynomials and finding the underlying finitely generated polynomial algebras. Those algebraic structures can then provide further information on sets of labeling operators. Another aspect of this method consists in exploiting the dual space and the symmetric algebra. Being independent of explicit realizations, it endows the algebraic relations with a universal character. We review the chains associated with $\mathfrak{su}(3) \supset \mathfrak{so}(3)$, $\mathfrak{so}(5) \supset \mathfrak{su}(2) \times \mathfrak{u}(1)$, $\mathfrak{su}(4) \supset \mathfrak{su}(2) \times \mathfrak{su}(2)$. Those chains are known as the Elliott, Seniority and Supermultiplet. We also provide new results and insights into the subalgebra chain $\mathfrak{so}(5) \supset \mathfrak{so}(3)$ of the Surfon model. For all chains, we present the related commutant, $\mathfrak{g}^{\prime}$-invariant polynomials and Poisson algebras.