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Subalgebra chains and nuclear physics: Commutant approach and construction of polynomial algebras

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.