The generalized Racah algebra as a commutant

Abstract

The Racah algebra $R(n)$ of rank $(n-2)$ is obtained as the commutant of the \mbox{$\mathfrak{o}(2{)}^{\oplus n}$} subalgebra of $\mathfrak{o}(2n)$ in oscillator representations of the universal algebra of $\mathfrak{o}(2n)$. This result is shown to be related in a Howe duality context to the definition of $R(n)$ as the algebra of Casimir operators arising in recouplings of $n$ copies of $\mathfrak{su}(1,1)$. These observations provide a natural framework to carry out the derivation by dimensional reduction of the generic superintegrable model on the $(n-1)$ sphere which is invariant under $R(n)$.

arXiv comment7 pages, based on a talk given by Luc Vinet at the 32nd International Colloquium on Group Theoretical Methods in Physics (Group 32)

References in corpus (4)

• Dual pairing of symmetry groups and dynamical groups in physics
• Superintegrability in two dimensions and the Racah-Wilson algebra
• A higher rank Racah algebra and the ${\mathbb{Z}}_2^n$ Laplace-Dunkl operator
• Hidden Symmetry of the Racah and Clebsch-Gordan Problems for the Quantum Algebra sl_q(2)

Cited by in corpus (17)

• Embedding of the Racah Algebra R($\mathbf{n}$) and Superintegrability
• The Higgs and Hahn algebras from a Howe duality perspective
• Algebraic (super-)integrability from commutants of subalgebras in universal enveloping algebras
• Racah Algebra $R(n)$ from Coalgebraic Structures and Chains of $R(3)$ Substructures
• Polynomial algebras from $su(3)$ and the generic model on the two sphere
• The general Racah algebra as the symmetry algebra of generic systems on pseudo--spheres
• The $q$-Higgs and Askey-Wilson algebras
• $N$-dimensional Smorodinsky-Winternitz model and related higher rank quadratic algebra $\mathcal{S}\mathcal{W}(N)$
• The dual pair $Pin(2n)\times \mathfrak{osp}(1|2)$, the Dirac equation and the Bannai-Ito algebra
• Superintegrable systems from block separation of variables and unified derivation of their quadratic algebras
• The dual pair $(U_q(\mathfrak{su}(1,1)),{\mathfrak{o}}_{q^{1/2}}(2n))$, $q$-oscillators and Askey-Wilson algebras
• Polynomial algebras from commutants: Classical and Quantum aspects of ${\mathcal{A}}_3$
• Howe duality and algebras of the Askey-Wilson type: an overview
• Generalized quadratic commutator algebras of PBW-type
• The Racah algebra and ${\mathfrak{sl}}_n$
• Generalized KS transformations, $ND$ singular oscillator and generalized MICZ-Kepler system
• A Howe correspondence for the algebra of the $\mathfrak{osp}(1|2)$ Clebsch-Gordan coefficients