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Dual pairs, spherical harmonics and a Capelli identity in quantum group theory. (English) Zbl 0930.17012

The aim of the paper is to extend to the quantum (\(q\)-deformed) case the results on dual pairing of the Lie algebras \(\text{sl}_2\) and \(\text{so}_n\). The authors introduce the oscillator representation of the quantized universal enveloping algebra \(U_{q^2}(\text{sl}_2)\) and realize its tensor product on the quantum vector space. It is observed that the non-standard \(q\)-deformed algebra \(U_q(\text{so}_n)\) appears in the commutant of the \(n\)-fold tensor power of the oscillator representation of \(U_{q^2}(\text{sl}_2)\). The algebra \(U_q(\text{so}_n)\) is an associative algebra generated by elements \(I_1\), \(I_2,\cdots ,I_{n-1}\), satisfying the defining relations \[ I_i^2I_{i\pm 1}-(q+q^{-1})I_iI_{i\pm 1}I_i+ I_{i\pm 1}I_i^2=-I_{i\pm 1} \quad \text{and} \quad I_iI_j-I_jI_i=0 \quad \text{for }|i-j|>1. \] The authors find explicit formulas for the Casimir element of the algebra \(U_q(\text{so}_n)\) and then prove the Capelli identity, which is an explicit identity between two central elements of the algebras \(U_q(\text{so}_n)\) and \(U_{q^2}(\text{sl}_2)\). The proof is representation theoretic and is done by the comparison of eigenvalues of the two operators in question.
Then the double commutant property is proved. The guiding principle here is the first fundamental theorem of invariants. The last one is established in a special form under the formulation of algebras with Hopf algebra symmetry. The authors find the explicit form of the central elements of the algebra \(U_q(\text{so}_n)\). Here the reflection equation is used which controls the commutation relations among the elements of \(U_q(\text{so}_n)\). Explicit formulas for the zonal spherical polynomials under the action of the algebra \(U_q(\text{so}_n)\) on the quantum vector space are derived.
Reviewer: A.Klimyk (Kyïv)

MSC:

17B37 Quantum groups (quantized enveloping algebras) and related deformations
33D80 Connections of basic hypergeometric functions with quantum groups, Chevalley groups, \(p\)-adic groups, Hecke algebras, and related topics
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