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Another look at the differential operators on quantum matrix spaces and its applications. (English) Zbl 0957.17023

Summary: The present paper gives a new viewpoint for differential operators with respect to the coordinates of quantum matrix spaces. Special emphasis is put on the inductive construction of these differential operators from the \(q\)-difference operators defined on each column (or row). This idea for understanding our operators provides us with two important applications, (1) a construction of the \(q\)-oscillator representation of the quantized enveloping algebra \(U_q ({\mathfrak {sp}}_{2N})\) of the symplectic Lie algebra \({\mathfrak {sp}}_{2N}\) and a construction of the quantum dual pair \(({\mathfrak {sp}}_{2N}, {\mathfrak o}_n)\) through the tensor power of the \(q\)-oscillator representation, and (2) a new definition of a quantum analogue of hypergeometric equations of many variables. In addition to these, we give an explanation of the spectral parameter in the quantum Capelli identity for \(GL_q (n)\) discussed by M. Noumi, T.Umeda and M. Wakayama [Duke Math. J. 76, 567-594 (1994; Zbl 0835.17013)].

MSC:

17B37 Quantum groups (quantized enveloping algebras) and related deformations
33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
39A70 Difference operators

Citations:

Zbl 0835.17013
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