Wick theorem and matrix Capelli identity for quantum differential operators on Reflection Equation Algebras

TL;DR

This paper develops quantum analogs of classical operators on Reflection Equation Algebras, proving invariance properties, establishing a Wick theorem, and deriving universal matrix Capelli identities in the context of quantum groups.

Contribution

It introduces quantum Laplace and Casimir operators, defines normal ordering for quantum differential operators, and proves a Wick theorem and Capelli identities for these operators.

Findings

Quantum operators preserve the central subalgebra.

Established a Wick theorem for quantum differential operators.

Derived universal matrix Capelli identities.

Abstract

Quantum differential operators on Reflection Equation Algebras, corresponding to Hecke symmetries R were introduced in previous publications. In the present paper we are mainly interested in quantum analogs of the Laplace and Casimir operators, which are invariant with respect to the action of the Quantum Groups U_q(sl(N)), provided R is the Drinfeld-Jimbo $R$-matrix. We prove that any such an operator maps the central characteristic subalgebra of a Reflection Equation algebra into itself. Also, we define the notion of normal ordering for the quantum differential operators and prove an analog of the Wick theorem for the product of partially ordered operators. As an important corollary we find a set of universal matrix Capelli identities generalizing the results of [Ok2] and [JLM]. Besides, we prove that the normal ordered form of any central differential operator from the characteristic subalgebra is also a central differential operator.