Matrix Capelli identities related to Reflection Equation algebra

TL;DR

This paper introduces matrix Capelli identities within Reflection Equation algebras using quantum doubles, generalizing classical Capelli identities and extending higher Capelli identities by Okounkov.

Contribution

It constructs a matrix identity in Reflection Equation algebras related to Hecke symmetries, generalizing classical and higher Capelli identities.

Findings

Established matrix Capelli identities in Reflection Equation algebras.

Derived scalar relations via quantum trace generalizing classical Capelli identities.

Extended higher Capelli identities to a broader algebraic framework.

Abstract

By using the notion of a quantum double we introduce analogs of partial derivatives on a Reflection Equation algebra, associated with a Hecke symmetry of GL(N) type. We construct the matrix L=MD, where M is the generating matrix of the Reflection Equation algebra and D is the matrix composed of the quantum partial derivatives and prove that the matrices M, D and L satisfy a matrix identity, called the matrix Capelli one. Upon applying the quantum trace, it becomes a scalar relation, which is a far-reaching generalization of the classical Capelli identity. Also, we get a generalization of the some higher Capelli identities defined by A.Okounkov.