Capelli identity on multiparameter quantum linear groups

TL;DR

This paper establishes a quantum Capelli identity for multiparameter quantum linear groups, introduces the quantum Pfaffian, and explores their transformations and generalizations within quantum algebra.

Contribution

It provides the first formulation of the quantum Capelli identity on multiparameter quantum groups and investigates related structures like the quantum Pfaffian and hyper-Pfaffian.

Findings

Quantum Capelli identity formulated for multiparameter quantum groups

Introduction of the multiparameter quantum Pfaffian and its transformation properties

Extension to the multiparameter hyper-Pfaffian and its relation to quantum minors

Abstract

A quantum Capelli identity is given on the multiparameter quantum general linear group based on the $(p_{ij}, u)$-condition. The multiparameter quantum Pfaffian of the $(p_{ij}, u)$-quantum group is also introduced and the transformation under the congruent action is given. Generalization to the multiparameter hyper-Pfaffian and relationship with the quantum minors are also investigated.