The center of the reflection equation algebra via quantum minors

TL;DR

This paper provides explicit formulas for the central elements in the reflection equation algebra related to quantum groups, resolving a longstanding question and connecting various algebraic descriptions.

Contribution

It introduces simple formulas for the central elements in the REA, linking them to quantum minors and offering a new presentation of $U_q(\mathfrak{gl}_N)$.

Findings

Explicit formulas for the $c_k$ elements in the REA

A quantum Girard-Newton identity relating $c_k$ to quantum traces

A new presentation of $U_q(\mathfrak{gl}_N)$ as a localization of the REA

Abstract

We give simple formulas for the elements $c_k$ appearing in a quantum Cayley-Hamilton formula for the reflection equation algebra (REA) associated to the quantum group $U_q(\mathfrak{gl}_N)$, answering a question of Kolb and Stokman. The $c_k$'s are certain canonical generators of the center of the REA, and hence of $U_q(\mathfrak{gl}_N)$ itself; they have been described by Reshetikhin using graphical calculus, by Nazarov-Tarasov using quantum Yangians, and by Gurevich, Pyatov and Saponov using quantum Schur functions; however no explicit formulas for these elements were previously known. As byproducts, we prove a quantum Girard-Newton identity relating the $c_k$'s to the so-called quantum power traces, and we give a new presentation for the quantum group $U_q(\mathfrak{gl}_N)$, as a localization of the REA along certain principal minors.