Centers in Generalized Reflection Equation algebras

TL;DR

This paper investigates the centers of generalized reflection equation algebras, defining quantum power sums, and identifies conditions under which these sums are central, especially focusing on algebras of RS type with spectral parameter dependence.

Contribution

It introduces the concept of quantum power sums in RS-type algebras and determines when these sums are central based on the charge parameter and bi-rank of the symmetry.

Findings

Lowest quantum power sum is central at critical charge value.

All quantum power sums are central for bi-rank (m|m) at critical charge.

Dependence of critical charge on the bi-rank of the initial R-matrix.

Abstract

In the Reflection Equation (RE) algebra associated with an involutive or Hecke symmetry $R$ the center is generated by elements ${\rm Tr}_R L^k$ (called the quantum power sums), where $L$ is the generating matrix of this algebra and ${\rm Tr}_R$ is the $R$-trace associated with $R$. We consider the problem: whether it is so in certain RE-like algebras depending on spectral parameters. Mainly, we deal with algebras similar to those considered in \cite{RS} (we call them algebras of RS type). These algebras are defined by means of some current (i.e. depending on parameters) $R$-matrices arising from involutive and Hecke symmetries via the so-called Baxterization procedure. We define quantum power sums in the algebras of RS type and show that the lowest quantum power sum in such an algebra is central iff the value of the "charge" $c$ entering its definition is critical. We exhibit the dependance of this critical value on the bi-rank of the initial symmetry $R$. Besides, we show that if the bi-rank of $R$ is $(m|m)$, and the value of $c$ is critical, then all quantum power sums are central.