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Centers of generalized reflection equation algebras
D. I. Gurevichab,
P. A. Saponovcd a Université Polytechnique Hauts-de-France, Valenciennes,
France
b Poncelet Interdisciplinary Scientific Center, Moscow, Russia
c National Research University "Higher School of Economics", Moscow, Russia
d Institute for High Energy Physics, Russian Research Center
"Kurchatov Institute", Protvino, Moscow Oblast, Russia
Abstract:
As is known, in the reflection equation (RE) algebra associated with an involutive or Hecke
$R$-matrix, the elements
$\operatorname{Tr}_RL^k$ (called quantum power sums) are central. Here,
$L$ is the generating matrix of this algebra, and
$\operatorname{Tr}_R$ is the operation of taking the
$R$-trace associated with a given
$R$-matrix. We consider the problem of whether this is true in certain RE-like algebras depending on a spectral parameter. We mainly study algebras similar to those introduced by Reshetikhin and Semenov-Tian-Shansky (we call them algebras of RS type). These algebras are defined using some current
$R$-matrices (i.e., depending on parameters) arising from involutive and Hecke
$R$-matrices by so-called Baxterization. In algebras of RS type. we define quantum power sums and show that the lowest quantum power sum is central iff the value of the “charge”
$c$ in its definition takes a critical value. This critical value depends on the bi-rank
$(m|n)$ of the initial
$R$-matrix. Moreover, if the bi-rank is equal to
$(m|m)$ and the charge
$c$ has a critical value, then all quantum power sums are central.
Keywords:
reflection equation algebra, algebra of Reshetikhin–Semenov-Tian-Shansky type, charge, quantum powers of the generating matrix, quantum power sum. Received: 21.12.2019
Revised: 24.03.2020
DOI:
10.4213/tmf9862