Centers of generalized reflection equation algebras
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CENTERS OF GENERALIZED REFLECTION EQUATION ALGEBRAS D. I. Gurevich∗ and P. A. Saponov†
As is known, in the reflection equation (RE) algebra associated with an involutive or Hecke R-matrix, the elements TrR Lk (called quantum power sums) are central. Here, L is the generating matrix of this algebra, and TrR 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 DOI: 10.1134/S0040577920090032
1. Introduction The best known quantum matrix (QM) algebras related to quantum groups (QGs) Uq (sl(N )) are the so-called RT T and reflection equation (RE) algebras. In the general case, any QM algebra is defined by a pair of compatible R-matrices (see [1]) that yield permutation relations for the algebra generators. The permutation relations can be written in matrix form by introducing the so-called generating matrix whose elements are generators of the given algebra. : V ⊗2 → V ⊗2 whose matrix (in some fixed basis) By definition, a braiding is an operator matrix R satisfies the so-called braid relation R12 R23 R12 = R23 R12 R23 ,
R12 = R ⊗ I,
R23 = I ⊗ R.
(1)
∗
Universit´e Polytechnique Hauts-de-France, Valenciennes, France; Poncelet Interdisciplinary Scientific Center, Moscow, Russia, e-mail: [email protected]. †
National Research University “Higher School of Economics,” Moscow, Russia; Institute for High Energy Physics, Russian Research Center “Kurchatov Institute,” Protvino, Moscow Oblast, Russia, e-mail: [email protected] (corresponding author). The research of P. A. Saponov was supported in part by the Russian Foundation for Basic Research (Grant No. 19-01-00726). Prepared from an English manuscript submitted by the authors; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 204, No. 3, pp. 355–366, September, 2020. Received December 21, 2019. Revised March 24, 2020. Accepted April 13, 2020. 1130
c 2020 Pleiades Publishing, Ltd. 0040-5779/20/2043-1130
Here, V is a finite-dimensional linear space of dimension N over the ground field C, and I denotes the identity operator. Moreover, the subscript on operators indicates the position of the factor(s) i
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