Centers of generalized reflection equation algebras

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.