Centers of generalized reflection equation algebras

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.