Centers of generalized reflection equation algebras.

As is known, in the reflection equation (RE) algebra associated with an involutive or Hecke -matrix, the elements (called quantum power sums) are central. Here, is the generating matrix of this algebra, and is the operation of taking the -trace associated with a given -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 -matrices (i.e., depending on parameters) arising from involutive and Hecke -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" in its definition takes a critical value. This critical value depends on the bi-rank of the initial -matrix. Moreover, if the bi-rank is equal to and the charge has a critical value, then all quantum power sums are central. [ABSTRACT FROM AUTHOR]