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 Tr_R(L^k) (called quantum power sums) are central. Here, L is the generating matrix of this algebra, and 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 birank (m|n) of the initial R-matrix. Moreover, if the birank is equal to (m|m) and the charge c has a critical value, then all quantum power sums are central.