Geodesic
Centers of generalized reflection equation algebras
Teoretičeskaâ i matematičeskaâ fizika, Tome 204 (2020) no. 3, pp. 355-366 Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Keywords: reflection equation algebra, algebra of Reshetikhin–Semenov-Tian-Shansky type, charge, quantum powers of the generating matrix, quantum power sum. 
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D. I. Gurevich; P. A. Saponov. Centers of generalized reflection equation algebras. Teoretičeskaâ i matematičeskaâ fizika, Tome 204 (2020) no. 3, pp. 355-366. http://geodesic.mathdoc.fr/item/TMF_2020_204_3_a2/

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