Centralizer construction for twisted Yangians

Abstract

For each of the classical Lie algebras $g(n)=o(2n+1), sp(2n), o(2n)$ of type B, C, D we consider the centralizer of the subalgebra $g(n-m)$ in the universal enveloping algebra $U(g(n))$. We show that the $n$th centralizer algebra can be naturally projected onto the $(n-1)$th one, so that one can form the projective limit of the centralizer algebras as $n\to\infty$ with $m$ fixed. The main result of the paper is a precise description of this limit (or stable) centralizer algebra, denoted by $A_m$. We explicitly construct an algebra isomorphism $A_m=Z\otimes Y_m$, where $Z$ is a commutative algebra and $Y_m$ is the so-called twisted Yangian associated to the rank $m$ classical Lie algebra of type B, C, or D. The algebra $Z$ may be viewed as the algebra of virtual Laplace operators; it is isomorphic to the algebra of polynomials with countably many indeterminates. The twisted Yangian $Y_m$ (and hence the algebra $A_m$) can be described in terms of a system of generators with quadratic and linear defining relations which are conveniently presented in R-matrix form involving the so-called reflection equation. This extends the earlier work on the type A case by the second author.