Centralizer construction for twisted Yangians

Abstract.

For each of the classical Lie algebras ${\frak g}(n)={\frak o}(2n+1)$,${\frak {sp}(2n),{\frak o}(2n)}$ of type B, C, D we consider the centralizer of the subalgebra ${\frak o}(2n-2m)$ or ${\frak {sp}}(2n-2m)$, respectively, in the universal enveloping algebra $\text{\rm U}({\frak g}(n))$. We show that the nth 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 $\text{\rm A}_m$. We explicitly construct an algebra isomorphism $\text{\rm A}_m=\text{\rm Z}\otimes \text{\rm Y}_m$, where Z is a commutative algebra and $\text{\rm Y}_m$ 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 $\text{\rm Y}_m$ (and hence the algebra $\text{\rm A}_m$) can be described in terms of a system of generators with quadratic and linear defining relations that are conveniently presented in R-matrix form involving the so-called reflection equation. This extends the earlier work on the type A case (i.e., ${\frak g}(n)={\frak gl}(n)$) by the second author.