Centralizer Construction for Twisted Yangians

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 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 → ∞ with m fixed. The main result of the paper is a precise description of this limit (or stable) centralizer algebra, denoted by Am. We explicitly construct an algebra isomorphism Am = Z ⊗ Ym, where Z is a commutative algebra and Ym 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 Ym (and hence the algebra Am) 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. Mathematics Subject Classifications (1991). 17B35, 17B37, 81R50 Postal addresses: Centre for Mathematics and its Applications Australian National University Canberra, ACT 0200, Australia (E-mail: [email protected]) Dobrushin Mathematics Laboratory Institute for Problems of Information Transmission Bolshoy Karetny 19, Moscow 101447, GSP-4, Russia (E-mail: [email protected], [email protected])