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→∞
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⊗
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.