For each of the classical Lie algebras 0(ra) = o(2n + 1), sp(2n), o(2n) of type B, C, D we consider the centralizer of the subalgebra o(2n -2m) or sp(2n -2m), respectively, in the universal enveloping algebra U (g(n)). We show that the rath centralizer algebra can be naturally projected onto the (n -l)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 that are conveniently presented in fi-matrix form involving the so-called reflection equation. This extends the earlier work on the type A case (i.e., g(n) = gl(n)) by the second author.
- Lie algebra
- Twisted yangian