## Abstract

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 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 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.

Original language | English |
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Pages (from-to) | 269-317 |

Number of pages | 49 |

Journal | Selecta Mathematica, New Series |

Volume | 6 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2000 |

Externally published | Yes |

## Keywords

- Centralizer
- Lie algebra
- Twisted yangian