Centralizer construction for twisted Yangians

• A. Molev ^[1] ; G. Olshanski ^[2]
  1. [1] University of Sydney

     University of Sydney

     Australia

     
  2. [2] Institute for Problems of Information Transmission, Rusia
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 6, Nº. 3, 2000, págs. 269-317
• Idioma: inglés
• DOI: 10.1007/pl00001390
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• Resumen

  • 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 o(2n−2m) or sp(2n−2m) , respectively, in the universal enveloping algebra \rm 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 \rm Am . We explicitly construct an algebra isomorphism \rm Am=\rm Z⊗\rm Ym , where Z is a commutative algebra and \rm 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 \rm Ym (and hence the algebra \rm Am ) 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., g(n)=gl(n) ) by the second author.