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Selecta Mathematica

, Volume 12, Issue 1, pp 1–38 | Cite as

Skew representations of twisted Yangians

  • A. I. MolevEmail author
Original Paper

Abstract.

Analogs of the classical Sylvester theorem have been known for matrices with entries in noncommutative algebras including the quantized algebra of functions on GL N and the Yangian for $$ \mathfrak{g}\mathfrak{l}_{{N}} $$ . We prove a version of this theorem for the twisted Yangians $$ {\text{Y(}}\mathfrak{g}_{N} {\text{)}} $$associated with the orthogonal and symplectic Lie algebras $$ \mathfrak{g}_{N} = \mathfrak{o}_{N} {\text{ or }}\mathfrak{s}\mathfrak{p}_{N} $$. This gives rise to representations of the twisted Yangian $$ {\text{Y}}{\left( {\mathfrak{g}_{{N - M}} } \right)} $$ on the space of homomorphisms $$ {\text{Hom}}_{{\mathfrak{g}_{M} }} {\left( {W,V} \right)} $$, where W and V are finite-dimensional irreducible modules over $$ \mathfrak{g}_{{M}} {\text{ and }}\mathfrak{g}_{{N}} $$, respectively. In the symplectic case these representations turn out to be irreducible and we identify them by calculating the corresponding Drinfeld polynomials.We also apply the quantum Sylvester theorem to realize the twisted Yangian as a projective limit of certain centralizers in universal enveloping algebras.

Mathematics Subject Classification (2000).

Primary 81R10 

Keywords.

Twisted Yangian centralizer Drinfeld polynomial 

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Copyright information

© Birkhäuser Verlag, Basel 2006

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsUniversity of SydneySydneyAustralia

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