Finite-dimensional irreducible representations of twisted Yangians

Research Report MRR97-047

Finite-dimensional irreducible representations of twisted Yangians

Alexander Molev

Abstract: We study quantized enveloping algebras called twisted Yangians. They are analogues of the Yangian $\text{Y}(\frak{gl}(N))$ for the classical Lie algebras of B, C, and D series. The twisted Yangians are subalgebras in $\text{Y}(\frak{gl}(N))$ and coideals with respect to the coproduct in $\text{Y}(\frak{gl}(N))$ . We give a complete description of their finite-dimensional irreducible representations. Every such representation is highest weight and we give necessary and sufficient conditions for an irreducible highest weight representation to be finite-dimensional. The result is analogous to Drinfeld's theorem for the ordinary Yangians. Its detailed proof for the A series is also reproduced. For the simplest twisted Yangians we construct an explicit realization for each finite-dimensional irreducible representation in tensor products of representations of the corresponding Lie algebras.

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