Poles of finite-dimensional representations of Yangians

Abstract

Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra over $\mathbb{C}$, and let $Y_{\hslash }(\mathfrak{g})$ be the Yangian of $\mathfrak{g}$. In this paper, we study the sets of poles of the rational currents defining the action of $Y_{\hslash }(\mathfrak{g})$ on an arbitrary finite-dimensional vector space $V$. Using a weak, rational version of Frenkel and Hernandez' Baxter polynomiality, we obtain a uniform description of these sets in terms of the Drinfeld polynomials encoding the composition factors of $V$ and the inverse of the $q$-Cartan matrix of $\mathfrak{g}$. We then apply this description to obtain a concrete set of sufficient conditions for the cyclicity and simplicity of the tensor product of any two irreducible representations, and to classify the finite-dimensional irreducible representations of the Yangian double.

arXiv comment58 pages. Significantly revised version. Updates include: simpler proof of the main theorem (5.2) based a weaker version of Baxter polynomiality; relation with polynomial R-matrices of Hernandez-Zhang via shifted Yangians; relation with denominator formulae; and updated references to the literature

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Cited by in corpus (4)

• Gelfand-Tsetlin bases of representations for super Yangian and quantum affine superalgebra
• Braid group actions, Baxter polynomials, and affine quantum groups
• Theta series for quantum loop algebras and Yangians
• The restricted quantum double of the Yangian