Poles of finite-dimensional representations of Yangians

On a Conjecture of Khoroshkin and Tolstoy

“…In more detail, that Θ z takes this form follows from Proposition 2.9 of [GW23] (or, more precisely, its proof). Next, let…”

The restricted quantum double of the Yangian

“…To expand on this, DY̵ h g can be realized as the ̵ h-adic completion of a Z-graded C[ ̵ h]-algebra DY̵ h g j defined by generators and relations (see Remark 4.2). One can further specialize ̵ h to any nonzero complex number ζ to obtain a C-algebra DY ζ g = DY̵ h g j /( ̵ h − ζ)DY̵ h g j , whose category of finite-dimensional representations was characterized in terms of that of the corresponding Yangian Y ζ (g) in [18]. Though this category has a tensor structure which corresponds to the Hopf structure on DY̵ h g, it is important to note that DY ζ g is not a Hopf algebra over C, and in particular it does not coincide with the (restricted) quantum double of Y ζ (g) defined in any reasonable sense.…”

“…For instance, it induces a family of isomorphisms between completions of DY̵ h g and Y̵ h g, realizes Y̵ h g as a degeneration of DY̵ h g, and is injective provided g is of finite type or of simply laced affine type. In addition, it was applied in [18] to characterize the category of finitedimensional representations of DY̵ h g, for ̵ h ∈ C × and g of finite type, as the tensorclosed Serre subcategory of that of the Yangian consisting of those representations which have no poles at zero.…”