For a complex finite-dimensional simple Lie algebra \(\mathfrak{g}\), consider its untwisted quantum loop algebra \(U_{q}(L\mathfrak{g})\) as a certain quantum affinization of the universal enveloping algebra \(U(\mathfrak{g})\). It is a Hopf algebra defined over \(\mathbf{k} = \overline{\mathbb{Q}(q)}\), where \(q\) is the generic quantum parameter. The structure of the monoidal abelian category \(\mathcal{C}\) of finite-dimensional \(U_{q}(L\mathfrak{g})\)-modules is much more complicated than that of \(U(\mathfrak{g})\). Indeed, the category \(\mathcal{C}\) is neither semisimple as an abelian category, nor braided as a monoidal category.
The normalized \(R\)-matrices are constructed as intertwining operators between tensor products of relatively generic simple objects of the category \(\mathcal{C}\), satisfying the quantum Yang–Baxter equation. They are matrix-valued rational functions in the spectral parameters, whose singularities strongly reflect the structure of tensor product modules, thus the singularities of normalized \(R\)-matrices carry important information on the monoidal structure of \(\mathcal{C}\).
R. Fujita presents a simple unified formula expressing the denominators of the normalized \(R\)-matrices between the fundamental modules over the quantum loop algebras of type \(\mathsf{ADE}\). It has an interpretation in terms of representations of Dynkin quivers and can be proved in a unified way using geometry of the graded quiver varieties. As a by-product, the author obtains a geometric interpretation of generalized quantum affine Schur-Weyl duality functor when it arises from a family of the fundamental modules. He also investigates several cases when the graded quiver varieties are isomorphic to unions of the graded nilpotent orbits of type \(\mathsf{A}\).

Reviewer: Mee Seong Im (Annapolis)

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