A functor for constructing \(R\)-matrices in the category \(\mathcal{O}\) of Borel quantum loop algebras. (English) Zbl 07808988
Summary: We tackle the problem of constructing \(R\)-matrices for the category \(\mathcal{O}\) associated to the Borel subalgebra of an arbitrary untwisted quantum loop algebra \(U_{q} ({\mathfrak{g}})\). For this, we define an invertible exact functor \(\mathcal{F}_{q}\) from the category \(\mathcal{O}\) linked to \(U_{q^{-1}}({\mathfrak{g}})\) to the one linked to \(U_{q}({\mathfrak{g}})\). This functor \(\mathcal{F}_{q}\) is compatible with tensor products, preserves irreducibility, and interchanges the subcategories \(\mathcal{O}^{+}\) and \(\mathcal{O}^{-}\) of D. Hernandez and B. Leclerc [Algebra Number Theory 10, No. 9, 2015–2052 (2016; Zbl 1390.17025)]. We construct \(R\)-matrices for \(\mathcal{O}^{+}\) by applying \(\mathcal{F}_{q}\) on the braidings already found for \(\mathcal{O}^{-}\) by D. Hernandez [Represent. Theory 26, 179–210 (2022; Zbl 1502.17013)]. We also use the factorization of the latter intertwiners in terms of stable maps to deduce an analogous factorization for our new braidings. We finally obtain as byproducts new relations for the Grothendieck ring \(K_{0}(\mathcal{O})\) as well as a functorial interpretation of a remarkable ring isomorphism \(K_{0}(\mathcal{O}^{+}) \simeq K_{0} (\mathcal{O}^{-})\) of Hernandez-Leclerc [loc. cit.].
MSC:
| 17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
| 13F60 | Cluster algebras |
| 16D90 | Module categories in associative algebras |
| 18M15 | Braided monoidal categories and ribbon categories |
| 81R10 | Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations |
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