| Autor: |
Pinet, Théo |
| Předmět: |
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| Zdroj: |
Journal of the London Mathematical Society; Jan2024, Vol. 109 Issue 1, p1-46, 46p |
| Abstrakt: |
We tackle the problem of constructing R$R$‐matrices for the category O$\mathcal {O}$ associated to the Borel subalgebra of an arbitrary untwisted quantum loop algebra Uq(g)$U_q({\mathfrak {g}})$. For this, we define an invertible exact functor Fq$\mathcal {F}_q$ from the category O$\mathcal {O}$ linked to Uq−1(g)$U_{q^{-1}}({\mathfrak {g}})$ to the one linked to Uq(g)$U_q({\mathfrak {g}})$. This functor Fq$\mathcal {F}_q$ is compatible with tensor products, preserves irreducibility, and interchanges the subcategories O+$\mathcal {O}^+$ and O−$\mathcal {O}^-$ of Hernandez and Leclerc (Algebra Number Theory10 (2016) 2015–2052). We construct R$R$‐matrices for O+$\mathcal {O}^+$ by applying Fq$\mathcal {F}_q$ on the braidings already found for O−$\mathcal {O}^-$ by Hernandez (Represent. Theory 26 (2022) 179–210). 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 K0(O)$K_0(\mathcal {O})$ as well as a functorial interpretation of a remarkable ring isomorphism K0(O+)≃K0(O−)$K_0(\mathcal {O}^+)\simeq K_0(\mathcal {O}^-)$ of Hernandez–Leclerc. [ABSTRACT FROM AUTHOR] |
| Databáze: |
Complementary Index |
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