Crystals from categorified quantum groups.(English)Zbl 1246.17017
Let \(\mathfrak{g}\) be a symmetrizable Kac-Moody algebra and denote by \(_{\mathcal{A}}\mathbf{U}^{-}_{q}\) the integral form of the negative part of the quantum enveloping algebra associated to \(\mathfrak{g}\). From previous works of the first author and M. Khovanov and R. Rouquier it is known that a family \(R=\bigoplus R(\nu)\) of graded algebras categorify \(_{\mathcal{A}}\mathbf{U}^{-}_{q}\), that is, there exists an isomorphism between this algebra and the Grothendieck group of finitely generated projective \(R\)-modules. In this paper, the authors give a new proof of this result by using crystal-theoretic methods.
By an algebraic treatment of the affine Hecke algebra and its cyclotomic quotients, they determine the size of the Grothendieck group for arbitrary cyclotomic quotients \(R^{\Lambda}(\nu)\) in order to introduce crystal structure on categories of modules. This allows to view cyclotomic quotients of the algebras \(R(\nu)\) as a categorification of the integrable highest weight representation \(V(\Lambda)\) of \(\mathbf{U}^{-}_{q}\), which proves partially a conjecture on cyclotomic quotients in the general setting stated by the first author and Khovanov. These results are then used to give the alternative proof of the categorification theorem which goes along entirely in the category of finitely generated modules.
The article is well-written and carefully organized. It contains all the preliminaries that are needed through the paper and ends with a section that contains the proof of the main theorems.
By an algebraic treatment of the affine Hecke algebra and its cyclotomic quotients, they determine the size of the Grothendieck group for arbitrary cyclotomic quotients \(R^{\Lambda}(\nu)\) in order to introduce crystal structure on categories of modules. This allows to view cyclotomic quotients of the algebras \(R(\nu)\) as a categorification of the integrable highest weight representation \(V(\Lambda)\) of \(\mathbf{U}^{-}_{q}\), which proves partially a conjecture on cyclotomic quotients in the general setting stated by the first author and Khovanov. These results are then used to give the alternative proof of the categorification theorem which goes along entirely in the category of finitely generated modules.
The article is well-written and carefully organized. It contains all the preliminaries that are needed through the paper and ends with a section that contains the proof of the main theorems.
Reviewer: Gastón Andrés García (Córdoba)
MSC:
| 17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
| 16T20 | Ring-theoretic aspects of quantum groups |
Cite
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