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Article Dans Une Revue Commun.Math.Phys. Année : 2021

Q-data and Representation Theory of Untwisted Quantum Affine Algebras

Résumé

For a complex finite-dimensional simple Lie algebra ${\mathfrak {g}}$, we introduce the notion of Q-datum, which generalizes the notion of a Dynkin quiver with a height function from the viewpoint of Weyl group combinatorics. Using this notion, we develop a unified theory describing the twisted Auslander–Reiten quivers and the twisted adapted classes introduced in Oh and Suh (J Algebra 535(1):53–132, 2019) with an appropriate notion of the generalized Coxeter elements. As a consequence, we obtain a combinatorial formula expressing the inverse of the quantum Cartan matrix of ${\mathfrak {g}}$, which generalizes the result of Hernandez and Leclerc (J Reine Angew Math 701:77–126, 2015) in the simply-laced case. We also find several applications of our combinatorial theory of Q-data to the finite-dimensional representation theory of the untwisted quantum affine algebra of ${\mathfrak {g}}$. In particular, in terms of Q-data and the inverse of the quantum Cartan matrix, (i) we give an alternative description of the block decomposition results due to Chari and Moura (Int Math Res Not 5:257–298, 2005) and Kashiwara et al. (Block decomposition for quantum affine algebras by the associated simply-laced root system, 2020. arXiv:2003.03265), (ii) we present a unified (partially conjectural) formula of the denominators of the normalized R-matrices between all the Kirillov–Reshetikhin modules, and (iii) we compute the invariants $\Lambda (V,W)$ and $\Lambda ^\infty (V, W)$ introduced in Kashiwara et al. (Compos Math 156(5):1039–1077, 2020) for each pair of simple modules V and W.

Dates et versions

hal-03235658 , version 1 (25-05-2021)

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Ryo Fujita, Se-Jin Oh. Q-data and Representation Theory of Untwisted Quantum Affine Algebras. Commun.Math.Phys., 2021, 384 (2), pp.1351-1407. ⟨10.1007/s00220-021-04028-8⟩. ⟨hal-03235658⟩
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