TY - JOUR

T1 - Q-data and Representation Theory of Untwisted Quantum Affine Algebras

AU - Fujita, Ryo

AU - Oh, Se jin

N1 - Funding Information:
We are deeply grateful to David Hernandez for helpful discussions and for pointing out that some of the results in the initial version of this paper were already known in the literature. We also wish to thank Masaki Kashiwara, Bernhard Keller, Myungho Kim, Bernard Leclerc and Euiyong Park for stimulating discussions and comments. Finally, we would like to thank the anonymous referees for many helpful suggestions about the exposition and the appropriate references.
Funding Information:
Ryo Fujita was supported in part by Grant-in-Aid for JSPS Research Fellow JP18J10669, by JSPS Grant-in-Aid for Scientific Research (B) JP19H01782, (A) JP17H01086 and also by JSPS Overseas Research Fellowships (during the revision). Se-jin Oh was supported by the Ministry of Education of the Republic of Korea and the National Research Foundation of Korea (NRF-2019R1A2C4069647).
Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2021/6

Y1 - 2021/6

N2 - For a complex finite-dimensional simple Lie algebra 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 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 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 Λ (V, W) and Λ ∞(V, W) introduced in Kashiwara et al. (Compos Math 156(5):1039–1077, 2020) for each pair of simple modules V and W.

AB - For a complex finite-dimensional simple Lie algebra 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 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 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 Λ (V, W) and Λ ∞(V, W) introduced in Kashiwara et al. (Compos Math 156(5):1039–1077, 2020) for each pair of simple modules V and W.

UR - http://www.scopus.com/inward/record.url?scp=85103179572&partnerID=8YFLogxK

U2 - 10.1007/s00220-021-04028-8

DO - 10.1007/s00220-021-04028-8

M3 - Article

AN - SCOPUS:85103179572

SN - 0010-3616

VL - 384

SP - 1351

EP - 1407

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

IS - 2

ER -