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Face functors for KLR algebras

Authors: Peter J. McNamara and Peter Tingley
Journal: Represent. Theory 21 (2017), 106-131
MSC (2010): Primary 17B37
Published electronically: July 12, 2017
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Abstract | References | Similar Articles | Additional Information

Abstract: Simple representations of KLR algebras can be used to realize the infinity crystal for the corresponding symmetrizable Kac-Moody algebra. It was recently shown that, in finite and affine types, certain sub-categories of ``cuspidal'' representations realize crystals for sub-Kac-Moody algebras. Here we put that observation on a firmer categorical footing by exhibiting a corresponding functor between the category of representations of the KLR algebra for the sub-Kac-Moody algebra and the category of cuspidal representations of the original KLR algebra.

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  • [BCP] Jonathan Beck, Vyjayanthi Chari, and Andrew Pressley, An algebraic characterization of the affine canonical basis, Duke Math. J. 99 (1999), no. 3, 455-487. MR 1712630,
  • [BN] Jonathan Beck and Hiraku Nakajima, Crystal bases and two-sided cells of quantum affine algebras, Duke Math. J. 123 (2004), no. 2, 335-402. MR 2066942
  • [BKM] Jonathan Brundan, Alexander Kleshchev, and Peter J. McNamara, Homological properties of finite-type Khovanov-Lauda-Rouquier algebras, Duke Math. J. 163 (2014), no. 7, 1353-1404. MR 3205728,
  • [CL] Sabin Cautis and Anthony Licata, Heisenberg categorification and Hilbert schemes, Duke Math. J. 161 (2012), no. 13, 2469-2547. MR 2988902,
  • [HK] Ruth Stella Huerfano and Mikhail Khovanov, A category for the adjoint representation, J. Algebra 246 (2001), no. 2, 514-542. MR 1872113,
  • [KKKO] Seok-Jin Kang, Masaki Kashiwara, Myungho Kim, and Se-jin Oh, Monoidal categorification of cluster algebras II. arXiv:1505.03241
  • [K] Masaki Kashiwara, Notes on parameters of quiver Hecke algebras, Proc. Japan Acad. Ser. A Math. Sci. 88 (2012), no. 7, 97-102. MR 2946856,
  • [KP] Masaki Kashiwara and Euiyong Park, Affinizations and R-matrices for quiver Hecke algebras. arXiv:1505.03241. To appear in JEMS.
  • [KM] A. Kleshchev and R. Muth, Affine zigzag algebras and imaginary strata for KLR algebras. arXiv:1505.03241
  • [KL] Mikhail Khovanov and Aaron D. Lauda, A diagrammatic approach to categorification of quantum groups. I, Represent. Theory 13 (2009), 309-347. MR 2525917,
  • [KL2] Mikhail Khovanov and Aaron D. Lauda, A diagrammatic approach to categorification of quantum groups II, Trans. Amer. Math. Soc. 363 (2011), no. 5, 2685-2700. MR 2763732,
  • [LV] Aaron D. Lauda and Monica Vazirani, Crystals from categorified quantum groups, Adv. Math. 228 (2011), no. 2, 803-861. MR 2822211,
  • [M] Peter. J. McNamara, Representations of Khovanov-Lauda-Rouquier Algebras III: Symmetric Affine Type. arXiv:1511.05905. To appear in Math Z.
  • [R] Raphael Rouquier, 2-Kac-Moody algebras. arXiv:1505.03241
  • [TW] Peter Tingley and Ben Webster, Mirković-Vilonen polytopes and Khovanov-Lauda-Rouquier algebras, Compos. Math. 152 (2016), no. 8, 1648-1696. MR 3542489,

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Additional Information

Peter J. McNamara
Affiliation: School of Mathematics and Physics, University of Queensland, St Lucia, QLD, Australia

Peter Tingley
Affiliation: Department of Mathematics and Statistics, Loyola University, Chicago, Illinois 60660

Received by editor(s): February 12, 2016
Received by editor(s) in revised form: October 3, 2016, March 13, 2017, and May 1, 2017
Published electronically: July 12, 2017
Article copyright: © Copyright 2017 American Mathematical Society

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