Face functors for KLR algebras

McNamara, Peter; Tingley, Peter

doi:10.1090/ert/496

Face functors for KLR algebras
HTML articles powered by AMS MathViewer

by Peter J. McNamara and Peter Tingley PDF

Represent. Theory 21 (2017), 106-131 Request permission

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.

References

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, DOI 10.1215/S0012-7094-99-09915-5
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, DOI 10.1215/S0012-7094-04-12325-2X
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, DOI 10.1215/00127094-2681278
Sabin Cautis and Anthony Licata, Heisenberg categorification and Hilbert schemes, Duke Math. J. 161 (2012), no. 13, 2469–2547. MR 2988902, DOI 10.1215/00127094-1812726
Ruth Stella Huerfano and Mikhail Khovanov, A category for the adjoint representation, J. Algebra 246 (2001), no. 2, 514–542. MR 1872113, DOI 10.1006/jabr.2001.8962
Seok-Jin Kang, Masaki Kashiwara, Myungho Kim, and Se-jin Oh, Monoidal categorification of cluster algebras II. arXiv:1505.03241
Masaki Kashiwara, Notes on parameters of quiver Hecke algebras, Proc. Japan Acad. Ser. A Math. Sci. 88 (2012), no. 7, 97–102. MR 2946856, DOI 10.3792/pjaa.88.97
Masaki Kashiwara and Euiyong Park, Affinizations and R-matrices for quiver Hecke algebras. arXiv:1505.03241. To appear in JEMS.
A. Kleshchev and R. Muth, Affine zigzag algebras and imaginary strata for KLR algebras. arXiv:1505.03241
Mikhail Khovanov and Aaron D. Lauda, A diagrammatic approach to categorification of quantum groups. I, Represent. Theory 13 (2009), 309–347. MR 2525917, DOI 10.1090/S1088-4165-09-00346-X
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, DOI 10.1090/S0002-9947-2010-05210-9
Aaron D. Lauda and Monica Vazirani, Crystals from categorified quantum groups, Adv. Math. 228 (2011), no. 2, 803–861. MR 2822211, DOI 10.1016/j.aim.2011.06.009
Peter. J. McNamara, Representations of Khovanov-Lauda-Rouquier Algebras III: Symmetric Affine Type. arXiv:1511.05905. To appear in Math Z.
Raphael Rouquier, 2-Kac-Moody algebras. arXiv:1505.03241
Peter Tingley and Ben Webster, Mirković-Vilonen polytopes and Khovanov-Lauda-Rouquier algebras, Compos. Math. 152 (2016), no. 8, 1648–1696. MR 3542489, DOI 10.1112/S0010437X16007338