## Face functors for KLR algebras

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- 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

## Additional Information

**Peter J. McNamara**- Affiliation: School of Mathematics and Physics, University of Queensland, St Lucia, QLD, Australia
- MR Author ID: 791816
- Email: maths@petermc.net
**Peter Tingley**- Affiliation: Department of Mathematics and Statistics, Loyola University, Chicago, Illinois 60660
- MR Author ID: 679482
- Email: ptingley@luc.edu
- 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
- © Copyright 2017 American Mathematical Society
- Journal: Represent. Theory
**21**(2017), 106-131 - MSC (2010): Primary 17B37
- DOI: https://doi.org/10.1090/ert/496
- MathSciNet review: 3670026