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Canonical bases and higher representation theory
Published online by Cambridge University Press: 07 October 2014
Abstract
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This paper develops a general theory of canonical bases and how they arise naturally in the context of categorification. As an application, we show that Lusztig’s canonical basis in the whole quantized universal enveloping algebra is given by the classes of the indecomposable 1-morphisms in a categorification when the associated Lie algebra is of finite type and simply laced. We also introduce natural categories whose Grothendieck groups correspond to the tensor products of lowest- and highest-weight integrable representations. This generalizes past work of the author’s in the highest-weight case.
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References
Achar, P. N. and Stroppel, C., Completions of Grothendieck groups, Bull. Lond. Math. Soc. (N.S.) 45 (2013), 200–212.CrossRefGoogle Scholar
Ágoston, I., Dlab, V. and Lukács, E., Quasi-hereditary extension algebras, Algebr. Represent. Theory 6 (2003), 97–117; MR 1960515 (2004c:16010).Google Scholar
Bao, H. and Wang, W., Canonical bases in tensor products revisited, Preprint (2014), arXiv:1403.0039.Google Scholar
Beilinson, A., Ginzburg, V. and Soergel, W., Koszul duality patterns in representation theory, J. Amer. Math. Soc. 9 (1996), 473–527.CrossRefGoogle Scholar
Braden, T., Licata, A., Proudfoot, N. and Webster, B., Gale duality and Koszul duality, Adv. Math. 225 (2010), 2002–2049.Google Scholar
Braden, T., Licata, A., Proudfoot, N. and Webster, B., Hypertoric category 𝓞, Adv. Math. 231 (2012), 1487–1545.Google Scholar
Brundan, J. and Kleshchev, A., Graded decomposition numbers for cyclotomic Hecke algebras, Adv. Math. 222 (2009), 1883–1942.Google Scholar
Brundan, J. and Kleshchev, A., Blocks of cyclotomic Hecke algebras and Khovanov–Lauda algebras, Invent. Math. 178 (2009), 451–484.CrossRefGoogle Scholar
Cautis, S. and Lauda, A., Implicit structure in 2-representations of quantum groups, Preprint (2011), arXiv:1111.1431.Google Scholar
Cline, E., Parshall, B. and Scott, L., Stratifying endomorphism algebras, Mem. Amer. Math. Soc. 124 (1996), no. 591.Google Scholar
Du, J., Parshall, B. and Scott, L., Stratifying endomorphism algebras associated to Hecke algebras, J. Algebra 203 (1998), 169–210; MR 1620729 (99e:20006).Google Scholar
Elias, B. and Williamson, G., The Hodge theory of Soergel bimodules, Preprint (2012), arXiv:1212.0791.Google Scholar
Grojnowski, I., Affine 
$\mathfrak{sl}_{p}$controls the representation theory of the symmetric group and related Hecke algebras, Preprint (1999), arXiv:math/9907129.Google Scholar
$\mathfrak{sl}_{p}$controls the representation theory of the symmetric group and related Hecke algebras, Preprint (1999), arXiv:math/9907129.Google ScholarJeffrey, L., Kiem, Y.-H. and Kirwan, F., On the cohomology of hyperKähler quotients, Transform. Groups 14 (2009), 801–823.Google Scholar
Kang, S.-J. and Kashiwara, M., Categorification of highest weight modules via Khovanov–Lauda–Rouquier algebras, Invent. Math. 190 (2012), 699–742.CrossRefGoogle Scholar
Kashiwara, M., Notes on parameters of quiver Hecke algebras, Preprint (2012), arXiv:1204.1776.Google Scholar
Kazhdan, D. and Lusztig, G., Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), 165–184.CrossRefGoogle Scholar
Khovanov, M. and Lauda, A. D., A diagrammatic approach to categorification of quantum groups. I, Represent. Theory 13 (2009), 309–347.Google Scholar
Khovanov, M. and Lauda, A. D., A categorification of quantum sl(n), Quantum Topol. 1 (2010), 1–92; MR 2628852 (2011g:17028).CrossRefGoogle Scholar
Khovanov, M. and Lauda, A. D., A diagrammatic approach to categorification of quantum groups II, Trans. Amer. Math. Soc. 363 (2011), 2685–2700; MR 2763732 (2012a:17021).Google Scholar
Lascoux, A., Leclerc, B. and Thibon, J.-Y., Hecke algebras at roots of unity and crystal bases of quantum affine algebras, Comm. Math. Phys. 181 (1996), 205–263.Google Scholar
Lauda, A. D., A categorification of quantum sl(2), Adv. Math. 225 (2010), 3327–3424; MR 2729010 (2012b:17036).Google Scholar
Losev, I., Proof of Varagnolo-Vasserot conjecture on cyclotomic categories ${\mathcal{O}}$, Preprint (2013), arXiv:1305.4894.Google Scholar
${\mathcal{O}}$, Preprint (2013), arXiv:1305.4894.Google ScholarLusztig, G., Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990), 447–498.Google Scholar
Lusztig, G., Canonical bases in tensor products, Proc. Natl. Acad. Sci. USA 89 (1992), 8177–8179.Google Scholar
Lusztig, G., Introduction to quantum groups, Progress in Mathematics, vol. 110 (Birkhäuser, Boston, 1993).Google Scholar
Rouquier, R., Shan, P., Varagnolo, M. and Vasserot, E., Categorifications and cyclotomic rational double affine Hecke algebras, Preprint (2013), arXiv:1305.4456.Google Scholar
Shan, P., Crystals of Fock spaces and cyclotomic rational double affine Hecke algebras, Ann. Sci. Éc. Norm. Supér. (4) 44 (2011), 147–182; MR 2760196 (2012c:20009).Google Scholar
Shan, P., Varagnolo, M. and Vasserot, E., Koszul duality of affine Kac–Moody algebras and cyclotomic rational DAHA, Preprint (2011), arXiv:1107.0146.Google Scholar
Soergel, W., The combinatorics of Harish-Chandra bimodules, J. Reine Angew. Math. 429 (1992), 49–74; MR 1173115 (94b:17011).Google Scholar
Springer, T. A., Quelques applications de la cohomologie d’intersection, in Bourbaki Seminar, Vol. 1981/1982, Astérisque, vol. 92 (Société Mathématique de France, Paris, 1982), 249–273; MR 689533 (85i:32016b).Google Scholar
Stošić, M., Indecomposable 1-morphisms of $\dot{{\mathcal{U}}}_{3}^{+}$ and the canonical basis of $U_{q}^{+}(\mathfrak{sl}_{3})$, Preprint (2011), arXiv:1105.4458.Google Scholar
$\dot{{\mathcal{U}}}_{3}^{+}$ and the canonical basis of 
$U_{q}^{+}(\mathfrak{sl}_{3})$, Preprint (2011), arXiv:1105.4458.Google ScholarStroppel, C. and Webster, B., Quiver Schur algebras and $q$-Fock space, Preprint (2011), arXiv:1110.1115.Google Scholar
$q$-Fock space, Preprint (2011), arXiv:1110.1115.Google ScholarSussan, J., Positivity and the canonical basis of tensor products of finite-dimensional irreducible representations of quantum $\mathfrak{sl}(k)$, Preprint (2008), arXiv:0804.2034.Google Scholar
$\mathfrak{sl}(k)$, Preprint (2008), arXiv:0804.2034.Google ScholarTingley, P. and Webster, B., Mirković–Vilonen polytopes and Khovanov–Lauda–Rouquier algebras, Preprint (2012), arXiv:1210.6921.Google Scholar
Uglov, D., Canonical bases of higher-level q-deformed Fock spaces and Kazhdan–Lusztig polynomials, in Physical combinatorics (Kyoto, 1999), Progress in Mathematics, vol. 191 (Birkhäuser, Boston, 2000), 249–299.Google Scholar
Varagnolo, M. and Vasserot, E., Canonical bases and KLR-algebras, J. Reine Angew. Math. 659 (2011), 67–100.Google Scholar
Webster, B., A categorical action on quantized quiver varieties, Preprint (2012), arXiv:1208.5957.Google Scholar
Webster, B., Weighted Khovanov–Lauda–Rouquier algebras, Preprint (2012), arXiv:1209.2463.Google Scholar
Webster, B., Knot invariants and higher representation theory, Preprint (2013), arXiv:1309.3796.Google Scholar
Webster, B., Rouquier’s conjecture and diagrammatic algebra, Preprint (2013), arXiv:1306.0074.Google Scholar
Williamson, G., On an analogue of the James conjecture, Preprint (2012), arXiv:1212.0794.Google Scholar
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