Abstract
In this paper we generalize cellular algebras by allowing different partial orderings relative to fixed idempotents. For these relative cellular algebras we classify and construct simple modules, and we obtain other characterizations in analogy to cellular algebras. We also give several examples of algebras that are relative cellular, but not cellular: most prominently, the restricted enveloping algebra and the small quantum group for \( \mathfrak{s}{\mathfrak{l}}_2 \), and an annular version of arc algebras.
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R. Anno, V. Nandakumar, Exotic t-structures for two-block Springer fibers, arXiv:1602.00768 (2016).
M. M. Asaeda, J. H. Przytycki, A. S. Sikora. Categorification of the Kauffman bracket skein module of I-bundles over surfaces, Algebr. Geom. Topol. 4 (2004), 1177–1210.
H. H. Andersen, C. Stroppel, D. Tubbenhauer, Cellular structures using Uq-tilting modules, Pacific J. Math. 292 (2018) no. 1, 21–59.
A. Beliakova, K. K. Putyra, S. M. Wehrli, Quantum link homology via trace functor I, Invent. Math. 215 (2019), no. 2, 383–492.
J. Brundan, C. Stroppel, Highest weight categories arising from Khovanov’s diagram algebra I: cellularity, Mosc. Math. J. 11 (2011), no. 4, 685–722, 821–822.
G. Bellamy, U. Thiel, Cores of graded algebras with triangular decomposition, arXiv:1711.00780 (2017).
G. Bellamy, U. Thiel, Highest weight theory for finite-dimensional graded algebras with triangular decomposition, Adv. Math. 330 (2018), 361–419.
K. Coulembier, R. Zhang, Borelic pairs for stratified algebras, Adv. Math. 345 (2019), 53–115.
J. Du, H. Rui, Based algebras and standard bases for quasi-hereditary algebras, Trans. Amer. Math. Soc. 350 (1998), no. 8, 3207–3235.
M. Ehrig, C. Stroppel, 2-row Springer fibres and Khovanov diagram algebras for type D, Canad. J. Math. 68 (2016), no. 6, 1285–1333.
M. Ehrig, C. Stroppel, Diagrammatic description for the categories of perverse sheaves on isotropic Grassmannians, Selecta Math. (N.S.) 22 (2016), no. 3, 1455–1536.
M. Ehrig, C. Stroppel, D. Tubbenhauer, The Blanchet–Khovanov algebras, in: Categorification and Higher Representation Theory, Contemp. Math., Vol. 683, Amer. Math. Soc., Providence, RI, 2017, pp. 183–226.
M. Ehrig, D. Tubbenhauer, Algebraic properties of zigzag algebras, Comm. Algebra, to appear, arXiv:1807.11173 (2018).
M. Ehrig, D. Tubbenhauer, A. Wilbert. Singular TQFTs, foams and type D arc algebras, Doc. Math., to appear, arXiv:1611.07444 (2016).
E. M. Friedlander, B. J. Parshall, Modular representation theory of Lie algebras, Amer. J. Math. 110 (1988). no. 6, 1055–1093.
F. M. Goodman, J. Graber, Cellularity and the Jones basic construction, Adv. in Appl. Math. 46 (2011), no. 1–4, 312–362.
J. J. Graham, G. Lehrer, Cellular algebras, Invent. Math. 123 (1996), no. 1, 1–34.
E. J. Grigsby, A. M. Licata, S. M. Wehrli, Annular Khovanov homology and knotted Schur–Weyl representations, Compos. Math. 154 (2018), no 3, 459–502.
A. Gadbled, A.-L. Thiel, E. Wagner, Categorical action of the extended braid group of affine type A, Commun. Contemp. Math. 19 (2017), no. 3, 1650024, 39 pp.
R. S. Huerfano, M. Khovanov, A category for the adjoint representation, J. Algebra 246 (2001), no. 2, 514–542.
J. Hu, A. Mathas, Graded cellular bases for the cyclotomic Khovanov–Lauda–Rouquier algebras of type A, Adv. Math. 225 (2010), no. 2, 598–642.
J. C. Jantzen, Representations of Lie algebras in positive characteristic, in: Representation Theory of Algebraic Groups and Quantum Groups, Adv. Stud. Pure Math., Vol. 40, Math. Soc. Japan, Tokyo, 2004, pp. 175–218.
M. Khovanov, A functor-valued invariant of tangles, Algebr. Geom. Topol. 2 (2002), 665–741.
S. König, C. Xi, On the structure of cellular algebras, in: Algebras and Modules, II (Geiranger, 1996), CMS Conf. Proc., Vol. 24, Amer. Math. Soc., Providence, RI, 1998, pp. 365–386.
S. König, C. Xi, When is a cellular algebra quasi-hereditary? Math. Ann. 315 (1999), no. 2, 281–293.
S. König, C. Xi, Affine cellular algebras, Adv. Math. 229 (2012), no. 1, 139–182.
G. Lusztig, Finite-dimensional Hopf algebras arising from quantized universal enveloping algebra, J. Amer. Math. Soc. 3 (1990), no. 1, 257–296.
M. Mackaay, The \( \mathfrak{s}{\mathfrak{l}}_n \)-web algebras and dual canonical bases, J. Algebra 409 (2014), 54–100.
M. Mackaay, W. Pan, D. Tubbenhauer, The \( \mathfrak{s}{\mathfrak{l}}_3 \)-web algebra, Math. Z. 277 (2014), no. 1–2, 401–479.
M. Mackaay, D. Tubbenhauer, Two-color Soergel calculus and simple transitive 2-representations, Canad. J. Math, to appear, arXiv:1609.00962 (2016).
H. Queffelec, D.E.V. Rose, Sutured annular Khovanov–Rozansky homology, Trans. Amer. Math. Soc. 370 (2018), no. 2, 1285–1319.
L. P. Roberts, On knot Floer homology in double branched covers, Geom. Topol. 17 (2013), no. 1, 413–467.
D. Tubbenhauer, \( \mathfrak{s}{\mathfrak{l}}_3 \)-web bases, intermediate crystal bases and categorification, J. Algebraic Combin. 40 (2014), no. 4, 1001–1076.
C. Xi, Standardly stratified algebras and cellular algebras, Math. Proc. Cambridge Philos. Soc. 133 (2002), no. 1, 37–53.
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EHRIG, M., TUBBENHAUER, D. RELATIVE CELLULAR ALGEBRAS. Transformation Groups 26, 229–277 (2021). https://doi.org/10.1007/s00031-019-09544-5
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DOI: https://doi.org/10.1007/s00031-019-09544-5