Hostname: page-component-68c7f8b79f-r8tb2 Total loading time: 0 Render date: 2025-12-26T02:45:51.240Z Has data issue: false hasContentIssue false
  • English
  • Français

A Basis Theorem for the Degenerate Affine Oriented Brauer–Clifford Supercategory

Part of: Lie algebras and Lie superalgebras Categories with structure

Published online by Cambridge University Press:  07 March 2019

Jonathan Brundan ,
Jonathan Comes  and
Jonathan Robert Kujawa
Jonathan Brundan
Affiliation:
Department of Mathematics, University of Oregon, Eugene, Oregon 97403-1222, USA Email: brundan@uoregon.edu
Jonathan Comes
Affiliation:
Department of Mathematics & Physical Sciences, The College of Idaho, Caldwell, Idaho 83605, United States Email: jonnycomes@gmail.com
Jonathan Robert Kujawa
Affiliation:
Department of Mathematics, University of Oklahoma, Norman, Oklahoma 73019-0315, United States Email: kujawa@math.ou.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We introduce the oriented Brauer–Clifford and degenerate affine oriented Brauer–Clifford supercategories. These are diagrammatically defined monoidal supercategories that provide combinatorial models for certain natural monoidal supercategories of supermodules and endosuperfunctors, respectively, for the Lie superalgebras of type Q. Our main results are basis theorems for these diagram supercategories. We also discuss connections and applications to the representation theory of the Lie superalgebra of type Q.

Keywords

monoidal categorysupercategoryLie superalgebratype Q

MSC classification

Primary: 17B10: Representations, algebraic theory (weights)
Secondary: 18D10: Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories

Information

Type
Article
Information
Canadian Journal of Mathematics , Volume 71 , Issue 5 , October 2019 , pp. 1061 - 1101
Copyright
© Canadian Mathematical Society 2019 

References

Balagovic, M., Daugherty, Z., Entova-Aizenbud, I., Halacheva, I., Hennig, J., Im, M. S., Letzter, G., Norton, E., Serganova, V., and Stroppel, C., The affine VW supercategory. 2018. arxiv:1801.04178.Google Scholar
Benkart, G., Guay, N., Jung, J. H., Kang, S.-J., and Wilcox, S., Quantum walled Brauer-Clifford superalgebras . J. Algebra 454(2016), 433–474. https://doi.org/10.1016/j.jalgebra.2015.04.038.Google Scholar
Benson, D. and Doty, S., Schur-Weyl duality over finite fields . Arch. Math. (Basel) 93(2009), 425–435. https://doi.org/10.1007/s00013-009-0066-8.Google Scholar
Brundan, J., Kazhdan-Lusztig polynomials and character formulae for the Lie superalgebra q (n) . Adv. Math. 182(2004), 28–77. https://doi.org/10.1016/S0001-8708(03)00073-2.Google Scholar
Brundan, J., On the definition of Heisenberg category . Algebr. Comb. 1(2018), 523–544. https://doi.org/10.5802/alco.26.Google Scholar
Brundan, J., Representations of oriented skein categories. 2017. arxiv:1712.08953.Google Scholar
Brundan, J., Comes, J., Nash, D., and Reynolds, A., A basis theorem for the affine oriented Brauer category and its cyclotomic quotients . Quantum Topol. 8(2017), 75–112. https://doi.org/10.4171/QT/87.Google Scholar
Brundan, J. and Davidson, N., Type A blocks of super category 𝓞 . J. Algebra 473(2017), 447–480. https://doi.org/10.1016/j.jalgebra.2016.11.022.Google Scholar
Brundan, J. and Davidson, N., Type C blocks of super category 𝓞. 2017. arxiv:1702.05055.Google Scholar
Brundan, J. and Ellis, A. P., Monoidal supercategories . Comm. Math. Phys. 351(2017), 1045–1089. https://doi.org/10.1007/s00220-017-2850-9.Google Scholar
Brundan, J. and Ellis, A. P., Super Kac-Moody 2-categories . Proc. Lond. Math. Soc. 115(2017), no. 5, 925–973. https://doi.org/10.1112/plms.12055.Google Scholar
Brundan, J. and Kleshchev, A., Hecke-Clifford superalgebras, crystals of type A 2l (2) and modular branching rules for Ŝ n . Represent. Theory 5(2001), 317–403. https://doi.org/10.1090/S1088-4165-01-00123-6.Google Scholar
Brundan, J. and Kleshchev, A., Projective representations of symmetric groups via Sergeev duality . Math. Z. 239(2002), 27–68. https://doi.org/10.1007/s002090100282.Google Scholar
Brundan, J. and Kleshchev, A., Modular representations of the supergroup Q (n). I . J. Algebra 260(2003), 64–98. https://doi.org/10.1016/S0021-8693(02)00620-8.Google Scholar
Chen, C.-W., Reduction method for representations of queer Lie superalgebras . J. Math. Phys. 57(2016), no. 5, 051703. https://doi.org/10.1063/1.4948744.Google Scholar
Cheng, S.-J., Kwon, J.-H., and Wang, W., Character formulae for queer Lie superalgebras and canonical bases of types A/C . Comm. Math. Phys. 352(2017), 1091–1119. https://doi.org/10.1007/s00220-016-2809-2.Google Scholar
Cheng, S.-J. and Wang, W., Dualities and representations of Lie superalgebras . Graduate Studies in Mathematics, 144, American Mathematical Society, Providence, RI, 2012. https://doi.org/10.1090/gsm/144.Google Scholar
Coulembier, K. and Mazorchuk, V., The G-centre and gradable derived equivalences. 2017. arxiv:1703.02623.Google Scholar
Du, J. and Wan, J., Presenting queer Schur superalgebras . Int. Math. Res. Not. IMRN (2015), no. 8, 2210–2272. https://doi.org/10.1093/imrn/rnt262.Google Scholar
Ellis, A. P. and Lauda, A. D., An odd categorification of U q(sl2) . Quantum Topol. 7(2016), 329–433. https://doi.org/10.4171/QT/78.Google Scholar
Gao, M., Rui, H., Song, L., and Su, Y., Affine walled Brauer–Clifford superalgebras. 2017. arxiv:1708.05135.Google Scholar
Gao, M., Rui, H., Song, L., and Su, Y., A proof of Comes-Kujawa’s conjecture. 2018. arxiv:1801.09071.Google Scholar
Hill, D., Kujawa, J. R., and Sussan, J., Degenerate affine Hecke-Clifford algebras and type Q Lie superalgebras . Math. Z. 268(2011), 1091–1158. https://doi.org/10.1007/s00209-010-0712-7.Google Scholar
Jung, J. and Kang, S., Mixed Schur-Weyl-Sergeev duality for queer Lie superalgebras . J. Algebra 399(2014), 516–545. https://doi.org/10.1016/j.jalgebra.2013.08.029.Google Scholar
Kang, S.-J., Kashiwara, M., and Oh, S.-j., Supercategorification of quantum Kac-Moody algebras . Adv. Math. 242(2013), 116–162. https://doi.org/10.1016/j.aim.2013.04.008.Google Scholar
Kang, S.-J., Kashiwara, M., and Oh, S.-j., Supercategorification of quantum Kac-Moody algebras II . Adv. Math. 265(2014), 169–240. https://doi.org/10.1016/j.aim.2014.07.036.Google Scholar
Kleshchev, A., Linear and projective representations of symmetric groups . Cambridge Tracts in Mathematics, 163, Cambridge University Press, Cambridge, 2005. https://doi.org/10.1017/CBO9780511542800.Google Scholar
Kujawa, J. R. and Tharp, B., The marked Brauer category . J. Lond. Math. Soci. 95(2017), 393–413. https://doi.org/10.1112/jlms.12015.Google Scholar
Nazarov, M., Young’s symmetrizers for projective representations of the symmetric group . Adv. Math. 127(1997), 190–257. https://doi.org/10.1006/aima.1997.1621.Google Scholar
Penkov, I. and Serganova, V., Characters of irreducible G-modules and cohomology of G/P for the Lie supergroup G = Q (N). Algebraic geometry, 7 , J. Math. Sci. (New York) 84(1997), 1382–1412. https://doi.org/10.1007/BF02399196.Google Scholar
Reynolds, A., Representations of the oriented Brauer category. PhD thesis, University of Oregon, Eugene, Oregon, 2015.Google Scholar
Rosso, D. and Savage, A., A general approach to Heisenberg categorification via wreath product algebras . Math. Z. 286(2017), 603–655. https://doi.org/10.1007/s00209-016-1776-9.Google Scholar
Rouquier, R., Quiver Hecke algebras and 2-Lie algebras . Algebra Colloq. 19(2012), 359–410. https://doi.org/10.1142/S1005386712000247.Google Scholar
Sergeev, A. N., The centre of enveloping algebra for Lie superalgebra Q (n, C) . Lett. Math. Phys. 7(1983), 177–179. https://doi.org/10.1007/BF00400431.Google Scholar
Sergeev, A. N., Tensor algebra of the identity representation as a module over the Lie superalgebras Gl (n, m) and Q (n) . Mat. Sb. (N.S.) 123(165), 422–430.Google Scholar