The nil-Brauer category

Brundan, Jonathan; Wang, Weiqiang; Webster, Benjamin

doi:10.5802/art.2

Brundan, Jonathan ¹; Wang, Weiqiang ² ; Webster, Benjamin ³

¹ Department of Mathematics, University of Oregon, 1222 University of Oregon, Eugene, OR 97403-1222, United States
² Department of Mathematics, University of Virginia, P. O. Box 400137, Charlottesville, VA 22904-4137, United States
³ Department of Pure Mathematics, University of Waterloo & Perimeter Institute for Theoretical Physics, 200 University Avenue West, Waterloo, ON N2L 3G1, Canada

Annals of Representation Theory, Volume 1 (2024) no. 1, pp. 21-58.

Abstract

We introduce the nil-Brauer category and prove a basis theorem for its morphism spaces. This basis theorem is an essential ingredient required to prove that nil-Brauer categorifies the split $ı$ -quantum group of rank one. As this $ı$ -quantum group is a basic building block for $ı$ -quantum groups of higher rank, we expect that the nil-Brauer category will play a central role in future developments related to the categorification of quantum symmetric pairs.

Received: 2023-05-18
Revised: 2023-06-27
Accepted: 2023-09-15
Published online: 2023-11-23

DOI: 10.5802/art.2

Classification: 17B10, 17B37
Keywords: Brauer category, string calculus, categorification

Author's affiliations:

Brundan, Jonathan ¹; Wang, Weiqiang ²; Webster, Benjamin ³

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     title = {The {nil-Brauer} category},
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Brundan, Jonathan; Wang, Weiqiang; Webster, Benjamin. The nil-Brauer category. Annals of Representation Theory, Volume 1 (2024) no. 1, pp. 21-58. doi : 10.5802/art.2. https://art.centre-mersenne.org/articles/10.5802/art.2/

References
Cited by

[1] Ariki, Susumu; Mathas, Andrew; Rui, Hebing Cyclotomic Nazarov–Wenzl algebras, Nagoya Math. J., Volume 182 (2006), pp. 47-134 | DOI | MR | Zbl

[2] Bao, Huanchen; Wang, Weiqiang Canonical bases arising from quantum symmetric pairs, Invent. Math., Volume 213 (2018) no. 3, pp. 1099-1177 | DOI | MR | Zbl

[3] Bao, Huanchen; Wang, Weiqiang A new approach to Kazhdan–Lusztig theory of type $B$ via quantum symmetric pairs, Astérisque, 402, 2018 | Zbl

[4] Berman, Collin; Wang, Weiqiang Formulae of $ı$ -divided powers in $U_{q} ({𝔰𝔩}_{2})$ , J. Pure Appl. Algebra, Volume 222 (2018) no. 9, pp. 2667-2702 | DOI | Zbl

[5] Brundan, Jonathan On the definition of Kac–Moody 2-category, Math. Ann., Volume 364 (2016), pp. 353-372 | DOI | MR | Zbl

[6] Brundan, Jonathan; Davidson, Nicholas Categorical actions and crystals, Categorification and higher representation theory (Contemporary Mathematics), Volume 683, American Mathematical Society, 2017, pp. 105-147 | DOI | MR | Zbl

[7] Brundan, Jonathan; Ellis, Alexander P. Monoidal supercategories, Commun. Math. Phys., Volume 351 (2017) no. 3, pp. 1045-1089 | DOI | MR | Zbl

[8] Brundan, Jonathan; Savage, Alistair; Webster, Ben Heisenberg and Kac–Moody categorification, Sel. Math., New Ser., Volume 26 (2020), Paper no. 74 | DOI | MR | Zbl

[9] Brundan, Jonathan; Savage, Alistair; Webster, Ben Foundations of Frobenius Heisenberg categories, J. Algebra, Volume 578 (2021), pp. 115-185 | DOI | MR | Zbl

[10] Brundan, Jonathan; Savage, Alistair; Webster, Ben The degenerate Heisenberg category and its Grothendieck ring, Ann. Sci. Éc. Norm. Supér., Volume 56 (2023), pp. 1517-1563

[11] Brundan, Jonathan; Webster, Ben; Wang, Weiqiang Nil–Brauer categorifies the split $ı$ -quantum group of rank one (2023) (arXiv:2305.05877)

[12] Carpentier, Rui Pedro From planar graphs to embedded graphs—a new approach to Kauffman and Vogel’s polynomial, J. Knot Theory Ramifications, Volume 9 (2000) no. 8, pp. 975-986 | DOI | MR | Zbl

[13] Daugherty, Zajj; Ram, Arun; Virk, Rahbar Affine and degenerate affine BMW algebras: actions on tensor space, Sel. Math., New Ser., Volume 19 (2013) no. 2, pp. 611-653 | DOI | MR | Zbl

[14] Ehrig, Michael; Stroppel, Catharina Nazarov–Wenzl algebras, coideal subalgebras and categorified skew Howe duality, Adv. Math., Volume 331 (2018), pp. 58-142 | DOI | MR | Zbl

[15] Khovanov, Mikhail; Lauda, Aaron D. A categorification of quantum $sl (n)$ , Quantum Topol., Volume 1 (2010), pp. 1-92 | DOI | MR | Zbl

[16] Lauda, Aaron D. A categorification of quantum $sl (2)$ , Adv. Math., Volume 225 (2010) no. 6, pp. 3327-3424 | DOI | MR | Zbl

[17] Lauda, Aaron D. Categorified quantum $sl (2)$ and equivariant cohomology of iterated flag varieties, Algebr. Represent. Theory, Volume 14 (2011) no. 2, pp. 253-282 | DOI | MR | Zbl

[18] Macdonald, Ian G. Symmetric Functions and Hall Polynomials, Oxford Classic Texts in the Physical Sciences, Clarendon Press, 2015 | Zbl

[19] Nazarov, Maxim Young’s orthogonal form for Brauer’s centralizer algebra, J. Algebra, Volume 182 (1996) no. 3, pp. 664-693 | DOI | MR | Zbl

[20] Rouquier, Raphaël $2$ -Kac–Moody algebras (2008) (arXiv:0812.5023)

[21] Rui, Hebing; Song, Linliang Affine Brauer category and parabolic category $𝒪$ in types $B$ , $C$ , $D$ , Math. Z., Volume 293 (2019), pp. 503-550 | DOI | MR | Zbl

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