Reflective centers of module categories and quantum K-matrices

Laugwitz, Robert ; Walton, Chelsea ; Yakimov, Milen

Forum of Mathematics, Sigma, Tome 13 (2025) no. 1, p. e95

Résumé

Our work is motivated by obtaining solutions to the quantum reflection equation (qRE) by categorical methods. To start, given a braided monoidal category ${\mathcal {C}}$ and ${\mathcal {C}}$-module category ${\mathcal {M}}$, we introduce a version of the Drinfeld center ${\mathcal {Z}}({\mathcal {C}})$ of ${\mathcal {C}}$ adapted for ${\mathcal {M}}$; we refer to this category as the reflective center ${\mathcal {E}}_{\mathcal {C}}({\mathcal {M}})$ of ${\mathcal {M}}$. Just like ${\mathcal {Z}}({\mathcal {C}})$ is a canonical braided monoidal category attached to ${\mathcal {C}}$, we show that ${\mathcal {E}}_{\mathcal {C}}({\mathcal {M}})$ is a canonical braided module category attached to ${\mathcal {M}}$; its properties are investigated in detail.Our second goal pertains to when ${\mathcal {C}}$ is the category of modules over a quasitriangular Hopf algebra H, and ${\mathcal {M}}$ is the category of modules over an H-comodule algebra A. We show that the reflective center ${\mathcal {E}}_{\mathcal {C}}({\mathcal {M}})$ here is equivalent to a category of modules over an explicit algebra, denoted by $R_H(A)$, which we call the reflective algebra of A. This result is akin to ${\mathcal {Z}}({\mathcal {C}})$ being represented by the Drinfeld double ${\operatorname {Drin}}(H)$ of H. We also study the properties of reflective algebras.Our third set of results is also in the Hopf setting above. We show that reflective algebras are quasitriangular H-comodule algebras, and we examine their corresponding quantum K-matrices; this yields solutions to the qRE. We also establish that the reflective algebra $R_H(\mathbb {k})$ is an initial object in the category of quasitriangular H-comodule algebras, where $\mathbb {k}$ is the ground field. The case when H is the Drinfeld double of a finite group is illustrated.

Détail

DOI : 10.1017/fms.2025.10055

Bibliographie

[AM07] Andruskiewitsch, N. and Mombelli, J. M., ‘On module categories over finite-dimensional Hopf algebras’, J. Algebra 314(1) (2007), 383–418.10.1016/j.jalgebra.2007.04.006 Google Scholar | DOI

[AV22] Appel, A. and Vlaar, B., Universal K-matrices for quantum Kac-Moody algebras, Representation Theory of the American Mathematical Society 26(26) (2022), 764–824.10.1090/ert/623 Google Scholar | DOI

[Bal19] Balagović, M., Universal K-matrices for quantum symmetric pairs (slide presentation), 2019, available at http://www.lpthe.jussieu.fr/~pzinn/ICR/balagovic.pdf. Retrieved June 20, 2023. Google Scholar

[BK19] Balagović, M. and Kolb, S. ‘Universal K-matrix for quantum symmetric pairs’, J. Reine Angew. Math. 747 (2019), 299–353.10.1515/crelle-2016-0012 Google Scholar | DOI

[Bro13] Brochier, A., ‘Cyclotomic associators and finite type invariants for tangles in the solid torus, Algebr. Geom. Topol. 13(6) (2013), 3365–3409.10.2140/agt.2013.13.3365 Google Scholar | DOI

[BW18a] Bao, H. and Wang, W., ‘Canonical bases arising from quantum symmetric pairs’, Invent. Math. 213(3) (2018), 1099–1177.10.1007/s00222-018-0801-5 Google Scholar | DOI

[BW18b] Bao, H. and Wang, W., ‘A new approach to Kazhdan-Lusztig theory of type B via quantum symmetric pairs’, Ast´erisque 402 (2018), vii+134. Google Scholar

[BZBJ18] Ben-Zvi, D., Brochier, A. and Jordan, D., ‘Quantum character varieties and braided module categories’, Selecta Math. (N.S.) 24(5) (2018), 4711–4748.10.1007/s00029-018-0426-y Google Scholar | DOI

[CMZ97] Caenepeel, S., Militaru, G. and Zhu, S., ‘Crossed modules and Doi-Hopf modules’, Israel J. Math. 100 (1997), 221–247.10.1007/BF02773642 Google Scholar | DOI

[DN13] Davydov, A. and Nikshych, D., ‘The Picard crossed module of a braided tensor category’, Algebra Number Theory 7(6) (2013), 1365–1403.10.2140/ant.2013.7.1365 Google Scholar | DOI

[DN21] Davydov, A. and Nikshych, D., ‘Braided Picard groups and graded extensions of braided tensor categories’, Selecta Math. (N.S.) 27(4) (2021), Paper No. 65, 87.10.1007/s00029-021-00670-1 Google Scholar | DOI

[Doi92] Doi, Y., Unifying Hopf modules, J. Algebra 153 (1992), no. 2, 373–385.10.1016/0021-8693(92)90160-N Google Scholar | DOI

[Dri87] Drinfel’D, V. G., ‘Quantum groups’, in Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) (1987) 798–820. Google Scholar

[EGNO15] Etingof, P., Gelaki, S., Nikshych, D. and Ostrik, V., Tensor Categories (Mathematical Surveys and Monographs) vol. 205 (American Mathematical Society, Providence, RI, 2015).10.1090/surv/205 Google Scholar | DOI

[ENO05] Etingof, P., Nikshych, D. and Ostrik, V., ‘On fusion categories’, Ann. of Math. (2) 162(2) (2005), 581–642.10.4007/annals.2005.162.581 Google Scholar | DOI

[EO04] Etingof, P. and Ostrik, V., ‘Finite tensor categories’, Mosc. Math. J. 4(3) (2004), 627–654, 782–783.10.17323/1609-4514-2004-4-3-627-654 Google Scholar | DOI

[Gre10] Greenough, J., ‘Monoidal 2-structure of bimodule categories’, J. Algebra 324(8) (2010), 1818–1859.10.1016/j.jalgebra.2010.06.018 Google Scholar | DOI

[HS20] Heckenberger, I. and Schneider, H.-J., Hopf Algebras and Root Systems (Mathematical Surveys and Monographs) vol. 247 (American Mathematical Society, Providence, RI, 2020).10.1090/surv/247 Google Scholar | DOI

[JS91] Joyal, A. and Street, R., ‘The geometry of tensor calculus. I’, Adv. Math. 88(1) (1991), 55–112.10.1016/0001-8708(91)90003-P Google Scholar | DOI

[Kas95] Kassel, C., Quantum Groups (Graduate Texts in Mathematics) vol. 155 (Springer-Verlag, New York, 1995).10.1007/978-1-4612-0783-2 Google Scholar | DOI

[Kol20] Kolb, S., ‘Braided module categories via quantum symmetric pairs’, Proc. Lond. Math. Soc. (3) 121(1) (2020), 1–31.10.1112/plms.12303 Google Scholar | DOI

[Kop95] Koppinen, M., ‘Variations on the smash product with applications to group-graded rings’, J. Pure Appl. Algebra 104(1) (1995), 61–80.10.1016/0022-4049(94)00124-2 Google Scholar | DOI

[KY20] Kolb, S. and Yakimov, M., ‘Symmetric pairs for Nichols algebras of diagonal type via star products’, Adv. Math. 365 (2020), Art. 107042, 69 pp.10.1016/j.aim.2020.107042 Google Scholar | DOI

[Let02] Letzter, G., ‘Coideal subalgebras and quantum symmetric pairs’, in New Directions in Hopf Algebras (MSRI Publications) vol. 43 (Cambridge University Press, Cambridge, 2002), 117–165. Google Scholar

[Let99] Letzter, G., ‘Symmetric pairs for quantized enveloping algebras’, J. Algebra 220(2) (1999), 729–767.10.1006/jabr.1999.8015 Google Scholar | DOI

[Maj00] Majid, S., Foundations of Quantum Group Theory (Cambridge University Press, Cambridge, 2000). Google Scholar

[Maj91] Majid, S., ‘Braided groups and algebraic quantum field theories’, Lett. Math. Phys. 22(3) (1991), 167–175.10.1007/BF00403542 Google Scholar | DOI

[Rad94] Radford, D. E., ‘On Kauffman’s knot invariants arising from finite-dimensional Hopf algebras’, in Advances in Hopf algebras (Chicago IL, 1992) (Lecture notes in pure and applied mathematics) vol. 158 (Dekker, New York, 1994), 205–266. Google Scholar

[Tak80] Takeuchi, M., (Sp R, , J. Algebra 67(2) (1980), 436–475.10.1016/0021-8693(80)90170-2 Google Scholar | DOI

[TV17] Turaev, V. and Virelizier, A., Monoidal Categories and Topological Field Theory (Progress in Mathematics) vol. 322 (Birkhauser/Springer, Cham, 2017).10.1007/978-3-319-49834-8 Google Scholar | DOI

[Wal24] Walton, C., Symmetries of Algebras, vol. 1 (619 Wreath Publishing, Oklahoma City, OK, 2024). Google Scholar