Characterisation and applications of $\Bbbk$-split bimodules

  • Volodymyr Mazorchuk
  • Vanessa Miemietz
  • Xiaoting Zhang

Abstract

We describe the structure of bimodules (over finite dimensional algebras) which have the property that the functor of tensoring with such a bimodule sends any module to a projective module. The main result is that all such bimodules are $\Bbbk $-split in the sense that they factor (inside the tensor category of bimodules) over $\Bbbk $-vector spaces. As one application, we show that any simple $2$-category has a faithful $2$-representation inside the $2$-category of $\Bbbk $-split bimodules. As another application, we classify simple transitive $2$-representations of the $2$-category of projective bimodules over the algebra $\Bbbk [x,y]/(x^2,y^2,xy)$.

References

Auslander, M. and Reiten, I., On a theorem of E. Green on the dual of the transpose, in “Representations of finite-dimensional algebras (Tsukuba, 1990)'', CMS Conf. Proc., vol. 11, Amer. Math. Soc., Providence, RI, 1991, pp. 53--65.

Bernstein, J., Frenkel, I., and Khovanov, M., A categorification of the Temperley-Lieb algebra and Schur quotients of $U(\mathfrak{sl}_2)$ via projective and Zuckerman functors, Selecta Math. (N.S.) 5 (1999), no. 2, 199–241. https://doi.org/10.1007/s000290050047

Chan, A. and Mazorchuk, V., Diagrams and discrete extensions for finitary $2$-representations, Math. Proc. Camb. Phil. Soc. (2017, online), 28 pp. https://doi.org/10.1017/S0305004117000858

Chuang, J. and Rouquier, R., Derived equivalences for symmetric groups and $\mathfrak{sl}_2$-categorification, Ann. of Math. (2) 167 (2008), no. 1, 245–298. https://doi.org/10.4007/annals.2008.167.245

Dugas, A. S. and Martínez-Villa, R., A note on stable equivalences of Morita type, J. Pure Appl. Algebra 208 (2007), no. 2, 421–433. https://doi.org/10.1016/j.jpaa.2006.01.007

Forsberg, L., Multisemigroups with multiplicities and complete ordered semi-rings, Beitr. Algebra Geom. 58 (2017), no. 2, 405–426. https://doi.org/10.1007/s13366-016-0320-8

Grensing, A.-L. and Mazorchuk, V., Categorification of the Catalan monoid, Semigroup Forum 89 (2014), no. 1, 155–168. https://doi.org/10.1007/s00233-013-9510-y

Grensing, A.-L. and Mazorchuk, V., Finitary $2$-categories associated with dual projection functors, Commun. Contemp. Math. 19 (2017), no. 3, 1650016, 40 pp. https://doi.org/10.1142/S0219199716500164

Khovanov, M., A categorification of the Jones polynomial, Duke Math. J. 101 (2000), no. 3, 359–426. https://doi.org/10.1215/S0012-7094-00-10131-7

Khovanov, M. and Lauda, A. D., A categorification of quantum $\mathrm{sl}(n)$, Quantum Topol. 1 (2010), no. 1, 1–92. https://doi.org/10.4171/QT/1

Kildetoft, T., Mackaay, M., Mazorchuk, V., and Zimmermann, J., Simple transitive $2$-representations of small quotients of Soergel bimodules, Trans. Amer. Math. Soc. (2018, online), 40 pp. https://doi.org/10.1090/tran/7456

Kildetoft, T. and Mazorchuk, V., Parabolic projective functors in type $A$, Adv. Math. 301 (2016), 785–803. https://doi.org/10.1016/j.aim.2016.06.026

Kildetoft, T. and Mazorchuk, V., Special modules over positively based algebras, Doc. Math. 21 (2016), 1171–1192.

König, S. and Xi, C., 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.

König, S. and Xi, C., Affine cellular algebras, Adv. Math. 229 (2012), no. 1, 139–182. https://doi.org/10.1016/j.aim.2011.08.010

Kudryavtseva, G. and Mazorchuk, V., On multisemigroups, Port. Math. 72 (2015), no. 1, 47–80. https://doi.org/10.4171/PM/1956

Leinster, T., Basic category theory, Cambridge Studies in Advanced Mathematics, vol. 143, Cambridge University Press, Cambridge, 2014. https://doi.org/10.1017/CBO9781107360068

Mac Lane, S., Categories for the working mathematician, second ed., Graduate Texts in Mathematics, vol. 5, Springer-Verlag, New York, 1998.

Mackaay, M. and Mazorchuk, V., Simple transitive $2$-representations for some $2$-subcategories of Soergel bimodules, J. Pure Appl. Algebra 221 (2017), no. 3, 565–587. https://doi.org/10.1016/j.jpaa.2016.07.006

Mackaay, M., Mazorchuk, V., Miemietz, V., and Tubbenhauer, D., Simple transitive $2$-representations via (co)algebra $1$-morphisms, Indiana Univ. Math. J. (to appear), 30 pp.

Mackaay, M. and Tubbenhauer, D., Two-color Soergel calculus and simple transitive $2$-representations, Canad. J. Math. (to appear), 39 pp. https://doi.org/10.4153/CJM-2017-061-2

Mazorchuk, V. and Miemietz, V., Cell $2$-representations of finitary $2$-categories, Compos. Math. 147 (2011), no. 5, 1519–1545. https://doi.org/10.1112/S0010437X11005586

Mazorchuk, V. and Miemietz, V., Additive versus abelian $2$-representations of fiat $2$-categories, Mosc. Math. J. 14 (2014), no. 3, 595–615.

Mazorchuk, V. and Miemietz, V., Endomorphisms of cell $2$-representations, Int. Math. Res. Not. IMRN (2016), no. 24, 7471–7498. https://doi.org/10.1093/imrn/rnw025

Mazorchuk, V. and Miemietz, V., Isotypic faithful $2$-representations of $\mathcal J$-simple fiat $2$-categories, Math. Z. 282 (2016), no. 1-2, 411–434. https://doi.org/10.1007/s00209-015-1546-0

Mazorchuk, V. and Miemietz, V., Morita theory for finitary $2$-categories, Quantum Topol. 7 (2016), no. 1, 1–28. https://doi.org/10.4171/QT/72

Mazorchuk, V. and Miemietz, V., Transitive $2$-representations of finitary $2$-categories, Trans. Amer. Math. Soc. 368 (2016), no. 11, 7623–7644. https://doi.org/10.1090/tran/6583

Mazorchuk, V., Miemietz, V., and Zhang, X., Pyramids and $2$-representations, preprint arXiv:1705.03174 [math.RT].

Mazorchuk, V. and Zhang, X., Simple transitive $2$-representations for two non-fiat $2$-categories of projective functors, Ukrainian Math. J. (to appear), preprint arXiv:1601.00097 [math.RT].

Rouquier, R., $2$-Kac-Moddy algebras, preprint arXiv:0812.5023 [math.RT].

Xantcha, Q. R., Gabriel $2$-quivers for finitary $2$-categories, J. Lond. Math. Soc. (2) 92 (2015), no. 3, 615–632. https://doi.org/10.1112/jlms/jdv037

Zhang, X., Duflo involutions for $2$-categories associated to tree quivers, J. Algebra Appl. 15 (2016), no. 3, 1650041, 25 pp. https://doi.org/10.1142/S0219498816500419

Zhang, X., Simple transitive $2$-representations and Drinfeld center for some finitary $2$-categories, J. Pure Appl. Algebra 222 (2018), no. 1, 97–130. https://doi.org/10.1016/j.jpaa.2017.03.006

Zimmermann, J., Simple transitive $2$-representations of Soergel bimodules in type $B_2$, J. Pure Appl. Algebra 221 (2017), no. 3, 666–690. https://doi.org/10.1016/j.jpaa.2016.07.011
Published
2019-06-17
How to Cite
Mazorchuk, V., Miemietz, V., & Zhang, X. (2019). Characterisation and applications of $\Bbbk$-split bimodules. MATHEMATICA SCANDINAVICA, 124(2), 161-178. https://doi.org/10.7146/math.scand.a-111146
Section
Articles