Article
Full entry |
PDF
(1.3 MB)
Feedback
PDF
(1.3 MB)
Feedback
Keywords:
Drinfeld center; monoidal adjunction; projection formula; braided (op)lax monoidal functor; monoidal Kleisli adjunction; monoidal Eilenberg–Moore adjunction; Yetter–Drinfeld module
Drinfeld center; monoidal adjunction; projection formula; braided (op)lax monoidal functor; monoidal Kleisli adjunction; monoidal Eilenberg–Moore adjunction; Yetter–Drinfeld module
Summary:
Given a monoidal adjunction, we show that the right adjoint induces a braided lax monoidal functor between the corresponding Drinfeld centers provided that certain natural transformations, called projection formula morphisms, are invertible. We investigate these induced functors on Drinfeld centers in more detail for the monoidal adjunction of restriction and (co-)induction along morphisms of Hopf algebras. The resulting functors are applied to examples related to affine algebraic groups, quantum groups at roots of unity, and Radford–Majid biproducts of Hopf algebras. Moreover, we use the projection formula morphisms to prove a characterization theorem for monoidal Kleisli adjunctions and a crude monoidal monadicity theorem. The functor on Drinfeld centers induced by the Eilenberg–Moore adjunction is given in terms of local modules over commutative central monoids.
References:
[1] Aguiar, Marcelo, Mahajan, Swapneel: Monoidal functors, species and Hopf algebras. CRM monograph series, American Mathematical Society, Providence, https://doi.org/10.1090/crmm/029, ISBN:978-0-8218-4776-3, DOI:10.1090/crmm/029 DOI 10.1090/crmm/029
[2] Andruskiewitsch, Nicolás, Schneider, Hans-Jürgen: Pointed Hopf algebras. New directions in Hopf algebras, pages 1-68, Math. Sci. Res. Inst. publ. 43
[3] Balan, Adriana: On Hopf adjunctions, Hopf monads and Frobenius-type properties. Appl. Categ. Structures, Vol. 25, Iss. 5, 747-774, https://doi.org/10.1007/s10485-016-9428-0, DOI:10.1007/s10485-016-9428-0 DOI 10.1007/s10485-016-9428-0
[4] Balmer, Paul, Dell'Ambrogio, Ivo, Sanders, Beren: Grothendieck-Neeman duality and the Wirthmüller isomorphism. Compos. Math., Vol. 152, Iss. 8, 1740-1776, https://doi.org/10.1112/S0010437X16007375, DOI:10.1112/S0010437X16007375 DOI 10.1112/S0010437X16007375
[5] Barr, Michael, Wells, Charles: Toposes, triples and theories. Repr. Theory Appl. Categ., Iss. 12, x+288
[6] Ben-Zvi, David, Francis, John, Nadler, David: Integral transforms and Drinfeld centers in derived algebraic geometry. J. Amer. Math. Soc., Vol. 23, Iss. 4, 909-966, https://doi.org/10.1090/S0894-0347-10-00669-7, DOI:10.1090/S0894-0347-10-00669-7 DOI 10.1090/S0894-0347-10-00669-7
[7] Borceux, Francis: Handbook of categorical algebra. 2. Encyclopedia of mathematics and its applications, Cambridge University Press, Cambridge, ISBN:0-521-44179-X
[8] Brown, Ken A., Goodearl, Ken R.: Lectures on algebraic quantum groups. Advanced courses in mathematics. CRM barcelona, Birkhäuser Verlag, Basel, https://doi.org/10.1007/978-3-0348-8205-7, ISBN:3-7643-6714-8, DOI:10.1007/978-3-0348-8205-7 DOI 10.1007/978-3-0348-8205-7
[9] Bruguières, Alain, Lack, Steve, Virelizier, Alexis: Hopf monads on monoidal categories. Adv. Math., Vol. 227, Iss. 2, 745-800, https://doi.org/10.1016/j.aim.2011.02.008, DOI:10.1016/j.aim.2011.02.008 DOI 10.1016/j.aim.2011.02.008
[10] Bruguières, Alain, Natale, Sonia: Exact sequences of tensor categories. Int. Math. Res. Not. IMRN, Iss. 24, 5644-5705, https://doi.org/10.1093/imrn/rnq294, DOI:10.1093/imrn/rnq294 DOI 10.1093/imrn/rnq294
[11] Bruguières, Alain, Virelizier, Alexis: Hopf monads. Adv. Math., Vol. 215, Iss. 2, 679-733, https://doi.org/10.1016/j.aim.2007.04.011, DOI:10.1016/j.aim.2007.04.011 DOI 10.1016/j.aim.2007.04.011
[12] Capucci, Matteo, Gavranović, Bruno: Actegories for the working amthematician. arXiv:2203.16351
[13] Chari, Vyjayanthi, Pressley, Andrew: A guide to quantum groups. Cambridge University Press, Cambridge
[14] Chriss, Neil, Ginzburg, Victor: Representation theory and complex geometry. Modern Birkhäuser classics, Birkhäuser, https://doi.org/10.1007/978-0-8176-4938-8, ISBN:978-0-8176-4937-1, DOI:10.1007/978-0-8176-4938-8 DOI 10.1007/978-0-8176-4938-8
[15] Cohn, P. M.: Algebra. Vol. 2. John Wiley and Sons, Chichester
[16] Davydov, Alexei: Modular invariants for group-theoretical modular data. I. J. Algebra, Vol. 323, Iss. 5, 1321-1348 DOI 10.1016/j.jalgebra.2009.11.041
[17] Davydov, Alexei, Müger, Michael, Nikshych, Dmitri, Ostrik, Victor: The Witt group of non-degenerate braided fusion categories. J. Reine Angew. Math., Vol. 677, 135-177
[18] Davydov, Alexei, Simmons, Darren: On Lagrangian algebras in group-theoretical braided fusion categories. J. Algebra, Vol. 471, 149-175, https://doi.org/10.1016/j.jalgebra.2016.09.016, DOI:10.1016/j.jalgebra.2016.09.016 DOI 10.1016/j.jalgebra.2016.09.016
[19] Day, Brian: On closed categories of functors. II. Category Seminar (Proc. Sem., Sydney, 1972/1973), pages 20-54, Lecture notes in math. Vol. 420
[20] De Concini, Corrado, Kac, Victor G.: Representations of quantum groups at roots of 1. Operator algebras, unitary representations, enveloping algebras, and invariant theory (Paris, 1989), pages 471-506, Progr. math. 92
[21] Deligne, P.: La catégorie des représentations du groupe symétrique $S_t$, lorsque $t$ n'est pas un entier naturel. Algebraic groups and homogeneous spaces, pages 209-273, Tata inst. Fund. Res. Stud. math. 19
[22] Etingof, Pavel, Gelaki, Shlomo, Nikshych, Dmitri, Ostrik, Victor: Tensor categories. Mathematical surveys and monographs, American Mathematical Society, Providence, https://doi.org/10.1090/surv/205, ISBN:978-1-4704-2024-6, DOI:10.1090/surv/205 DOI 10.1090/surv/205
[23] Fausk, H., Hu, P., May, J. P.: Isomorphisms between left and right adjoints. Theory Appl. Categ., Vol. 11, No. 4, 107-131
[24] Flake, Johannes, Harman, Nate, Laugwitz, Robert: The indecomposable objects in the center of Deligne's category $\underline{\rm Re}{\rm p}S_t$. Proc. Lond. Math. Soc. (3), Vol. 126, Iss. 4, 1134-1181, https://doi.org/10.1112/plms.12509, DOI:10.1112/plms.12509 DOI 10.1112/plms.12509
[25] Fröhlich, Jürg, Fuchs, Jürgen, Runkel, Ingo, Schweigert, Christoph: Correspondences of ribbon categories. Adv. Math., Vol. 199, Iss. 1, 192-329 DOI 10.1016/j.aim.2005.04.007
[26] Gelaki, Shlomo, Naidu, Deepak, Nikshych, Dmitri: Centers of graded fusion categories. Algebra Number Theory, Vol. 3, Iss. 8, 959-990, https://doi.org/10.2140/ant.2009.3.959, DOI:10.2140/ant.2009.3.959 DOI 10.2140/ant.2009.3.959
[27] Greenough, Justin: Relative centers and tensor products of tensor and braided fusion categories. J. Algebra, Vol. 388, 374-396, https://doi.org/10.1016/j.jalgebra.2013.04.014, DOI:10.1016/j.jalgebra.2013.04.014 DOI 10.1016/j.jalgebra.2013.04.014
[28] Hannah, Samuel, Laugwitz, Robert, Ros Camacho, Ana: Frobenius monoidal functors of Dijkgraaf-Witten categories and rigid Frobenius algebras. SIGMA Symmetry Integrability Geom. Methods Appl., Vol. 19, Paper No. 075, https://doi.org/10.3842/SIGMA.2023.075, DOI:10.3842/SIGMA.2023.075 DOI 10.3842/SIGMA.2023.075
[29] Hartshorne, Robin: Algebraic geometry. Graduate texts in mathematics, no. 52, Springer-Verlag, New York, ISBN:0-387-90244-9
[30] Huang, Yi-Zhi, Kirillov, Alexander, Lepowsky, James: Braided tensor categories and extensions of vertex operator algebras. Comm. Math. Phys., Vol. 337, Iss. 3, 1143-1159, https://doi.org/10.1007/s00220-015-2292-1, DOI:10.1007/s00220-015-2292-1 DOI 10.1007/s00220-015-2292-1
[31] Johnson, Niles, Yau, Donald: 2-dimensional categories. Oxford University Press, Oxford, DOI:10.1093/oso/9780198871378.001.0001 DOI 10.1093/oso/9780198871378.001.0001
[32] Johnstone, Peter T.: Sketches of an elephant: A topos theory compendium. Vol. 1. Oxford logic guides, Clarendon Press, Oxford University Press, New York, ISBN:0-19-853425-6
[33] Joyal, André, Street, Ross: Tortile Yang-Baxter operators in tensor categories. J. Pure Appl. Algebra, Vol. 71, Iss. 1, 43-51, https://doi.org/10.1016/0022-4049(91)90039-5, DOI:10.1016/0022-4049(91)90039-5 DOI 10.1016/0022-4049(91)90039-5
[34] Kassel, Christian: Quantum groups. Graduate texts in mathematics, Springer, New York, https://doi.org/10.1007/978-1-4612-0783-2, ISBN:0-387-94370-6, DOI:10.1007/978-1-4612-0783-2 DOI 10.1007/978-1-4612-0783-2
[35] Kelly, G. M.: Doctrinal adjunction. Category Seminar (Proc. Sem., Sydney, 1972/1973), pp 257-280
[36] Kelly, G. M., Street, Ross: Review of the elements of 2-categories. Category Seminar (Proc. Sem., Sydney, 1972/1973), pages 75-103, Lecture notes in math. Vol. 420 DOI 10.1007/BFb0063101
[37] Kirillov, Alexander, Ostrik, Viktor: On a $q$-analogue of the McKay correspondence and the ADE classification of ${sl}_2$ conformal field theories. Adv. Math., Vol. 171, Iss. 2, 183-227 DOI 10.1006/aima.2002.2072
[38] Kong, Liang, Yuan, Wei, Zheng, Hao: Pointed Drinfeld center functor. Comm. Math. Phys., Vol. 381, Iss. 3, 1409-1443, https://doi.org/10.1007/s00220-020-03922-x, DOI:10.1007/s00220-020-03922-x DOI 10.1007/s00220-020-03922-x
[39] Kong, Liang, Zheng, Hao: The center functor is fully faithful. Adv. Math., Vol. 339, 749-779, https://doi.org/10.1016/j.aim.2018.09.031, DOI:10.1016/j.aim.2018.09.031 DOI 10.1016/j.aim.2018.09.031
[40] Laugwitz, Robert, Walton, Chelsea: Constructing non-semisimple modular categories with local modules. Comm. Math. Phys., Vol. 403, Iss. 3, 1363-1409, https://doi.org/10.1007/s00220-023-04824-4, DOI:10.1007/s00220-023-04824-4 DOI 10.1007/s00220-023-04824-4
[41] Lawvere, F. William: Equality in hyperdoctrines and comprehension schema as an adjoint functor. Applications of Categorical Algebra (Proc. Sympos. Pure Math., Vol. XVII, New York, 1968), pp 1-14
[42] Lyubashenko, V.: Modular transformations for tensor categories. J. Pure Appl. Algebra, Vol. 98, Iss. 3, 279-327, https://doi.org/10.1016/0022-4049(94)00045-K, DOI:10.1016/0022-4049(94)00045-K DOI 10.1016/0022-4049(94)00045-K
[43] Mac Lane, Saunders: Categories for the working mathematician. Graduate texts in mathematics, Springer-Verlag, New York, ISBN:0-387-98403-8
[44] Majid, Shahn: Representations, duals and quantum doubles of monoidal categories. Proceedings of the Winter School on Geometry and Physics (Srnı́, 1990), pp 197-206
[45] Majid, Shahn: Cross products by braided groups and bosonization. J. Algebra, Vol. 163, Iss. 1, 165-190, https://doi.org/10.1006/jabr.1994.1011, DOI:10.1006/jabr.1994.1011 DOI 10.1006/jabr.1994.1011
[46] Majid, Shahn: Foundations of quantum group theory. Cambridge University Press, Cambridge, DOI:10.1017/CBO9780511613104 DOI 10.1017/CBO9780511613104
[47] Moerdijk, I.: Monads on tensor categories. J. Pure Appl. Algebra, pages 189-208, 168
[48] Montgomery, Susan: Hopf algebras and their actions on rings. CBMS regional conference series in mathematics, Published for the Conference Board of the Mathematical Sciences, Washington, American Mathematical Society, Providence, https://doi.org/10.1090/cbms/082, ISBN:0-8218-0738-2, DOI:10.1090/cbms/082 DOI 10.1090/cbms/082
[49] nLab authors: Frobenius reciprocity. Retrieved from https://ncatlab.org/nlab/revision/Frobenius+reciprocity/56, December 19, 2023
[50] Pareigis, Bodo: On braiding and dyslexia. J. Algebra, Vol. 171, Iss. 2, 413-425 DOI 10.1006/jabr.1995.1019
[51] Radford, David E.: The structure of Hopf algebras with a projection. J. Algebra, Vol. 92, Iss. 2, 322-347, https://doi.org/10.1016/0021-8693(85)90124-3, DOI:10.1016/0021-8693(85)90124-3 DOI 10.1016/0021-8693(85)90124-3
[52] Radford, David E.: Hopf algebras. Series on knots and everything, World Scientific, Hackensack, ISBN:978-981-4335-99-7; 981-4335-99-1
[53] Riehl, Emily: Category theory in context. Dover Publications
[54] Safronov, Pavel: A categorical approach to quantum moment maps. Theory Appl. Categ., Vol. 37, Paper No. 24, 818-862
[55] Sanders, Beren: A characterization of finite étale morphisms in tensor triangular geometry. Épijournal Géom. Algébrique, Vol. 6, Art. 18, 25
[56] Schanuel, Stephen, Street, Ross: The free adjunction. Cahiers Topologie Géom. Différentielle Catég., Vol. 27, Iss. 1, 81-83
[57] Schauenburg, Peter: The monoidal center construction and bimodules. J. Pure Appl. Algebra, Vol. 158, Iss. 2-3, 325-346, https://doi.org/10.1016/S0022-4049(00)00040-2, DOI:10.1016/S0022-4049(00)00040-2 DOI 10.1016/S0022-4049(00)00040-2
[58] Seal, Gavin J.: Tensors, monads and actions. Theory Appl. Categ., Vol. 28, No. 15, 403-433
[59] Serre, Jean-Pierre: Linear representations of finite groups. Springer-Verlag, New York
[60] Shimizu, Kenichi: Further results on the structure of (co)ends in finite tensor categories. Appl. Categ. Structures, Vol. 28, Iss. 2, 237-286, https://doi.org/10.1007/s10485-019-09577-7, DOI:10.1007/s10485-019-09577-7 DOI 10.1007/s10485-019-09577-7
[61] Szlachányi, Kornél: The monoidal Eilenberg-Moore construction and bialgebroids. J. Pure Appl. Algebra, Vol. 182, Iss. 2-3, 287-315, https://doi.org/10.1016/S0022-4049(03)00018-5, DOI:10.1016/S0022-4049(03)00018-5 DOI 10.1016/S0022-4049(03)00018-5
[62] Taft, Earl J.: The order of the antipode of finite-dimensional Hopf algebra. Proc. Nat. Acad. Sci. U.S.A., Vol. 68, 2631-2633, https://doi.org/10.1073/pnas.68.11.2631, DOI:10.1073/pnas.68.11.2631 DOI 10.1073/pnas.68.11.2631
[63] Takeuchi, Mitsuhiro: Finite Hopf algebras in braided tensor categories. J. Pure Appl. Algebra, Vol. 138, Iss. 1, 59-82, https://doi.org/10.1016/S0022-4049(97)00207-7, DOI:10.1016/S0022-4049(97)00207-7 DOI 10.1016/S0022-4049(97)00207-7
[64] Turaev, Vladimir, Virelizier, Alexis: Monoidal categories and topological field theory. Progress in mathematics, Birkhäuser/Springer, Cham, https://doi.org/10.1007/978-3-319-49834-8, ISBN:978-3-319-49833-1; 978-3-319-49834-8, DOI:10.1007/978-3-319-49834-8 DOI 10.1007/978-3-319-49834-8
[65] Willerton, Simon: The twisted Drinfeld double of a finite group via gerbes and finite groupoids. Algebr. Geom. Topol., Vol. 8, Iss. 3, 1419-1457, https://doi.org/10.2140/agt.2008.8.1419, DOI:10.2140/agt.2008.8.1419 DOI 10.2140/agt.2008.8.1419
[66] Yadav, Harshit: Frobenius monoidal functors from (co)Hopf adjunctions. Proc. Amer. Math. Soc., Vol. 152, Iss. 2, 471-487, https://doi.org/10.1090/proc/16494, DOI:10.1090/proc/16494 DOI 10.1090/proc/16494
[67] Yetter, David N.: Quantum groups and representations of monoidal categories. Math. Proc. Cambridge Philos. Soc., Vol. 108, Iss. 2, 261-290, https://doi.org/10.1017/S0305004100069139, DOI:10.1017/S0305004100069139 DOI 10.1017/S0305004100069139
[68] Zawadowski, Marek: The formal theory of monoidal monads. J. Pure Appl. Algebra, Vol. 216, Iss. 8-9, 1932-1942, https://doi.org/10.1016/j.jpaa.2012.02.030, DOI:10.1016/j.jpaa.2012.02.030 DOI 10.1016/j.jpaa.2012.02.030
Search
Partner of
