Algebra & Number Theory
- Algebra Number Theory
- Volume 3, Number 2 (2009), 149-207.
Weak Hopf monoids in braided monoidal categories
Full-text: Open access
Abstract
We develop the theory of weak bimonoids in braided monoidal categories and show that they are in one-to-one correspondence with quantum categories with a separable Frobenius object-of-objects. Weak Hopf monoids are shown to be quantum groupoids. Each separable Frobenius monoid leads to a weak Hopf monoid .
Article information
Source
Algebra Number Theory, Volume 3, Number 2 (2009), 149-207.
Dates
Received: 26 January 2008
Revised: 14 November 2008
Accepted: 23 December 2008
First available in Project Euclid: 20 December 2017
Permanent link to this document
https://projecteuclid.org/euclid.ant/1513797355
Digital Object Identifier
doi:10.2140/ant.2009.3.149
Mathematical Reviews number (MathSciNet)
MR2491942
Zentralblatt MATH identifier
1185.16035
Subjects
Primary: 16W30
Secondary: 18B40: Groupoids, semigroupoids, semigroups, groups (viewed as categories) [See also 20Axx, 20L05, 20Mxx] 18D10: Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories [See also 19D23]
Keywords
weak bimonoids (weak bialgebra) weak Hopf monoids (weak Hopf algebra) quantum category quantum groupoid monoidal category
Citation
Pastro, Craig; Street, Ross. Weak Hopf monoids in braided monoidal categories. Algebra Number Theory 3 (2009), no. 2, 149--207. doi:10.2140/ant.2009.3.149. https://projecteuclid.org/euclid.ant/1513797355
References
- J. N. Alonso Álvarez, R. González Rodríguez, J. M. Fernández Vilaboa, M. P. López López, and E. Villanueva Novoa, “Weak Hopf algebras with projection and weak smash bialgebra structures”, J. Algebra 269:2 (2003), 701–725. Mathematical Reviews (MathSciNet): MR2005b:16062
Zentralblatt MATH: 1042.16020
Digital Object Identifier: doi:10.1016/j.jalgebra.2003.05.003 - J. N. Alonso Álvarez, J. M. Fernández Vilaboa, and R. González Rodríguez, “Weak braided Hopf algebras”, preprint, 2008. To appear in Indiana University Math. J. Mathematical Reviews (MathSciNet): MR2463974
Zentralblatt MATH: 1165.16021
Digital Object Identifier: doi:10.1512/iumj.2008.57.3294 - J. N. Alonso Álvarez, J. M. Fernández Vilaboa, and R. González Rodríguez, “Weak Hopf algebras and weak Yang–Baxter operators”, J. Algebra 320:5 (2008), 2101–2143. Mathematical Reviews (MathSciNet): MR2437645
Zentralblatt MATH: 1142.18006
Digital Object Identifier: doi:10.1016/j.jalgebra.2008.02.026 - I. Bálint, Quantum groupoid symmetry in low dimensional quantum field theory, Ph.D. thesis, Eötvös Loránd University, Budapest, 2008.
- I. Bálint, “Scalar extension of bicoalgebroids”, Appl. Categ. Structures 16:1-2 (2008), 29–55. Mathematical Reviews (MathSciNet): MR2383276
Zentralblatt MATH: 05265525
Digital Object Identifier: doi:10.1007/s10485-007-9098-z - M. Barr, “Nonsymmetric ${}\sp \ast$-autonomous categories”, Theoret. Comput. Sci. 139:1-2 (1995), 115–130. Mathematical Reviews (MathSciNet): MR96e:03077
Zentralblatt MATH: 0874.18004
Digital Object Identifier: doi:10.1016/0304-3975(94)00089-2 - G. Bòhm and K. Szlachónyi, “A coassociative $C\sp *$-quantum group with nonintegral dimensions”, Lett. Math. Phys. 38:4 (1996), 437–456. Mathematical Reviews (MathSciNet): MR97k:46080
Zentralblatt MATH: 0872.16022
Digital Object Identifier: doi:10.1007/BF01815526 - G. B öhm and K. Szlachányi, “Weak Hopf algebras, II: Representation theory, dimensions, and the Markov trace”, J. Algebra 233:1 (2000), 156–212. Mathematical Reviews (MathSciNet): MR2001m:16059
Zentralblatt MATH: 0980.16028
Digital Object Identifier: doi:10.1006/jabr.2000.8379 - G. B öhm and K. Szlachányi, “Hopf algebroids with bijective antipodes: axioms, integrals, and duals”, J. Algebra 274:2 (2004), 708–750. Mathematical Reviews (MathSciNet): MR2004m:16047
Zentralblatt MATH: 1080.16035
Digital Object Identifier: doi:10.1016/j.jalgebra.2003.09.005 - G. B öhm, F. Nill, and K. Szlachányi, “Weak Hopf algebras, I: Integral theory and $C\sp *$-structure”, J. Algebra 221:2 (1999), 385–438. Mathematical Reviews (MathSciNet): MR2001a:16059
Zentralblatt MATH: 0949.16037
Digital Object Identifier: doi:10.1006/jabr.1999.7984 - T. Brzeziński and G. Militaru, “Bialgebroids, $\times\sb A$-bialgebras and duality”, J. Algebra 251:1 (2002), 279–294. Mathematical Reviews (MathSciNet): MR2003c:16044
Zentralblatt MATH: 1003.16033
Digital Object Identifier: doi:10.1006/jabr.2001.9101 - B. Day and R. Street, “Quantum categories, star autonomy, and quantum groupoids”, pp. 187–225 in Galois theory, Hopf algebras, and semiabelian categories, edited by G. Janelidze et al., Fields Institute Communications 43, Amer. Math. Soc., Providence, RI, 2004.
- B. Day, P. McCrudden, and R. Street, “Dualizations and antipodes”, Appl. Categ. Structures 11:3 (2003), 229–260. Mathematical Reviews (MathSciNet): MR2004b:18013
Zentralblatt MATH: 1137.18302
Digital Object Identifier: doi:10.1023/A:1024236601870 - T. Hayashi, “Face algebras, I: A generalization of quantum group theory”, J. Math. Soc. Japan 50:2 (1998), 293–315. Mathematical Reviews (MathSciNet): MR99d:16043
Zentralblatt MATH: 0908.16032
Digital Object Identifier: doi:10.2969/jmsj/05020293
Project Euclid: euclid.jmsj/1225376609 - A. Joyal and R. Street, “The geometry of tensor calculus, I”, Adv. Math. 88:1 (1991), 55–112. Mathematical Reviews (MathSciNet): MR92d:18011
Zentralblatt MATH: 0738.18005
Digital Object Identifier: doi:10.1016/0001-8708(91)90003-P - A. Joyal and R. Street, “Braided tensor categories”, Adv. Math. 102:1 (1993), 20–78. Mathematical Reviews (MathSciNet): MR94m:18008
Zentralblatt MATH: 0817.18007
Digital Object Identifier: doi:10.1006/aima.1993.1055 - J.-H. Lu, “Hopf algebroids and quantum groupoids”, Internat. J. Math. 7:1 (1996), 47–70. Mathematical Reviews (MathSciNet): MR97a:16073
Zentralblatt MATH: 0884.17010
Digital Object Identifier: doi:10.1142/S0129167X96000050 - D. Nikshych and L. Vainerman, “Finite quantum groupoids and their applications”, pp. 211–262 in New directions in Hopf algebras, edited by S. Montgomery and H.-J. Schneider, Math. Sci. Res. Inst. Publ. 43, Cambridge Univ. Press, 2002.
- F. Nill, “Axioms for weak bialgebras”, preprint, 1998. arXiv: math/9805104
- P. Schauenburg, “Face algebras are $\times\sb R$-bialgebras”, pp. 275–285 in Rings, Hopf algebras, and Brauer groups (Antwerp/Brussels, 1996), edited by S. Caenepeel and A. Verschoren, Lecture Notes in Pure and Appl. Math. 197, Dekker, New York, 1998.
- P. Schauenburg, “Duals and doubles of quantum groupoids ($\times\sb R$-Hopf algebras)”, pp. 273–299 in New trends in Hopf algebra theory (La Falda, 1999), edited by N. Andruskiewitsch et al., Contemporary Mathematics 267, Amer. Math. Soc., Providence, RI, 2000.
- P. Schauenburg, “Weak Hopf algebras and quantum groupoids”, pp. 171–188 in Noncommutative geometry and quantum groups (Warsaw, 2001), edited by P. M. Hajac and W. Pusz, Banach Center Publ. 61, Polish Acad. Sci., Warsaw, 2003.
- M. E. Sweedler, “Groups of simple algebras”, Inst. Hautes Études Sci. Publ. Math. 44 (1974), 79–189. Mathematical Reviews (MathSciNet): MR51:587
Zentralblatt MATH: 0314.16008
Digital Object Identifier: doi:10.1007/BF02685882 - K. Szlachányi, “Weak Hopf algebras”, pp. 621–632 in Operator algebras and quantum field theory (Rome, 1996), edited by S. Doplicher et al., Int. Press, Cambridge, MA, 1997.
- M. Takeuchi, “Groups of algebras over $A\otimes \overline A$”, J. Math. Soc. Japan 29:3 (1977), 459–492. Mathematical Reviews (MathSciNet): MR58:22151
Zentralblatt MATH: 0349.16012
Digital Object Identifier: doi:10.2969/jmsj/02930459
Project Euclid: euclid.jmsj/1240432948 - P. Xu, “Quantum groupoids”, Communications in Mathematical Physics 216:3 (2001), 539–581. Mathematical Reviews (MathSciNet): MR2002f:17033
Zentralblatt MATH: 0986.17003
Digital Object Identifier: doi:10.1007/s002200000334 - T. Yamanouchi, “Duality for generalized Kac algebras and a characterization of finite groupoid algebras”, J. Algebra 163:1 (1994), 9–50. Mathematical Reviews (MathSciNet): MR95c:22010
Zentralblatt MATH: 0830.46047
Digital Object Identifier: doi:10.1006/jabr.1994.1002
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