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When weak Hopf algebras are Frobenius
Author(s):
Miodrag
Cristian
Iovanov;
Lars
Kadison
Journal:
Proc. Amer. Math. Soc.
MSC (2000):
Primary 18D10;
Secondary 16W30, 16S50, 16D90, 16L30
Posted:
October 22, 2009
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Abstract:
We investigate when a weak Hopf algebra  is Frobenius. We show this is not always true, but it is true if the semisimple base algebra  has all its matrix blocks of the same dimension. However, if  is a semisimple algebra not having this property, there is a weak Hopf algebra  with base  which is not Frobenius (and consequently, it is not Frobenius ``over''  either). Moreover, we give a categorical counterpart of the result that a Hopf algebra is a Frobenius algebra for a noncoassociative generalization of a weak Hopf algebra.
References:
-
- [BC]
- D. Bulacu, S. Caenepeel, Integrals for (dual) quasi-Hopf algebras. Applications, J. Algebra 266 (2003), no. 2, 552-583. MR 1995128 (2004k:16095)
- [BNS]
- G. Böhm, F. Nill, K. Szlachányi, Weak Hopf algebras. I. Integral theory and C
 -structures, J. Algebra 221 (1999), 385-438. MR 1726707 (2001a:16059) - [BoSz]
- G. Böhm, K. Szlachányi, A coassociative C
 -quantum group with nonintegral dimensions, Lett. Math. Phys. 38 (1996), no. 4, 437-456. MR 1421688 (97k:46080) - [BK]
- B. Bakalov, A. Kirillov, Jr., Lectures on tensor categories and modular functors, Univ. Lect. Ser. 21, Amer. Math. Soc., Providence, RI, 2001. MR 1797619 (2002d:18003)
- [CE]
- D. Calaque, P. Etingof, Lectures on tensor categories, in Quantum Groups (B. Enriquez, V. Turaev, eds.), IRMA Lectures in Mathematics and Theoretical Physics, 12, EMS, 2008. MR 2432988 (2009g:18010)
- [CDG]
- S. Caenepeel, E. De Groot, Galois theory for weak Hopf algebras, Rev. Roumaine Math. Pures Appl. 52 (2007), 151-176. MR 2337749 (2008g:16061)
- [D]
- V.G. Drinfel'd, Quasi-Hopf algebras, Leningrad Math. J. 1 (1990), 1419-1457. MR 1047964 (91b:17016)
- [EO]
- P. Etingof, V. Ostrik, Finite tensor categories, Mosc. Math. J. 4, No. 3 (July-Sept. 2004), 627-654. MR 2119143 (2005j:18006)
- [ENO]
- P. Etingof, D. Nikshych, V. Ostrik, On fusion categories, Ann. of Math. (2) 162 (2005), no. 2, 581-642. MR 2183279 (2006m:16051)
- [FFRS]
- J. Frölich, J. Fuchs, I. Runkel, C. Schweigert, Picard groups in rational conformal field theory, Contemp. Math., 391, Amer. Math. Soc., Providence, RI, 2005, 85-100. MR 2184014 (2007a:81131)
- [HO]
- R. Haring-Oldenburg, Reconstruction of weak quasi-Hopf algebras, J. Algebra 194 (1997), no. 1, 14-35. MR 1461480 (98h:16062)
- [K]
- L. Kadison, New Examples of Frobenius Extensions, Univ. Lect. Ser., 14, Amer. Math. Soc., Providence, RI, 1999. MR 1690111 (2001j:16024)
- [L]
- T.-Y. Lam, Lectures on Modules and Rings, Grad. Texts in Math., 189, Springer, 1999. MR 1653294 (99i:16001)
- [MS]
- G. Mack, V. Schomerus, Quasi-Hopf quantum symmetry in quantum theory, Nuclear Phys. B 370 (1992), no. 1, 185-230. MR 1148544 (93c:81081)
- [N]
- D. Nikshych, On the structure of weak Hopf algebras, Adv. Math. 170 (2002), 257-286. MR 1932332 (2003f:16063)
- [S]
- K. Szlachányi, Weak Hopf algebras, Operator algebras and quantum field theory (Rome, 1996), 621-632, Int. Press, Cambridge, MA, 1997. MR 1491146 (99a:46105)
- [V]
- P. Vecsernyés, Larson-Sweedler theorem and the role of grouplike elements in weak Hopf algebras, J. Algebra 270 (2003), 471-520. MR 2019628 (2005a:16058)
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Additional Information:
Miodrag
Cristian
Iovanov
Affiliation:
Faculty of Mathematics, University of Bucharest, Str. Academiei 14, RO-70109, Bucharest, Romania - and - State University of New York, Buffalo, 244 Mathematics Building, Buffalo, New York 14260-2900
Email:
yovanov@gmail.com, e-mail@yovanov.net
Lars
Kadison
Affiliation:
Department of Mathematics, University of Pennsylvania, David Rittenhouse Laboratory, 209 S. 33rd Street, Philadelphia, Pennsylvania 19104
Address at time of publication:
Department of Mathematics, University of California at San Diego, 9500 Gilman Drive, \#0112, La Jolla, California 92093
Email:
lkadison@math.upenn.edu
DOI:
10.1090/S0002-9939-09-10121-1
PII:
S 0002-9939(09)10121-1
Keywords:
Weak Hopf algebra,
quasi-Hopf algebra,
Frobenius algebra,
quasi-Frobenius algebra,
tensor category,
Tannakian reconstruction
Received by editor(s):
November 20, 2008,
Received by editor(s) in revised form:
July 15, 2009
Posted:
October 22, 2009
Additional Notes:
The first author was partially supported by contract no. 24/28.09.07 with UEFISCU ``Groups, quantum groups, corings and representation theory'' of CNCIS, PN II (ID\_1002).
Communicated by:
Martin Lorenz
Copyright of article:
Copyright
2009,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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