# zbMATH — the first resource for mathematics

On the structure of weak Hopf algebras. (English) Zbl 1010.16041
Weak Hopf algebras were introduced by G. Böhm and K. Szlachányi [Lett. Math. Phys. 38, No. 4, 437-456 (1996; Zbl 0872.16022)]. They generalize ordinary Hopf algebras and groupoid algebras. A weak Hopf algebra $$H$$ has an algebra and a coalgebra structure and the comultiplication preserves products. But the Hopf algebra conditions that the comultiplication preserves the unit and the counit preserves the multiplication are replaced by weaker versions. Also the conditions for an antipode $$S$$ are replaced by weaker versions. It turns out that $$S$$ is an antimorphism of both the algebra and the coalgebra structures. If $$H$$ is finite-dimensional, then $$H^*$$ also has the structure of a weak Hopf algebra. The Hopf algebra definition of group-like elements is weakened. They are invertible and form a group $$G(H)$$. If $$H_{\min}$$ is a minimal weak Hopf subalgebra of $$H$$ (it is unique), then $$G(H_{\min})$$ (denoted $$G_0(H)$$) is a normal subgroup of $$G(H)$$, and the quotient group $$G^\sim(H)$$ plays an important role in studying $$H$$. $$G^\sim(H)$$ may be infinite, even if $$H$$ is finite-dimensional. When $$H$$ is finite-dimensional, $$G(H)$$ and $$G(H^*)$$ are used to get an extension of D. E. Radford’s formula for $$S^4$$ [Am. J. Math. 98, 333-355 (1976; Zbl 0332.16007)]. $$S$$ may have infinite order, but the order is finite modulo a trivial automorphism of $$H$$ (too technical to describe here). Also, the R. G. Larson and D. E. Radford formula for $$\text{Trace}(S^2)$$ [J. Algebra 117, No. 2, 267-289 (1988; Zbl 0649.16005)] is extended and used to give a sufficient condition for $$H$$ to be semisimple. Relations between semisimplicity and cosemisimplicity are discussed. These results are applied to show that a dynamical twisting deformation of a semisimple Hopf algebra is cosemisimple.
Reviewer’s note: The term “weak Hopf algebra” has been used in a different way as a bialgebra with a weak antipode, for example by F. Li [J. Algebra 208, No. 1, 72-100 (1998; Zbl 0916.16020)].

##### MSC:
 16W30 Hopf algebras (associative rings and algebras) (MSC2000)
Full Text:
##### References:
 [1] Böhm, G.; Nill, F.; Szlachányi, K., Weak Hopf algebras I. integral theory and C*-structure, J. algebra, 221, 385-438, (1999) · Zbl 0949.16037 [2] Böhm, G.; Szlachányi, K., A coassociative C*-quantum group with nonintegral dimensions, Lett. in math. phys., 35, 437-456, (1996) · Zbl 0872.16022 [3] Böhm, G.; Szlachányi, K., Weak Hopf algebras II. representation theory, dimensions and the Markov trace, J. algebra, 233, 156-212, (2000) · Zbl 0980.16028 [4] Etingof, P.; Nikshych, D., Dynamical quantum groups at roots of 1, Duke math. J., 108, 135-168, (2001) · Zbl 1023.17007 [5] P. Etingof, and, O. Schiffmann, Lectures on the dynamical Yang-Baxter equations, preprint, 1999, math.QA/9908064. · Zbl 1036.17013 [6] G. Felder, Elliptic quantum groups, inXIth International Congress of Mathematical Physics1994 pp. 211-218, International Press, Cambridge, 1995. [7] Kadison, L.; Nikshych, D., Frobenius extensions and weak Hopf algebras, J. algebra, 244, 312-342, (2001) · Zbl 1018.16020 [8] Larson, R.; Radford, D., Finite dimensional cosemisimple Hopf algebras in characteristic 0 are semisimple, J. algebra, 117, 267-289, (1988) · Zbl 0649.16005 [9] Larson, R.; Sweedler, M., An associative orthogonal bilinear form for Hopf algebras, Amer. J. math., 91, 75-93, (1969) · Zbl 0179.05803 [10] Montgomery, S., Hopf algebras and their actions on rings, (1993), AMS Providence · Zbl 0804.16041 [11] D. Nikshych, Quantum Groupoids, their Representation Categories, Symmetries of von Neumann Factors, and Dynamical Quantum Groups, Ph.D. Thesis, U.C.L.A., 2001. [12] Nikshych, D.; Turaev, V.; Vainerman, L., Quantum groupoids and invariants of knots and 3-manifolds, J. topology appl., (2000) [13] Nikshych, D.; Vainerman, L., Algebraic versions of a finite dimensional quantum groupoid, 209, (2000), Marcel Dekker New York, p. 189-221 · Zbl 1032.46537 [14] Nikshych, D.; Vainerman, L., A characterization of depth 2 subfactors of II1 factors, J. funct. anal., 171, 278-307, (2000) · Zbl 1010.46063 [15] Nikshych, D.; Vainerman, L., A Galois correspondence for II1 factors and quantum groupoids, J. funct. anal., 178, 113-142, (2000) · Zbl 0995.46041 [16] Nikshych, D.; Vainerman, L., Finite quantum groupoids and their applications, New directions in Hopf algebras, 43, (2002), MSRI PublicationsCambridge University Press Cambridge, p. 211-262 · Zbl 1026.17017 [17] V. Ostrik, Module categories, weak Hopf algebras, and modular invariants, preprint, 2001, math.QA/0111139. [18] Szlachányi, K., Weak Hopf algebras, () · Zbl 1098.16504 [19] Radford, D., The order of the antipode of a finite dimensional Hopf algebra is finite, Amer. J. math., 98, 333-355, (1976) · Zbl 0332.16007 [20] Stefan, D., The set of types of n-dimensional semisimple and cosemisimple Hopf algebras is finite, J. algebra, 193, 571-580, (1997) · Zbl 0882.16029 [21] Sweedler, M.E., Hopf algebras, (1969), W. A. Benjamin, Inc New York [22] P. Vecsernyes, Larson-Sweedler theorem, grouplike elements, and invertible modules in weak Hopf algebras, preprint, 2001, math.QA/0111045. [23] Zhu, Y., Hopf algebras of prime dimension, Internat. math. res. notices, 1, 53-59, (1994) · Zbl 0822.16036
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.