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On the structure of weak Hopf algebras. (English) Zbl 1010.16041
Weak Hopf algebras were introduced by {\it G. Böhm} and {\it 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 {\it 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 {\it R. G. Larson} and {\it 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 {\it F. Li} [J. Algebra 208, No. 1, 72-100 (1998; Zbl 0916.16020)].

16W30Hopf algebras (associative rings and algebras) (MSC2000)
Full Text: DOI arXiv
[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
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[9] Larson, R.; Sweedler, M.: An associative orthogonal bilinear form for Hopf algebras. Amer. J. Math. 91, 75-93 (1969) · Zbl 0179.05803
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[12] Nikshych, D.; Turaev, V.; Vainerman, L.: Quantum groupoids and invariants of knots and 3-manifolds. J. topology appl. (2000) · Zbl 1021.16026
[13] Nikshych, D.; Vainerman, L.: Algebraic versions of a finite dimensional quantum groupoid. 209 (2000) · 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) · 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. Operator algebras and quantum field theory (1996) · Zbl 0872.16022
[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
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[23] Zhu, Y.: Hopf algebras of prime dimension. Internat. math. Res. notices 1, 53-59 (1994) · Zbl 0822.16036