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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 $G\left(H\right)$. If ${H}_{min}$ is a minimal weak Hopf subalgebra of $H$ (it is unique), then $G\left({H}_{min}\right)$ (denoted ${G}_{0}\left(H\right)$) is a normal subgroup of $G\left(H\right)$, and the quotient group ${G}^{\sim }\left(H\right)$ plays an important role in studying $H$. ${G}^{\sim }\left(H\right)$ may be infinite, even if $H$ is finite-dimensional. When $H$ is finite-dimensional, $G\left(H\right)$ and $G\left({H}^{*}\right)$ 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 $\text{Trace}\left({S}^{2}\right)$ [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 (assoc. rings and algebras) (MSC2000)