Representation theory of semisimple Hopf algebras (Q2759665)

The paper under review contains a nice up-to-date exposition of the main aspects connected with the representation theory of semisimple Hopf algebras over an algebraically closed field of characteristic zero. The author starts by revisiting the main features of a finite-dimensional Hopf algebra \(H\), with emphasis on the results of Larson and Radford connecting the semisimplicity of \(H\) and \(H^*\) with the square of the antipode of \(H\); a short proof of the fact that the semisimplicity of \(H\) and \(H^*\) are equivalent is given. Then she studies the character algebra of the semisimple Hopf algebra \(H\), the orthogonality relations for irreducible characters and the class equation of G. I. Kac and Y. Zhu and its applications, including the classification of Hopf algebras of prime dimension and the normality of Hopf subalgebras whose index is the smallest prime number dividing the dimension of \(H\). As another application of the class equation, the author presents Schneider's proof of the result of Etingof and Gelaki stating that the dimension of an irreducible module over the Drinfeld double of \(H\) divides the dimension of \(H\). Finally, the paper discusses Schur indicators, following the paper by the author and \textit{V. Linchenko} [in Algebr. Represent. Theory 3, No. 4, 347-355 (2000; Zbl 0971.16018)]. A related reference, by the same author, is the survey on the classification of semisimple Hopf algebras [in Contemp. Math. 229, 265-279 (1998; Zbl 0921.16025)]. Further related references are the lecture notes by \textit{H.-J. Schneider} [Lectures on Hopf algebras, Trabajos de Matemática 31 (1995; Zbl 0935.16030)] and by \textit{N. Andruskiewitsch} [in ``About finite dimensional Hopf algebras'', Lectures given at the CIMPA School ``Quantum symmetries in theoretical physics and mathematics'', Bariloche 2000, to appear in Contemp. Math.].NEWLINENEWLINEFor the entire collection see [Zbl 0969.00049].