Representations of the quantum double of quasitriangular Hopf algebras (Q6551456)

In a series of previous works, [in: Ring and module theory. Selected papers of the international conference, Ankara, Turkey, August 18--22, 2008. Basel: Birkhäuser. 91--114 (2010; Zbl 1211.16022); Commun. Algebra 39, No. 12, 4618--4633 (2011; Zbl 1259.16038); Bull. Math. Soc. Sci. Math. Roum., Nouv. Sér. 56(104), No. 3, 299--313 (2013; Zbl 1349.16038); J. Pure Appl. Algebra 218, No. 10, 1845--1866 (2014; Zbl 1298.16017)] the authors have extended and investigated to the representation theory of Hopf algebras some concepts and results that extends their counterparts in the representation theory of group algebras.\N\NThe main result of the paper is Theorem 1.7: let \((H,R)\) be a semisimple quasitriangular Hopf algebra. Then each irreducible \(D(H)\)-representation is isomorphic to a direct summand of \(\mathfrak{C}^i\otimes V_\ell\), where \(V_\ell\) is an irreducible representation of \(H\), \(\mathfrak{C}^i\) some conjugacy class and \(D(H)\) acts on \(\mathfrak{C}^i\) and \(V_\ell\) by determined actions. The key results go toward relating a partition on the set of indices of the primitive idempotents of the algebra generated by the irreducible characters of \(D(H)\) and the decomposition of \(D(H)^*\) determined by \textit{W. D. Nichols} and \textit{M. B. Richmond} equivalence relation with respect to a certain Hopf subalgebra of \(D(H)^*\) [Commun. Algebra 26, No. 4, 1081--1095 (1998; Zbl 0901.16018)].\N\NAlthough short, the paper is very readable and self-contained. There is a preliminary section dedicated to recalling the concepts used throughout the text. In particular, when \(V_\ell=V\) is one dimensional, the authors present a characterization for \(D(H)\)-representation \(\mathfrak{C}^i\otimes V\) as a twist of a determined action of \(D(H)\) on \(\mathfrak{C}^i\), and then they finish the paper presenting the decomposition of each \(\mathfrak{C}^i\otimes V_\ell\) into its irreducible components for two Hopf algebras: \(k S_3\) and \(k D_4\).\N\NFor the entire collection see [Zbl 1539.11006].