Harmonic analysis on semisimple Hopf algebras (Q2761453)

Let \(H\) be a semisimple Hopf algebra. The authors study, via induction and restriction, the relation between the character algebra of \(H\), \(R(H)\), and the character algebra \(R(K)\), where \(K\) is a Hopf subalgebra, extending the classical finite group case. Let \(K\subseteq H\) be an inclusion of Hopf algebras and \(e\in K\) be an idempotent, the Hecke algebra associated with \(e\) is the subalgebra \({\mathcal H}(H,K,e):=eHe\) of \(H\). If \(e\) is a central idempotent of \(K\), it is shown that the Hecke algebra is isomorphic to \(L^2_e(K\backslash H/K):=\{\alpha\in H^*:e\rightharpoonup\alpha\leftharpoonup e=\alpha\}\), where \(\rightharpoonup\) and \(\leftharpoonup\) are the transposes of right and left multiplication. Spherical functions for semisimple Hopf algebras are studied and described. \((H,K)\) is said to be a Gelfand pair if \(L^2(K\backslash H/K)\) if a commutative subalgebra of \(H^*\) with convolution product. In this case, the quotient \(H\)-module coalgebra \(H/HK^+\) is called a symmetric space. The authors obtains several characterizations of Gelfand pairs and two interesting sufficient conditions. A Fourier expansion in \(L^2(K\backslash H/K)\) is given.NEWLINENEWLINENEWLINEFinally, the authors consider two examples: the biproducts and the Drinfeld double of \(H\), \(D(H)\). It is shown that \((D(H),H)\) is a Gelfand pair if and only if \(H\) is an almost cocommutative Hopf algebra.