Solvability for semisimple Hopf algebras via integrals

We use integrals of left coideal subalgebras to develop Harmonic analysis for semisimple Hopf algebras. We show how \(N^*,\) the space of functional on \(N,\) is embedded in \(H^*.\) We define a bilinear form on \(N^*\) and show that irreducible \(N\)-characters are orthogonal with respect to that form. We then give an explicit formula for induced characters of \(N\) and show how the induced characters are embedded in \(R(H).\) In the second part we give an intrinsic definition for solvable semisimple Hopf algebras via left coideal subalgebras and their integrals. We show how this definition generalizes solvability for finite groups. In particular, commutative and nilpotent Hopf algebras are solvable. We finally prove an analogue of Burnside theorem: A semisimple quasitriangular Hopf algebras of dimension \(p^aq^b\) is solvable.