Cherednik, Hecke and quantum algebras as free Frobenius and Calabi–Yau extensions

Abstract

We show how the existence of a PBW-basis and a large enough central subalgebra can be used to deduce that an algebra is Frobenius. We apply this to rational Cherednik algebras, Hecke algebras, quantised universal enveloping algebras, quantum Borels and quantised function algebras. In particular, we give a positive answer to [R. Rouquier, Representations of rational Cherednik algebras, in: Infinite-Dimensional Aspects of Representation Theory and Applications, Amer. Math. Soc., 2005, pp. 103–131] stating that the restricted rational Cherednik algebra at the value t = 0 is symmetric.