Frobenius and quasi-Frobenius left Hopf algebroids

Abstract

We study when left (op)Hopf algebroids in the sense of Takeuchi-Schauenburg give rise to a Frobenius or quasi-Frobenius extension. The case of Hopf algebroids in the sense of Böhm was treated by G. Böhm ([4]). Contrary to Hopf algebroids, (op)Hopf left algebroids don't necessarily have an antipode but their Hopf-Galois map is invertible. We make use of recent results about left Hopf algebroids ([18], [35], [25]). Our results are applied to the restricted enveloping algebra of a restricted Lie-Rinehart algebra.