Rigid dualizing complexes over quantum homogeneous spaces

Abstract

A quantum homogeneous space of a Hopf algebra is a right coideal subalgebra over which the Hopf algebra is faithfully flat. It is shown that the Auslander–Gorenstein property of a Hopf algebra is inherited by its quantum homogeneous spaces. If the quantum homogeneous space B of a pointed Hopf algebra H is AS-Gorenstein of dimension d, then B has a rigid dualizing complex B ν [ d ] . The Nakayama automorphism ν is given by ν = ad ( g ) ∘ S 2 ∘ Ξ [ τ ] , where ad ( g ) is the inner automorphism associated to some group-like element g ∈ H and Ξ [ τ ] is the algebra map determined by the left integral of B. The quantum homogeneous spaces of U q ( sl 2 ) are classified and all of them are proved to be Auslander-regular, AS-regular and Cohen–Macaulay.