Rigid dualizing complexes over quantum homogeneous spaces

摘要

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 Bν[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 Uq(sl2) are classified and all of them are proved to be Auslander-regular, AS-regular and Cohen-Macaulay.