Cohomological dimension of braided Hopf algebras

Abstract

We show that for a braided Hopf algebra in the category of comodules over a cosemisimple coquasitriangular Hopf algebra, the Hochschild cohomological dimension, the left and right global dimensions and the projective dimensions of the trivial left and right module all coincide. We also provide convenient criteria for smoothness and the twisted Calabi–Yau property for such braided Hopf algebras (without the cosemisimplicity assumption on $H$), in terms of properties of the trivial module. These generalize well-known results in the case of ordinary Hopf algebras. As an illustration, we study the case of the coordinate algebra on the two-parameter braided quantum group ${\mathrm{SL}}_2$.