Cohomological dimension of braided Hopf algebras

TL;DR

This paper investigates the homological dimensions of braided Hopf algebras within a specific categorical framework, establishing their equivalence and providing criteria for smoothness and Calabi-Yau properties, with applications to quantum groups.

Contribution

It generalizes known results for ordinary Hopf algebras to braided Hopf algebras, establishing the equivalence of various homological dimensions and criteria for smoothness and Calabi-Yau properties.

Findings

Homological dimensions coincide for certain braided Hopf algebras.

Provided criteria for smoothness and twisted Calabi-Yau property.

Applied results to the coordinate algebra of a braided quantum group.

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 a well-known result 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 $\textrm{SL}_{2}$.