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We classify pointed Hopf algebras with finite Gelfand–Kirillov dimension whose infinitesimal braiding has dimension 2 but is not of diagonal type, or equivalently is a block. These Hopf algebras are new and turn out to be liftings of either a Jordan or a super Jordan plane over a nilpotent-by-finite group.
Let G be a connected, simply connected, simple complex algebraic group and let ϵ be a primitive ℓth root of one, ℓ odd and 3∤ℓ if G is of type G2. We determine all Hopf algebra quotients of the quantized coordinate algebra 𝒪ϵ(G).
We construct explicit examples of weak Hopf algebras (actually face algebras in the sense of Hayashi [H]) via vacant double groupoids as explained in [AN]. To this end, we first study the Kac exact sequence for matched pairs of groupoids and show that it can be computed via group cohomology. Then we describe explicit examples of finite vacant double groupoids.