Cohomology and coquasi-bialgebra extensions associated to a matched pair of bialgebras

AbstractBy using braid diagrams, we explicitly reconstruct the cohomology associated to a matched pair of cocommutative bialgebras, in order to give a method of constructing coquasi-bialgebras, which generalize bialgebras, and classifying them up to monoidal equivalence of their comodule categories. An alternative, homological proof is given for Schauenburg's generalized Kac Sequence involving the abelian group Opext(H,K) of bialgebra extensions. We define an abelian group, Opext″(H,K), of coquasi-bialgebra extensions associated to a Singer pair (H,K) of bialgebras, and prove a variant of Schauenburg's sequence which involves the group. It is also proved that there is a natural isomorphism Opext″(H1,K)≅Opext″(H2,K) that preserves monoidal equivalence classes, if (H1,K) and (H2,K) arise from such matched pairs that are shifts of each other