Matched pairs of groups and bismash products of hopf algebras

On Mirković-Vilonen cycles and crystal combinatorics

“…The first assertion in Proposition 5.1 (ii) means that Stab + (F ) has the structure of a bicrossed product Stab + (F , F ) Stab + (F ) (see [30]) whenever F and F are two faces in the Coxeter complex such that F ⊆ F . Suppose now that F is an alcove and that F is a facet of F .…”

Bicrossed Products for Finite Groups

“…It was rediscovered by Szép [11] and yet again by Takeuchi [12]. The terminology bicrossed product is taken from Takeuchi, other terms referring to this construction used in the literature are knit product and Zappa-Szép product.…”

“…Assume for simplicity that k is a field of characteristic zero. Let E be a finite group that is a bicrossed product of the groups H and G. A noncommutative noncocommutative Hopf algebra k[H] * #k[G] that is both semisimple and cosemisimple can be constructed [12]. This is the easiest way to construct semisimple cosemisimple finite dimensional Hopf algebras.…”

“…The main motivation behind the definition of matched pair is the following result (we refer to [12] or [5, section IX.1] for the proof).…”

Hopf cohomology vanishing via approximation by Hochschild cohomology

“…Theorem 3.1 is the starting point of our new approach, given in Section 3, to Takeuchi's and Sullivan's Theorems. The same idea will be used in Section 5 to prove vanishing of the cohomology associated to an abelian matched pair [Si;T3] (or a Singer pair) of Hopf algebras.…”