Exact sequences of representation categories of weak Hopf algebras

Abstract

We study exact sequences of representation categories of weak Hopf algebras over an arbitrary field. Given a sequence $A\overset{k}{\to} B\oversetπ{\to} H$, where $A$ and $B$ are weak Hopf algebras and $H$ is a Hopf algebra, we develop verifiable algebraic conditions on $k$ and $π$ under which there is an exact sequence $\mathrm{Rep}(H)\to\mathrm{Rep}(B)\to\mathrm{Rep}(A)$ of tensor categories in the sense of Bruguières and Natale (2011). Along the way, we develop a generalization of the restriction of scalars functor for maps $π:B\to H$ between associative algebras satisfying a weakened multiplicativity constraint depending on a relatively separable subalgebra $B_r\subseteq B$, as well as a theory of kernels and cokernels for weak Hopf algebras. We, in particular, find that the cokernel of a weak Hopf algebra homomorphism $k:A\to B$ always exists, is a Hopf algebra, and the cokernel map is surjective when $A$ is connected. We conclude by studying examples of such exact sequences built from groupoids, formal ribbon extensions of quasitriangular weak Hopf algebras, and cocycled crossed products of a Hopf algebra acting weakly on a weak Hopf algebra.