Exact sequences of representation categories of weak Hopf algebras

@misc{kolt2026exact,
  title = {{Exact sequences of representation categories of weak Hopf algebras}},
  author = {Quinn T. Kolt},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.40298},
  eprint = {2609.40298},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.40298},
}

TY  - UNPB
TI  - Exact sequences of representation categories of weak Hopf algebras
AU  - Quinn T. Kolt
PY  - 2026
DA  - 2026/10/14
PB  - arXiv
DO  - 10.48550/arxiv.2609.40298
UR  - https://arxiv.org/abs/2609.40298
AB  - We study exact sequences of representation categories of weak Hopf algebras over an arbitrary field. Given a sequence Aoversetk→ Boversetπ→ 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 Rep(H)→Rep(B)→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→ H between associative algebras satisfying a weakened multiplicativity constraint depending on a relatively separable subalgebra B_r⊆ 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→ 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.
ER  -