2023/05/21 by Han Xiao, Shahn Majid, Han, Xiao +1 · 4 citations
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA)
paper · pdf · doi:10.48550/arxiv.2305.12465
openalex publication_date 2023/05/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce and study the group B(L) of bisections of a Hopf algebroid L and show that they form a group crossed module or 2-group with the group \Aut(L) of automorphisms. Moreover, the group of vertical bisections turns out to be part of a certain non-Abelian cohomology of which H2(L,B) governs cotwisting of a Hopf algebroid with base B. For the Ehresmann-Schauenburg Hopf algebroid L(P,H) of a quantum principal bundle or Hopf-Galois extension, B(L(P,H)) reduces to the group \AutH(P) of bundle automorphisms and vertical bisections to the group of `gauge transformations' of the bundle. The general H2(L(P,H),B) reduces to a known non-Abelian cohomology in the case where P is a trivial principal bundle or cleft extension. Parallel characterisations are obtained for the bisections and non-Abelian cohomology of the action Hopf algebroid B# Hop associated to a braided-commutative algebra B in the category of Drinfeld-Yetter modules over a Hopf algebra H. Examples include the Heisenberg double or Weyl Hopf algebroid of a Hopf algebra and a canonical action Hopf algebroid \underline H# Hop when H is coquasitriangular and \underlineH is its transmutation.