00896nas a2200109   4500008004100000245003200041210002800073260001200101520062200113100001400735856003700749       2020                        eng d00aOn coherent Hopf 2-algebras0 acoherent Hopf 2algebras  c05/20203 aWe construct a coherent Hopf 2-algebra as quantization of a coherent 2-group, which consists of two Hopf coquasigroups and a coassociator. For this constructive method, if we replace Hopf coquasigroups by Hopf algebras, we can construct a strict Hoft 2-algebra, which is a quantisation of 2-group. We also study the crossed comodule of Hopf algebras, which is shown to be a strict Hopf 2-algebra under some conditions. As an example, a quasi coassociative Hopf coquasigroup is employed to build a special coherent Hopf 2-algebra with nontrivial coassociator. Following this we study functions on Cayley algebra basis.1 aHan, Xiao  uhttps://arxiv.org/abs/2005.1120700996nas a2200121   4500008004100000245004100041210003400082260001200116520067500128100001400803700002000817856003700837       2020                        eng d00aOn the gauge group of Galois objects0 agauge group of Galois objects  c03/20203 aWe study the Ehresmann--Schauenburg bialgebroid of a noncommutative principal bundle as a quantization of the classical gauge groupoid of a principal bundle. When the base algebra is in the centre of the total space algebra, the gauge group of the noncommutative principal bundle is isomorphic to the group of bisections of the bialgebroid. In particular we consider Galois objects (non-trivial noncommutative bundles over a point in a sense) for which the bialgebroid is a Hopf algebra. For these we give a crossed module structure for the bisections and the automorphisms of the bialgebroid. Examples include Galois objects of group Hopf algebras and of Taft algebras.1 aHan, Xiao1 aLandi, Giovanni  uhttps://arxiv.org/abs/2002.0609701165nas a2200109   4500008004100000245004700041210004700088260001200135520085700147100001401004856003701018       2020                        eng d00aTwisted Ehresmann Schauenburg bialgebroids0 aTwisted Ehresmann Schauenburg bialgebroids  c09/20203 aWe construct an invertible normalised 2 cocycle on the Ehresmann Schauenburg bialgebroid of a cleft Hopf Galois extension under the condition that the corresponding Hopf algebra is cocommutative and the image of the unital cocycle corresponding to this cleft Hopf Galois extension belongs to the centre of the coinvariant subalgebra. Moreover, we show that any Ehresmann Schauenburg bialgebroid of this kind is isomorphic to a 2-cocycle twist of the Ehresmann Schauenburg bialgebroid corresponding to a Hopf Galois extension without cocycle, where comodule algebra is an ordinary smash product of the coinvariant subalgebra and the Hopf algebra (i.e. $\C(B/#_{\sigma}H, H)\simeq \C(B\#H, H)^{\tilde{\sigma}}$). We also study the theory in the case of a Galois object where the base is trivial but without requiring the Hopf algebra to be cocommutative.1 aHan, Xiao  uhttps://arxiv.org/abs/2009.0276400970nas a2200145   4500008004100000245005300041210005300094300001200147490000700159520055900166100002800725700001400753700002000767856003700787       2018                        eng d00aPrincipal fibrations over noncommutative spheres0 aPrincipal fibrations over noncommutative spheres  a18500200 v303 aWe present examples of noncommutative four-spheres that are base spaces of $SU(2)$-principal bundles with noncommutative seven-spheres as total spaces. The noncommutative coordinate algebras of the four-spheres are generated by the entries of a projection which is invariant under the action of $SU(2)$. We give conditions for the components of the Connes–Chern character of the projection to vanish but the second (the top) one. The latter is then a non-zero Hochschild cycle that plays the role of the volume form for the noncommutative four-spheres.1 aDubois-Violette, Michel1 aHan, Xiao1 aLandi, Giovanni  uhttps://arxiv.org/abs/1804.07032