%0 Report
%D 2020
%T On coherent Hopf 2-algebras
%A Xiao Han
%X We 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.
%8 05/2020
%G eng
%U https://arxiv.org/abs/2005.11207
%0 Report
%D 2020
%T On the gauge group of Galois objects
%A Xiao Han
%A Giovanni Landi
%X We 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.
%8 03/2020
%G eng
%U https://arxiv.org/abs/2002.06097
%0 Report
%D 2020
%T Twisted Ehresmann Schauenburg bialgebroids
%A Xiao Han
%X We 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.
%8 09/2020
%G eng
%U https://arxiv.org/abs/2009.02764
%0 Journal Article
%J Reviews in Mathematical Physics
%D 2018
%T Principal fibrations over noncommutative spheres
%A Michel Dubois-Violette
%A Xiao Han
%A Giovanni Landi
%X We 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 Connesâ€“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.
%B Reviews in Mathematical Physics
%V 30
%P 1850020
%G eng
%U https://arxiv.org/abs/1804.07032
%R 10.1142/S0129055X18500204