Gauge groups and bialgebroids

Abstract

We study the Ehresmann–Schauenburg bialgebroid of a noncommutative principal bundle as a quantization of the gauge groupoid of a classical principal bundle. We show that the gauge group of the noncommutative bundle is isomorphic to the group of bisections of the bialgebroid, and we give a crossed module structure for the bisections and the automorphisms of the bialgebroid. Examples include: Galois objects of Taft algebras, a monopole bundle over a quantum sphere and a not faithfully flat Hopf–Galois extension of commutative algebras. For each of the latter two examples, there is in fact a suitable invertible antipode for the bialgebroid making it a Hopf algebroid.

Quantum Inverse Semigroups

“…It is not clear, at this point, how to extend this approach to cover Hopf algebroids with noncommutative bases; the balance between noncommutativity of the Hopf algebroid and the locality of the bisections is, so far, the most difficult problem to solve. It is worth noting that the noncommutativity of a Hopf algebroid and of its base algebra was adequately addressed at the level of global biretractions by Xiao Han and Giovanni Landi in their work on the Eheresmann-Schauenburg bialgebroid associated to a noncommutative principal bundle [17]. The group of gauge transformations of a quantum principal bundle (a Hopf-Galois extension) was proved to be isomorphic to the group of global bisections of the Ehresmann-Schauenburg bialgebroid (see [17], Proposition 4.6).…”

“…It is worth noting that the noncommutativity of a Hopf algebroid and of its base algebra was adequately addressed at the level of global biretractions by Xiao Han and Giovanni Landi in their work on the Eheresmann-Schauenburg bialgebroid associated to a noncommutative principal bundle [17]. The group of gauge transformations of a quantum principal bundle (a Hopf-Galois extension) was proved to be isomorphic to the group of global bisections of the Ehresmann-Schauenburg bialgebroid (see [17], Proposition 4.6).…”

“…Let End Alg (A) denote the k-algebra of multiplicative (not necessarily unital) k-linear endomorphisms of A. Since (α * β) • t = β • t • α • t for all α, β ∈ Brt(H, A) (see Equation (17)), there exists an antimultiplicative map ξ : Brt(H, A) → End Alg (A) given by α → α • t. Now consider the subset J(A) ⊂ End Alg (A) of the multiplicative endomorphisms ϕ : A → A such that there exists an idempotent e ∈ A satisfying (i) ϕ(A) = Aϕ(e);…”

Braided Hopf algebras and gauge transformations

“…The following theorem improves previous results in [6,Prop. 3.6] (where A was quasi-commutative) and [23,Prop. 3.3] (where the faithfully flat condition was used), see also [34,Rem.…”

Hopf-Galois extensions and twisted Hopf algebroids

“…Moreover, B is a B e -ring with the product and unit It was shown in [9], [13] that the B-bimodule B is isomorphic to the B-bimodule of coinvariant elements, B ≃ (P ⊗ P ) coH := {p ⊗ q ∈ P ⊗ P | p (0) ⊗ q (0) ⊗ p (1) q (1) = p ⊗ q ⊗ 1 H }.…”