Coalgebra Bundles

Abstract

We develop a generalised gauge theory in which the role of gauge group is played by a coalgebra and the role of principal bundle by an algebra. The theory provides a unifying point of view which includes quantum group gauge theory, embeddable quantum homogeneous spaces and braided group gauge theory, the latter being introduced now by these means. Examples include ones in which the gauge groups are the braided line and the quantum plane.

Deformation of Algebra Factorisations

“…Thus the second of equations (1) holds. Similarly one shows that the equations (2) hold for Ψ t modulo t 2 if and only if d A µ (1) B −d B Ψ (1) = 0. Therefore the necessary and sufficient condition for X(B, A) t to be an infinitesimal deformation of X(B, A) is that µ (1) A ⊕ Ψ (1) ⊕ µ (1) B be a 2-cocycle in C 2 (X(B, A)) as required.…”

“…This is precisely the statement that d B µ (1) A +d A Ψ (1) = 0. Evaluating this condition at 1 A ⊗1 A ⊗b one easily finds that Ψ (1) (1 A ⊗b) = −b⊗µ (1) A…”

“…For an infinitesimal deformation it is enough to consider µ At = µ A + tµ (1) A , µ Bt = µ B + tµ (1) B , Ψ t = Ψ + tΨ (1) where X(B, A)). First we show that the triple (µ (1) A , Ψ (1) , µ (1) B ) defines an infinitesimal deformation if and only if µ (1) A ⊕ Ψ (1) ⊕ µ (1) B is a cocycle. As the first row and the first column in C (X(B, A)…”

“…Let X t (B t , A t ) andX t (B t ,Ã t ) be two infinitesimal deformations of an algebra factorisation X(B, A) given by the cocycles µ (1) A…”

“…This definition of a deformation of an algebra factorisation generalises the definition introduced in [4], where only the map Ψ was allowed to be deformed. The need for such a generalisation arises from the theory of quantum and coalgebra principal bundles [1]. As explained in [2] the structure of a classical principal bundle is encoded in the factorisation built on the algebra of functions on the total space of a bundle and the group algebra of the structure group.…”