Abstract
Given a Hopf algebra H and an algebra A that is an H-module algebra we consider the category of left H-modules and A-bimodules [Formula: see text], where morphisms are just right A-linear maps (not necessarily H-equivariant). Given a twist [Formula: see text] of H we then quantize (deform) H to [Formula: see text], A to A⋆ and correspondingly the category [Formula: see text] to [Formula: see text]. If we consider a quasitriangular Hopf algebra H, a quasi-commutative algebra A and quasi-commutative A-bimodules, we can further construct and study tensor products over A of modules and of morphisms, and their twist quantization. This study leads to the definition of arbitrary (i.e., not necessarily H-equivariant) connections on quasi-commutative A-bimodules, to extend these connections to tensor product modules and to quantize them to A⋆-bimodule connections. Their curvatures and those on tensor product modules are also determined.
Highlights
Consider the basic algebraic structures underlying the differential geometry of a manifold M : the algebra A = C∞(M ) of complex valued functions on M ; the A-module of sections of the tangent bundle, and that of oneforms; the algebra of tensor fields (T, ⊗A) and the exterior algebra (Ω, ∧)
The universal enveloping algebra of the Lie algebra of vector fields is a Hopf algebra H and it acts on all the above structures
Concerning morphisms, in Refs. 1,2 we study in particular the deformations of the Lie derivative and inner derivative
Summary
A-module morphisms; their tensor product will be defined for a subclass of noncommutative algebras A and A-bimodules: quasi-commutative algebras and bimodules carrying an action of a quasitriangular Hopf algebra H. Given two connections on two quasi-commutative A-bimodules V and W we define their sum as the connection on the tensor product module V ⊗A W (physically this is relevant for example when considering the covariant derivative of composite fields) This differs from the sum of bimodule connections discussed in the literature.[7,8,9] We further study the corresponding quantized connections as well as their curvatures. For example we have that the bimodule of one forms Ω on a manifold M is an element in the category UΞC∞(M)MC∞(M), where U Ξ is the universal enveloping algebra of the Lie algebra of vector fields and their action is given by the Lie derivative. (this result follows from the equivalence of the tensor categories of H-modules and twisted H-modules.3)
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