Twist deformations of module homomorphisms and connections

Abstract

Let H be a Hopf algebra, A a left H-module algebra and V a left H-module A-bimodule. We study the behavior of the right A-linear endomorphisms of V under twist deformation. We in particular construct a bijective quantization map to the right A -linear endomorphisms of V , with A ,V denoting the usual twist deformations of A,V . The quantization map is extended to right A-linear homomorphisms between left H-module A-bimodules and to right connections on V . We then investigate the tensor product of linear maps between left H-modules. Given a quasitriangular Hopf algebra we can define an H-covariant tensor product of linear maps, which restricts for left H-module A-bimodules to a well-defined tensor product of right A-linear homomorphisms on tensor product modules over A. This also requires a quasi-commutativity condition on the algebra and bimodules. Using this tensor product we can construct a new lifting prescription of connections to tensor product modules, generalizing the usual prescription to also include nonequivariant connections.

Nonassociative geometry in quasi-Hopf representation categories I: Bimodules and their internal homomorphisms

“…As a consequence, there seems to be no good replacement for the concept of bimodule connections in the case where the algebra A and the bimodules V are generic. However, a suitable substitute for bimodule connections and their tensor products can be developed when we restrict ourselves to certain classes of algebras and bimodules, namely those which are commutative up to a braiding; these techniques have been developed in [8] (see also [9,10] for brief summaries). Given any quasitriangular Hopf algebra H, one considers algebras and bimodules on which there is an action of the Hopf algebra.…”

Nonassociative geometry in quasi-Hopf representation categories II: Connections and curvature

“…H-equivariance) are very restrictive and hence the framework in [BM10] allows for only very special geometric structures. Our internal homomorphism approach is inspired by the formalism of [AS14] (see [Sch11,Asc12] for overviews), and it clarifies these ideas and constructions in categorical terms.…”

Noncommutative connections on bimodules and Drinfeld twist deformation

“…In the special case of H-equivariant connections we recover the usual notion of bimodule connections [Mou94, DVM96, Mad00, DV01]. An early account of our results appeared in the PhD thesis [Sch11a] and the proceedings articles [Sch11b,Asc12].…”