Javier López Peña

General twisting of algebras

Abstract We introduce the concept of pseudotwistor (with particular cases called twistor and braided twistor) for an algebra ( A , μ , u ) in a monoidal category, as a morphism T : A ⊗ A → A ⊗ A satisfying a list of axioms ensuring that ( A , μ ○ T , u ) is also an algebra in the category. This concept provides a unifying framework for various deformed (or twisted) algebras from the literature, such as twisted tensor products of algebras, twisted bialgebras and algebras endowed with Fedosov products. Pseudotwistors appear also in other topics from the literature, e.g. Durdevichs braided quantum groups and ribbon algebras. We also focus on the effect of twistors on the universal first order differential calculus, as well as on lifting twistors to braided twistors on the algebra of universal differential forms.

Dihalo(imidazolium)sulfuranes: A Versatile Platform for the Synthesis of New Electrophilic Group-Transfer Reagents

The syntheses of imidazolium thiocyanates and imidazolium thioalkynes from dihalo(imidazolium) sulfuranes are reported and their reactivities as CN(+) and R-CC(+) synthons evaluated, respectively. The easy and scalable preparation of these electrophilic reagents, their operationally simple handling, broad substrate scope, and functional group tolerance clearly illustrate the potential of these species to become a reference for the direct electrophilic cyanation and alkynylation of organic substrates.

A Connection for Product Manifolds in Noncommutative Geometry

Motivated from some results in classical differential geometry, we give a constructive procedure for building up a connection over a (twisted) tensor product of two algebras, starting from connections defined on the factors. The curvature for the product connection is explicitly calculated, and shown to be independent of the choice of the twisting map and the module twisting map used to define the product connection. As a consequence, we obtain that a product of two flat connections is again a flat connection. We show that our constructions also behves well with respesct to bimodule structures, namely being the product of two bimodule connections again a bimodule connection. As an application of our theory, all the product connections on the quantum plane are computed.

Factorization structures with a two-dimensional factor

We introduce the notion of quantum duplicates of an (associative, unital) algebra, motivated by the problem of constructing toy-models for quantizations of certain configuration spaces in quantum mechanics. The proposed (algebraic) model relies on the classification of factorization structures with a two-dimensional factor. In the present paper, main properties of this particular kind of structure are determined, and we present a complete description of quantum duplicates of finite set algebras. As an application, we obtain a classification (up to isomorphism) of all the algebras of dimension 4 (over an arbitrary field) that can be factorized as a product of two factors.