After introducing a noncommutative counterpart of commutative algebraic geometry based on monoidal categories of quasi-coherent sheaves we show that various constructions in noncommutative geometry (e.g. Morita equivalences, Hopf-Galois extensions) can be given geometric meaning extending their geometric interpretations in the commutative case. On the other hand, we show that some constructions...

We generalise the electric-magnetic duality in standard Maxwell theory to its non-commutative version. Both space-space and space-time noncommutativity are necessary. The duality symmetry is then extended to a general class of non-commutative gauge theories that goes beyond non-commutative electrodynamics. As an application of this symmetry, plane wave solutions are analysed. Dispersion relatio...

Let k be a commutative ring and let R be a commutative k−algebra. Given a positive integer n and a R−algebra A one can consider three functors of points from the category CR of commutative R−algebras to the small category of sets. All these functors are representable, namely • RepA represents the functor induced by B → homR(A,Mn(B)), where Mn(B) are the n× n matrices over B, for all B ∈ CR. • t...

The purpose of this paper is to discuss the relationship between commutative and non-commutative integrability of Hamiltonian systems and to construct new examples of integrable geodesic flows on Riemannian manifolds. In particular, we prove that the geodesic flow of the biinvariant metric on any bi-quotient of a compact Lie group is integrable in the non-commutative sense by means of polynomia...

We study strong coupling phenomena in U(1) gauge theory on the noncommutative plane. To do so, we make use of a T-dual description in terms of an N → ∞ limit of U(N) gauge theory on a commutative torus. The magnetic flux on this torus is taken to be m = N − 1, while the area scales like 1/N , keeping ΛQCD fixed. With a few assumptions, we argue that the speed of high frequency light in pure non...

Proof nets can be seen as a multiple conclusion natural deduction system for Linear Logic (LL) and form a good formalism to analyze some computation mechanisms, for instance in type-theoretic interpretations. This paper presents an algorithm for automated proof nets construction in the non-commutative multiplicative linear logic that is useful for applications including planning, concurrency or...

A non–commutative analogue of the classical differential forms is constructed on the phase–space of an arbitrary quantum system. The non– commutative forms are universal and are related to the quantum mechanical dynamics in the same way as the classical forms are related to classical dynamics. They are constructed by applying the Weyl–Wigner symbol map to the differential envelope of the linear...

This paper establishes the normalisation of natural deduction or lambda calculus formulation of Intuitionistic Non Commutative Logic — which involves both commutative and non commutative connectives. This calculus first introduced by de Groote and as opposed to the classical version by Abrusci and Ruet admits a full entropy which allow order to be relaxed into any suborder. Our result also incl...

Abstract The theory of R. Crowell on derived modules is approached within the non-commutative differential modules. We also seek analogies to cotangent complex from differentials in commutative ring setting. Finally, we give examples motivated Galois coverings curves.

it is shown that a commutative reduced ring r is a baer ring if and only if it is a cs-ring; if and only if every dense subset of spec (r) containing max (r) is an extremally disconnected space; if and only if every non-zero ideal of r is essential in a principal ideal generated by an idempotent.