Related papers: Noncommutative Algebra and Noncommutative Geometry

A scheme for constructing quantum mechanics is given that does not have Hilbert space and linear operators as its basic elements. Instead, a version of algebraic approach is considered. Elements of a noncommutative algebra (observables) and…

In this article, gentle algebras are realised as tiling algebras, which are associated to partial triangulations of unpunctured surfaces with marked points on the boundary. This notion of tiling algebras generalise the notion of Jacobian…

In this paper, we investigate the properties of $A$-coherent and $A$-quasi-coherent sheaves within the framework of algebraic geometry over non-algebraically closed fields. We define an $\mathcal{O}_X$-module to be $A$-coherent (resp.…

In this paper we introduce congruence spaces, which are topological spaces that are canonically attached to monoid schemes and that reflect closed topological properties. This leads to satisfactory topological characterizations of closed…

Several classes of *-algebras associated to the action of an affine transformation are considered, and an investigation of the interplay between the different classes of algebras is initiated. Connections are established that relate…

A discretisation scheme that preserves topological features of a physical problem is extended so that differential geometric structures can be approximated in a consistent way thus giving access to the study of physical systems which are…

Smooth parametrization consists in a subdivision of the mathematical objects under consideration into simple pieces, and then parametric representation of each piece, while keeping control of high order derivatives. The main goal of the…

This paper is a contribution to the development of a framework, to be used in the context of semiclassical canonical quantum gravity, in which to frame questions about the correspondence between discrete spacetime structures at "quantum…

In this note the noncommutative geometry is interpreted as a functor, whose range is a family of the operator algebras. Some examples are given and a program is sketched.

In arXiv:math/0606241v2 M. Kontsevich and Y. Soibelman argue that the category of noncommutative (thin) schemes is equivalent to the category of coalgebras. We propose that under this correspondence the affine scheme of a k-algebra A is the…

We briefly sketch the noncommutative geometry approach to the Standard Model, with attention to what can be inferred about particle masses.

A Riemannian geometry of noncommutative n-dimensional surfaces is developed as a first step towards the construction of a consistent noncommutative gravitational theory. Historically, as well, Riemannian geometry was recognized to be the…

The 4-dimensional space-time is extended to pseudo-complex coordinates. Proposing the standard quantization rules in this extended space, the ones for the 4-dimensional sub-space acquire, as one solution, the commutation relations with…

As the first step in an approach to the solution of Hilbert's sixth problem, a general scheme of mechanics, called `supmech', is developed integrating noncommutative symplectic geometry and noncommutative probability theory in an algebraic…

I hesitated for a long time before giving shape to these notes, originally intended for preliminary reading by the attendees to the Summer School "New paths towards quantum gravity" (Holbaek Bay, Denmark, May 2008). At the end, I decide…

The mathematical apparatus of non commutative geometry and operator algebras which Connes has brought to bear to construct a rational scheme for the internal symmetries of the standard model is presented from the physicist's point of view.…

We examine several algebraic properties of the noncommutive $z$-plane and Riemann surfaces. The starting point of our investigation is a two-dimensional noncommutative field theory, and the framework of the theory will be converted into…

In [1], Connes presented axioms governing noncommutative geometry. He went on to claim that when specialised to the commutative case, these axioms recover spin or spin^c geometry depending on whether the geometry is ''real'' or not. We…

Under very strong axioms, there is precisely one real noncommutative geometry between the classical one and the free one, namely the half-classical one, coming from the relations $abc=cba$. We discuss here the complex analogues of this…

A new field of discrete differential geometry is presently emerging on the border between differential and discrete geometry. Whereas classical differential geometry investigates smooth geometric shapes (such as surfaces), and discrete…