Related papers: Non-Commutative Geometry, Categories and Quantum P…
The "quantum-event / prime ideal in a category/ noncommutative-point" alternative to "classical-event / commutative prime ideal/ point" is suggested. Ideals in additive categories, prime spectra and representation of quivers are considered…
The topic of this thesis is the development of a versatile and geometrically motivated differential calculus on non-commutative or quantum spaces, providing powerful but easy-to-use mathematical tools for applications in physics and related…
A detailed study is made of the noncommutative geometry of $R^3_q$, the quantum space covariant under the quantum group $SO_q(3)$. For each of its two $SO_q(3)$-covariant differential calculi we find its metric, the corresponding frame and…
Looking to the history of mathematics one could find out two outer approaches to Geometry. First one (algebraic) is due to Descartes and second one (group-theoretic)--to Klein. We will see that they are not rivalling but are tied (by…
We aim to construct a non-commutative algebraic geometry by using generalised valuations. To this end, we introduce groupoid valuation rings and associate suitable value functions to them. We show that these objects behave rather like their…
We discuss the noncommutative generalizations of polynomial algebras which after appropriate completions can be used as coordinate algebras in various noncommutative settings, (noncommutative differential geometry, noncommutative algebraic…
Expansion of the categorical point of view on many areas of the mathematics and mathematical physics will cause to deeper understanding of genuine features of these problems. New applications of categorical methods are connected with new…
We apply one of the formalisms of noncommutative geometry to $R^N_q$, the quantum space covariant under the quantum group $SO_q(N)$. Over $R^N_q$ there are two $SO_q(N)$-covariant differential calculi. For each we find a frame, a metric and…
Non-commutative geometry (NCG) is a mathematical discipline developed in the 1990s by Alain Connes. It is presented as a new generalization of usual geometry, both encompassing and going beyond the Riemannian framework, within a purely…
In this paper we introduce the semi-graded rings, which extend graded rings and skew PBW extensions. For this new type of non-commutative rings we will discuss some basic problems of non-commutative algebraic geometry. In particular, we…
We develop an approach to noncommutative algebraic geometry ``in the perturbative regime" around ordinary commutative geometry. Let R be a noncommutative algebra and A=R/[R,R] its commutativization. We describe what should be the formal…
Quantum mechanics in its presently known formulation requires an external classical time for its description. A classical spacetime manifold and a classical spacetime metric are produced by classical matter fields. In the absence of such…
This informal introduction is an extended version of a three hours lecture given at the 6th Peyresq meeting ``Integrable systems and quantum field theory''. In this lecture, we make an overview of some of the mathematical results which…
An algebraic approach to the study of quantum mechanics on configuration spaces with a finite fundamental group is presented. It uses, in an essential way, the Gelfand-Naimark and Serre-Swan equivalences and thus allows one to represent…
What is quantum geometry? This question is becoming a popular leitmotiv in theoretical physics and in mathematics. Conformal field theory may catch a glimpse of the right answer. We review global aspects of the geometry of conformal fields,…
MSc thesis of the author offering an introduction to the operator algebraic approach to noncommutative geometry, with a treatment of some more advanced elements such as the noncommutative geometry of quantum groups, fuzzy physics, and…
A motivation of using noncommutative and nonarchimedean geometry on very short distances is given. Besides some mathematical preliminaries, we give a short introduction in adelic quantum mechanics. We also recall to basic ideas and tools…
Gelfand duality between unital commutative C*-algebras and Compact Hausdorff spaces is extended to all unital C*-algebras, where the dual objects are what we call compact Hausdorff quantum spaces. We apply this result to obtain, a…
These notes are intended as an introduction to a study of applications of noncommutative calculus to quantum statistical Physics. Centered on noncommutative calculus we describe the physical concepts and mathematical structures appearing in…
A generalization of classical gauge theory is presented, in the framework of a noncommutative-geometric formalism of quantum principal bundles over smooth manifolds. Quantum counterparts of classical gauge bundles, and classical gauge…