相关论文: An introduction to quantum groups and non-commutat…
This is a set of lecture notes suitable for a Master's course on quantum computation and information from the perspective of theoretical computer science. The first version was written in 2011, with many extensions and improvements in…
This paper is an overview of my recent work on abstract homomorphisms of algebraic groups. It is based on a talk given at the Conference on Group Actions and Applications in Geometry, Topology, and Analysis held in Kunming in July 2012.
We explore the differential geometry of finite sets where the differential structure is given by a quiver rather than as more usual by a graph. In the finite group case we show that the data for such a differential calculus is described by…
Two differential calculi are developped on an algebra generalizing the usual q-oscillator algebra and involving three generators and three parameters. They are shown to be invariant under the same quantum group that is extended to a…
We present a bicovariant differential calculus on the quantum Poincare group in two dimensions. Gravity theories on quantum groups are discussed.
Noncommutative or `quantum' differential geometry has emerged in recent years as a process for quantizing not only a classical space into a noncommutative algebra (as familiar in quantum mechanics) but also differential forms, bundles and…
Lecture notes on an introductory course on arithmetic lattices (EPFL 2014).
Notes from a course given at Oujda university, Morocco, october 2002 - march 2003 within the support of a fellowship from the Agence Universitaire de la Francophonie. These notes present a brief introduction to Connes' non commutative…
Noncommutative Euclidean spaces -- otherwise known as Moyal spaces or quantum Euclidean spaces -- are a standard example of a non-compact noncommutative geometry. Recent progress in the harmonic analysis of these spaces gives us the…
We show that bicovariant bimodules as defined by Woronowicz are in one to one correspondence with the Drinfeld quantum double representations. We then prove that a differential calculus associated to a bicovariant bimodule of dimension n is…
Some derivation-based differential calculi which have been used to construct models of noncommutative gauge theories are presented and commented. Some comparisons between them are made.
Differential calculi are obtained for quantum homogeneous spaces by extending Woronowicz' approach to the present context. Representation theoretical properties of the differential calculi are investigated. Connections on quantum…
The article provides an introduction to infinite-dimensional differential calculus over topological fields and surveys some of its applications, notably in the areas of infinite-dimensional Lie groups and dynamical systems.
We show that a semi-commutative Galois extension of a unital associative algebra can be endowed with the structure of a graded q-differential algebra. We study the first and higher order noncommutative differential calculus of…
Two lectures given at the UK-Japan Winter School on 'Geometry and Analysis Towards Quantum Theory', Durham, January 2004.
The main notions of the quantum groups: coproduct, action and coaction, representation and corepresentation are discussed using simplest examples: $GL_q(2)$, $sl_q(2)$, $q$-oscillator algebra ${\cal A}(q)$, and reflection equation algebra.…
In high school physics and physics at university level anti-commutative and non-commutative quantities play an outstanding role at the theoretical description of physical relations. But while using commutative quantities, the basic formal…
These notes are an elaboration on: (i) a short course that I gave at the IPhT-Saclay in May-June 2012; (ii) a previous letter on reversibility in quantum mechanics. They present an introductory, but hopefully coherent, view of the main…
This is a first of our papers devoted to "noncommutative topology and graph theory". Its origin is the paper math.QA/0002238 by I. Gelfand, V. Retakh, and R.L. Wilson where a new class of noncommutative algebras $Q_n$ was introduced. The…
This paper is the second part of a series that develops the mathematical framework necessary for studying field theories on ``T-Minkowski'' noncommutative spacetimes. These spacetimes constitute a class of noncommutative geometries,…