Related papers: An introduction to quantum groups and non-commutat…
These lecture notes are an expanded write-up of my short lecture series "Noncommutative Resolutions" given to the MSRI Graduate Student Workshop "Noncommutative Algebraic Geometry" during June 2012. The notes include five chapters, an…
These notes are an introduction to the theory of quantum symmetries of finite and infinite sets, graphs, and locally compact spaces.
These are lecture notes for an introductory course on noncommutative field and gauge theory. We begin by reviewing quantum mechanics as the prototypical noncommutative theory, as well as the geometrical language of standard gauge theory.…
The present note surveys my research related to generalizing notions of abelian group theory to non-commutative case and applying them particularly to investigate fundamental groups.
In this expository paper we present a brief introduction to the geometrical modeling of some quantum computing problems. After a brief introduction to establish the terminology, we focus on quantum information geometry and ZX-calculus,…
These notes are an expanded version of the author's lectures at the graduate workshop "Noncommutative Algebraic Geometry" at the Mathematical Sciences Research Institute in June 2012. The main topics discussed are Artin-Schelter regular…
We explore a differential calculus on the algebra of smooth functions on a manifold. The former is `noncommutative' in the sense that functions and differentials do not commute, in general. Relations with bicovariant differential calculus…
A constructive approach to differential calculus on quantum principal bundles is presented. The calculus on the bundle is built in an intrinsic manner, starting from given graded (differential) *-algebras representing horizontal forms on…
Noncommutative geometry, in its many incarnations, appears at the crossroad of various researches in theoretical and mathematical physics: from models of quantum space-time (with or without breaking of Lorentz symmetry) to loop gravity and…
Application of the noncommutative geometry to several physical models is considered.
The aim of this lecture is to give a pedagogical explanation of the notion of a Poisson Lie structure on the external algebra of a Poisson Lie group which was introduced in our previous papers. Using this notion as a guide we construct…
A generalization of the differential geometry of forms and vector fields to the case of quantum Lie algebras is given. In an abstract formulation that incorporates many existing examples of differential geometry on quantum spaces we combine…
Talk given at NATO ARW in Kiev (September 2000) "Non-commutative Structures in Mathematics and Physics".
We describe recent attempts at discretizing canonical quantum gravity in four dimensions in terms of a connection formulation. This includes a general introduction, a comparison between the real and complex connection approach, and a…
Explicit general constructions of paragrassmann calculus with one and many variables are given. Relations of the paragrassmann calculus to quantum groups are outlined and possible physics applications are briefly discussed. This paper is…
These notes reflect the contents of three lectures given at the workshop of the 14th International Conference on Representations of Algebras (ICRA XIV), held in August 2010 in Tokyo. We first provide an introduction to quantum loop algebras…
These lectures notes are an intoduction for physicists to several ideas and applications of noncommutative geometry. The necessary mathematical tools are presented in a way which we feel should be accessible to physicists. We illustrate…
We study some aspects of the theory of non-commutative differential calculi over complex algebras, especially over the Hopf algebras associated to compact quantum groups in the sense of S.L. Woronowicz. Our principal emphasis is on the…
This paper defines and examines the basic properties of noncommutative analogues of almost complex structures, integrable almost complex structures, holomorphic curvature, cohomology, and holomorphic sheaves. The starting point is a…
The book covers basics of noncommutative geometry and its applications in topology, algebraic geometry and number theory. A brief survey of main parts of noncommutative geometry with historical remarks, bibliography and a list of exercises…