Advanced Summer School Integrable systems Quantum Symmetries Prague 2007 Title: Hopf algebras and Combinatorial Physics Abstract: The course will begin by two versions of the exponential formula: a (general) "practical" and a "categorical" form. Examples will be given and the link with vector fields on the line and germs of Lie groups will be made clear. Then, the pairing of two exponentials, by means of the product formula of the QFTP and then the Feynman-like - diagrammatic - expansion of the theory yields very effective (i.e. drawable and computable) data structures which generate a Hopf Algebra with the (multivariate) polynomials as an epimorphic image. In order to connect this Hopf Algebra with others relevant in modern physics, one must first define a non-commutative analogue and then construct a three parameter "true" Hopf deformation. It turns out that the "combinatorial" idea of counting crossings and overlappings works perfectly, connecting this diagrammatic construction to Hopf Algebras of Foissy, Connes-Moscovici, Connes-Kreimer and Matrix Noncommutative Symmetric Functions. Many of these algebras are free (as algebras), we will then detail how to harness the computation in their Sweedlers duals as well as the link with representation theory: classical (a modern view of Peter-Weyl's theorem), elements of Automata Theory, rational calculus and applications to NonCommutative Geometry. If time permits, the geometric/combinatorial meaning of the laws in deformed examples will be presented as well as some new general purpose constructions. Aborder : Rationalité, automates et Géométrie non-commutative ---------- Forwarded message ---------- From: Gérard H. E. Duchamp Date: 05-Apr-2007 03:18 Subject: Re: ISQS16 To: "Cestmir. Burdik" < burdik@kmlinux.fjfi.cvut.cz> Cc: Richard Kerner , Penson Karol Dear Casimir, Please find below title and an abridged plan of the mini-course I plan to give in your summer School. Best regards Gérard H. E. Duchamp %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Gérard H. E. Duchamp: Title : Hopf algebras and Combinatorial Physics Abstract 1) Exponential formula - origins - a modern categorical version - applications to the parametric moment problem and composition of combinatorial structures. 2) Hopf algebras and deformations - examples coming from physics and noncommutative character theory - fonctorial implications 3) Finite orbits methods - Analyse of Sweedlers duals - Link with representation theory : classical (a modern view of Peter-Weyl's theorem), elements of Automata Theory, rational calculus and applications to NonCommutative Geometry. If time permits, the geometric/combinatorial meaning of the laws in deformed examples of (2) will be presented as well as some general purpose Hopf constructions. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%