Hopf algebras in physics

Uvodimo Hopfove algebre i primjenjujemo ih u teoriji polja na nekomutativnom prostoru, višečestičnom anyonskom sustavu, renormalizaciji Feynmanovih dijagrama te pokazujemo bicrossproduct strukturu κ-Poincareove algebre. Ilustriramo neke fenomene koji se pojavljuju u teoriji polja na nekomutativnom prostoru poput UV/IR miješanja i neplanarnih dijagrama, dajemo vezu između kvazitriangularne strukture i cistične statistike te izvodimo Yang- Baxterovu jednadžbu. Feynmanove dijagrame opisujemo pomoću Hopfove algebre ukorijenjenih stabala slijedeći pristup Connesa i Kreimera. Na kraju faktoriziramo κ-Poincare algebru analogno semidirektnom produktu Poincareove algebre.We introduce Hopf algebras and apply them to non-commutative field theory, multiparticle anyonic system, renormalization of Feynman diagrams and bicrossproduct structure of the κ-Poincare Hopf algebra. Several phenomena arising from the spacetime non-commutativity are illustrated, such as UV/IR mixing and non-planar diagrams. The connection between the quasitriangular structure and particle statistics is demonstrated via derivation of the Yang-Baxter equation. Feynman diagrams are renormalized using the Connes-Kreimer technique of rooted trees. In the end we show how κ-Poincare algebra factorises in the way analogous to the semidirect structure of the Poincare group