HIGHLIGHTS OF SYMMETRY GROUPS

The concepts of symmetry and symmetry groups are at the heart of several developments in modern theoretical and mathematical physics. The present paper is devoted to a number of selected topics within this framework: Euclidean and rotation groups; the properties of fullerenes in physical chemistry; Galilei, Lorentz and Poincaré groups; conformal transformations and the Laplace equation; quantum groups and Sklyanin algebras. For example, graphite can be vaporized by laser irradiation, producing a remarkably stable cluster consisting of 60 carbon atoms. The corresponding theoretical model considers a truncated icosahedron, i.e., a polygon with 60 vertices and 32 faces, 12 of which are pentagonal and 20 hexagonal. The C60 molecule obtained when a carbon atom is placed at each vertex of this structure has all valences satisfied by two single bonds and one double bond. Thus, a 'cage' in which all 12 pentagons are completely surrounded by hexagons has optimum stability. On a more formal side, the exactly solvable models of quantum and statistical physics can be studied with the help of the quantum inverse problem method. This analysis leads in turn to the discovery of Sklyanin algebras.