Hopf algebras and quantum groups

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The study of symmetry is central to the mathematical and physical sciences and symmetry is expressed mathematically in the language of groups. More subtle forms of symmetry require more sophisticated mathematical notions. One of these is the notion of quantum symmetry, so that for example, rather than having a simple relation of the form ab equals ba, we have that ab is qba, in other words ab and ba are equal up to a factor of q. This notion occurs naturally in the physical world. This project will continue research in the algebraic theory of Hopf algebras and quantum groups. Hopf algebras, named from work of H. Hopf in the 1940's, were studied by a few ring theorists such as Milnor and Moore, Sweedler, Kaplansky and Abe forty years ago. But, in the last twenty years, since V.G.Drinfel'd spoke of quantum groups in his address to the International Congress of Mathematicians, describing work of L.D.Faddeev and his school of physics, there has been greatly increased awareness of and interest in the study of quantum groups and in the general area of the algebraic theory of Hopf algebras. An ongoing problem then, is to describe all Hopf algebras of a given type-- pointed, copointed, semisimple, etc.-- in order to understand the links with physics and with other branches of mathematics such as knot theory, in which these Hopf algebras appear naturally.