Title: Hopfs algebras and integral forms
Abstract: Quantum groups introduced byDrinfeld and Jimbo are Hopf algebras defined over the field of functions in onevariable, and can be thought of as deformations of universal envelopingalgebras of semisimple Lie algebras.
In the 1990s, two integral forms over the algebra of polynomials in onevariable were considered, which give rise, via proper evaluation of thevariable in complex numbers, to two types of algebras: the quantum groups of deConcini-Kac-Procesi and those of Lusztig. Both give rise to extensions of Hopfalgebras when the parameter is evaluated at a root of unity.
In this talk, we will describe these two contexts, together with the propertiesof the associated Hopf extensions and the generalizations of these results tothe context of Lie superalgebras, in the first case in a joint work with N.Andruskiewitsch and M. Yakimov, and in the second coming from a work inprogress with C. Vay.