Quantum approximations of algebraic representations

Algebraic Geometry and Number Theory Seminar

To a fixed root system one can associate various representation theories. Among the more challenging are the representations of algebraic groups over a ground field of positive characteristic (the modular case), and of quantum groups at a root of unity (the quantum case). Lusztig and Lusztig-Williamson conjectured the existence of infinitely many generations of representation theories, the quantum case being the first generation, and the modular case being the limit at infinity (for big enough prime characteristics). For a fixed root system we provide a framework that includes both the modular and the quantum case, and that allows to define infinitely many approximations in between. These approximations  might serve as a candidate for the Lusztig-Williamson generations.