PBW (Poincaré-Birkhoff-Witt) deformations of smash products are moduli spaces of algebra deformations with a regularity condition containing many prominent cases, such as various types of Hecke and Cherednik algebras, which are closely related to representations of algebraic groups and of finite groups. Both the parametrization of the moduli spaces, and the algebra structure and representation theory of the deformed objects pose many important open problems which we will study constructively and algorithmically using Gröbner basis techniques. Further applications include current algebras and algebraic systems theory.
