The Rational Cherednik Algebra via PBW Theorems

The original Poincaré-Birkhoff-Witt (PBW) theorem originated in Lie theory, but the phrase "PBW theorem" now refers to a myriad of similar results in various areas. We will begin with a nuts-and-bolts PBW theorem for the Weyl algebra. Then, seeking compatibility with a group action, we construct a broad class of algebras and describe precisely which ones have a PBW theorem. Time permitting, we conclude by defining the rational Cherednik algebra, and understanding it as the "universal equivariant Weyl algebra" that admits a PBW theorem.