TITLE: PBW in the wake of Poincaré

ABSTRACT: After a brief sketch of the historic background to Poincaré’s work on PBW [1900], we discuss in detail what moved Poincaré to invent the universal enveloping algebra, and how he approached the PBW problem, i.e., to exhibit a basis of the universal enveloping algebra. Then we outline some applications and developments in the direction laid out in Poincaré’s paper.