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Many noncommutative rings of current interest share a fortuitous property: They behave in many ways like polynomial rings. Classically, such behavior is described by the Poincare-Birkhoff-Witt Theorem on Lie algebras. Similar behavior is observed for many quantum groups. This greatly simplifies the study of Lie algebras and their quantum cousins.

In this talk we will survey these ubiquitous topics from modern algebra. We will define Lie algebras, and touch on the classification of a large class of them via some graphs called Dynkin diagrams. We will define some quantum groups, which are not groups, but are Hopf algebras that first arose in statistical mechanics. We will explain the Poincare-Birkhoff-Witt Theorem as it applies to Lie algebras and to quantum groups, and mention applications in representation theory.