Pointed Hopf Algebras—from Enveloping Algebras to Quantum Groups and Beyond

A fundamental example of a pointed Hopf algebra is the universal enveloping algebra of a Lie algebra. This example finds its generalization in quantum groups. The class of pointed Hopf algebras is rather extensive and has been the subject of intense study. We recount some of the basic ideas in the development of the theory. A fundamental structure in the general theory is an analog of the enveloping algebra in a certain category, the Nichols algebra.