In [Bialgebras of type one, 1978], Warren Nichols introduced a family of algebras that initially received little attention. Perhaps MathSciNet gives a clue as to why, calling the paper, "a fairly technical investigation into the structure of certain sorts of Hopf algebras." No surprise that it took more than a decade, and independent discovery by others, for Nichols' algebras to find purchase in the Hopf algebra community.
Today, Nichols algebras play a critical role in the classification of finite dimensional pointed Hopf algebras. They also have a perfectly natural (and non-technical!) definition. I will start here, then give an overview of the ongoing classification program. The ubiquitous Cartan matrices make an appearance.
