Hopf algebras : definitions and examples

Abstract

We define a Hopf algebra and give a variety of examples of varying complexity. To facilitate the definition, we first define the commutative diagram, the tensor product, and an algebra/coalgebra/bialgebra. We briefly discuss the duality between algebras and coalgebras. Prior to introducing the non-commutative Hopf algebras of Sweedler and Taft, we define the q-binomial coefficient and prove a related lemma from q-series which allows an explicit formula for the coproduct of a Taft algebra.