Here we collect the definitions and basic constructions that will be needed throughout the book. This material is mostly quite standard for Hopf algebraists, but developed now with simplified or streamlined new proofs. The last section contains newer material. The content of the various definitions will become apparent after numerous examples and exercises. Once the unfamiliar notation is mastered, working with Hopf algebras is no harder than working with algebras alone, and is more fun. More practice will be provided when we turn to the advanced theory in Chapter 2. I would recommend that any reader should work through the present elementary chapter and at least the first part of Chapter 2 in detail since these sections are central to much of the later development. The main exception is Chapter 4 on matrix quantum groups, where one can get quite a long way purely by analogy with ordinary matrices, and perhaps Chapter 3 by analogy with Lie algebras. Thus, a viable alternative is to begin with Chapters 3 or 4 and use the present chapter and its sequel as reference or for clarification.

Algebras

This section is included to fix our notation and terminology. Any serious attempt to apply the extensive literature on Hopf algebras to physical situations will require familiarity with the basic ideas given here.

Recall that a group (G, ·) is a set G on which an associative product is defined, with a unit element e such that e · u = u = u · e for all u ∈ G, and such that every element u has an inverse, denoted u−1.