In this chapter we sketch the basic material – primarily algebra – needed in later chapters. As mentioned in the Introduction, the aspiration of this book isn't to ‘Textbookhood’. There are plenty of good textbooks on the material of this chapter (e.g. ). What is harder to find are books that describe the ideas beneath and the context behind the various definitions, theorems and proofs. This book, and this chapter, aspire to that. What we lose in depth and detail, we hope to gain in breadth and conceptual content. The range of readers in mind is diverse, from mathematicians expert in other areas to physicists, and the chosen topics, examples and explanations try to reflect this range.
Finite groups (Section 1.1) and lattices (Section 1.2.1) appear as elementary examples throughout the book. Lie algebras (Section 1.4), more than their nonlinear partners the Lie groups, are fundamental to us, especially through their representations (Section 1.5). Functional analysis (Section 1.3), category theory (Section 1.6) and algebraic number theory (Section 1.7) play only secondary roles. Section 1.2 provides some background geometry, but for proper treatments consult , , , , .
Note the remarkable unity of algebra. Algebraists look at mathematics and science and see structure; they study form rather than content. The foundations of a new theory are laid by running through a fixed list of questions; only later, as the personality quirks of the new structure become clearer, does the theory become more individual.