Groups, Lie Groups, and Lie Algebras

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Abstract

Chapter 4 introduces abstract groups and Lie groups, which are a formalization of the notion of a physical transformation. The chapter begins with the definition of an abstract group along with examples, then specializes to a discussion of the groups that arise most often in physics, particularly the rotation group O(3) and the Lorentz group SO(3,1) o . These groups are discussed in coordinates and in great detail, so that the reader gets a sense of what they look like in action. Then we discuss homomorphisms of groups, which allows us to make precise the relationship between the rotation group O(3) and its quantum-mechanical ‘double-cover’ SU(2). We then define matrix Lie groups and demonstrate how the so-called ‘infinitesimal’ elements of the group give rise to a Lie algebra, whose properties we then explore. We discuss many examples of Lie algebras in physics, and then show how homomorphisms of matrix Lie groups induce homomorphisms of their associated Lie algebras.