This is a collection of my notes from chapter 3 of Brian Hall’s “Lie Groups, Lie Algebras, and Representations”. In this chapter we are finally introduced to the definition of a Lie algebra, and learn how they are related to Lie groups. In the literature, Lie algebras are often expressed using lowercase Gothic (Fraktur) characters. In latex, these characters can be written using the command “\mathfrak”.
Definitions and First Examples
Definition 1: A finite-dimensional real or complex Lie algebra is a finite-dimensional real or complex vector space \(\frak{g}\), together with a map \([\cdot, \cdot]\) from \(\frak{g} \times \frak{g}\) into \(\frak{g}\), with the following properties: 1. \([\cdot, \cdot]\) is bilinear. 2. \([\cdot, \cdot]\) is skew-symmetric: \([X, Y] = - [Y, X]\) for all \(X, Y \in \frak{g}\). 3. The Jacobi identity holds: \[[X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y]] = 0\] for all \(X, Y, Z \in \frak{g}\).
Two elements \(X, Y\) of a Lie algebra $ $ commute if \([X, Y] = 0\). A Lie algebra \(\frak{g}\) is commutative if \([X, Y] = 0\) for all \(X, Y \in \frak{g}\).