Linear Lie Groups

Abstract

We call a closed subgroup $G \subseteq \mathop {\mathrm {GL}}\nolimits _{n}({\mathbb{K}})$ a linear Lie group. In this section, we shall use the exponential function to assign to each linear Lie group G a vector space

$$\mathop {\bf L{}}\nolimits (G) := \bigl\{ x \in M_n({\mathbb{K}}) \colon \exp({\mathbb{R}}x) \subseteq G\bigr\},$$

called the Lie algebra of G. This subspace carries an additional algebraic structure because, for $x,y \in \mathop {\bf L{}}\nolimits (G)$, the commutator [x,y]=xy−yx is contained in $\mathop {\bf L{}}\nolimits (G)$, so that [⋅,⋅] defines a skew-symmetric bilinear operation on $\mathop {\bf L{}}\nolimits (G)$. As a first step, we shall see how to calculate $\mathop {\bf L{}}\nolimits (G)$ for concrete groups and to use it to generalize the polar decomposition to a large class of linear Lie groups.