Linear Lie Groups

Chapter
Part of the Springer Monographs in Mathematics book series (SMM)

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]=xyyx 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.

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