4 Linear Lie groups and their Lie algebras 4 Linear Lie groups and their Lie algebras 4.2 The exponential map

4.1 Linear Lie groups

We fix some notation:

•

$\mathbb{K}$ denotes either the field $\mathbb{R}$ or $\mathbb{C}$ ;
•

$\mathfrak{gl}_{n,\mathbb{K}}=M_{n}(\mathbb{K})$ is the vector space of all $n\times n$ matrices over $\mathbb{K}$ .

Definition 4.1.

A (linear) Lie group is a closed subgroup of $\operatorname{GL}_{n}(\mathbb{C})$ , for some $n$ .

Remark 4.2.

The usual definition of a Lie group is a smooth manifold together with a group structure such that the group operations are smooth functions. It is a theorem (Cartan’s theorem, or the closed subgroup theorem) that every linear Lie group in the sense of definition 4.1 is a Lie group in this sense. Not every Lie group is a linear Lie group, but we will only be studying linear Lie groups; sometimes I might omit the word ‘linear’.

We give various examples (note that any subgroup defined by equalities of continuous functions will be closed):

•

the real general linear group $\operatorname{GL}_{n}(\mathbb{R})$ : we simply impose the closed condition that all the entries of the matrix are real;
•

the (real or complex) special linear groups $\operatorname{SL}_{n}(\mathbb{K})$ ;
•

if $\left\langle\cdot,\cdot\right\rangle$ is a bilinear form on $\mathbb{R}^{n}$ then we obtain a linear Lie group

$\{g\in\operatorname{GL}_{n}(\mathbb{R}):\left\langle gv,gw\right\rangle=\left% \langle v,w\right\rangle\}.$

There is a matrix $A$ such that $\left\langle v,w\right\rangle=v^{T}Aw$ for all $v, w$ ; the bilinear form is symmetric if and only $A$ is symmetric, alternating if and only if $A$ is skew-symmetric ( $A^{T}=-A$ ), and nondegenerate if and only if $\det A$ is nonzero. Then the group is:

$\{g\in GL_{n}(\mathbb{R}):g^{T}Ag=A\}.$

Some special cases follow.
•

The orthogonal and special orthogonal groups

$\mathrm{O}(n)=\{g\in\operatorname{GL}_{n}(\mathbb{R}):gg^{T}=I\}$

and $\mathrm{SO}(n)=\operatorname{O}(n)\cap\operatorname{SL}_{n}(\mathbb{R})$ ;
•

the unitary and special unitary groups

$\mathrm{U}(n)=\{A\in\operatorname{GL}_{n}(\mathbb{C}):gg^{\dagger}=I\}$

and $\mathrm{SU}(n)=\mathrm{U}(n)\cap\operatorname{SL}_{n}(\mathbb{C})$ (not strictly a special case of the above, but closely related);
•

the symplectic groups

$\operatorname{Sp}(2n)=\{g\in GL_{n}(\mathbb{R}):gJg^{T}=J\}$

where $J=\begin{pmatrix}0&I^{\prime}\\ -I^{\prime}&0\end{pmatrix}$ and $I^{\prime}$ is the $n\times n$ matrix with 1s on the antidiagonal and 0s elsewhere. This corresponds to a nondegenerate alternating bilinear form.
•

the Heisenberg group

$\left\{\begin{pmatrix}1&x&y\\ 0&1&z\\ 0&0&1\end{pmatrix}:x,y,z\in\mathbb{R}\right\};$

Example 4.3.

Non-examples are $\operatorname{GL}_{n}(\mathbb{Q})$ (this is a subgroup of $\operatorname{GL}_{n}(\mathbb{C})$ , but not closed), or (if $\alpha$ is an irrational real number) the subgroup

\left\{\begin{pmatrix}e^{ix}&0\\ 0&e^{i\alpha x}\end{pmatrix}:x\in\mathbb{R}\right\}\subset\operatorname{GL}_{2% }(\mathbb{C}).

This is a subgroup, isomorphic — as a group — to $\mathbb{R}$ , but not closed. You should picture it as a string wound infinitely densely around a torus.

The idea of Lie theory is to simplify the study of these groups by just studying their structure ’very close to the identity’. This crucially uses that they are groups with a topology. By looking at the tangent spaces of these groups at the origin, you obtain Lie algebras; the group operation then turns into a structure called the Lie bracket.