Topological groups, Lie groups Classical Lie groups Example

Table of Lie groups

Builds on General linear group Special linear group Orthogonal group Special orthogonal group Unitary group Special unitary group SU(n)Compact symplectic group Symplectic group Spin group Circle group Fundamental group Compact space

Example

The table lists common real Lie groups with a few invariants. These are the real dimension, compactness, the number of connected components, and the fundamental group $\pi_1$ of the identity component. Here $n \ge 3$ , to keep the entries uniform.

Group	Dimension	Compact	Components	$\pi_1$
$\R^n$	$n$	no	$1$	$0$
$U(1)$	$1$	yes	$1$	$\Z$
$GL_n(\R)$	$n^2$	no	$2$	$\Z/2$
$SL_n(\R)$	$n^2 - 1$	no	$1$	$\Z/2$
$O(n)$	$n(n-1)/2$	yes	$2$	$\Z/2$
$SO(n)$	$n(n-1)/2$	yes	$1$	$\Z/2$
$\mathrm{Spin}(n)$	$n(n-1)/2$	yes	$1$	$0$
$U(n)$	$n^2$	yes	$1$	$\Z$
$SU(n)$	$n^2 - 1$	yes	$1$	$0$
$GL_n(\C)$	$2n^2$	no	$1$	$\Z$
$SL_n(\C)$	$2n^2 - 2$	no	$1$	$0$
$Sp(2n, \R)$	$n(2n+1)$	no	$1$	$\Z$
$Sp(n)$	$n(2n+1)$	yes	$1$	$0$

The rows are the circle group $U(1)$ , the general linear group and the special linear group. Then come the orthogonal group, the special orthogonal group, the spin group, the unitary group and the special unitary group. The last rows are the real symplectic group $Sp(2n, \R)$ of $2n \times 2n$ real matrices. They also include the compact symplectic group $Sp(n)$ of quaternionic $n \times n$ matrices.

Check of the row $O(n)$ . Its Lie algebra is the space of real skew-symmetric $n \times n$ matrices, which has dimension $n(n-1)/2$ . The determinant of an orthogonal matrix is $\pm 1$ , and both values occur, so $O(n)$ has at least two components. Each fibre of $\det$ is a translate of the connected group $SO(n)$ , so there are exactly two. $O(n)$ is closed and bounded in $\R^{n^2}$ , since $A^T A = I$ forces every entry into $[-1, 1]$ . So $O(n)$ is compact.

Remarks

For $n = 2$ some entries change: $SO(2) \cong U(1)$ and the identity components of $GL_2(\R)$ and $SL_2(\R)$ have fundamental group $\Z$ . The $\pi_1$ entries for the noncompact groups use one fact. A connected real Lie group deformation retracts onto a maximal compact subgroup, for example $SO(n)$ in $SL_n(\R)$ and $U(n)$ in $Sp(2n, \R)$ . For $n \ge 3$ , $\mathrm{Spin}(n)$ is the universal cover of $SO(n)$ , a double cover.

Going deeper

For a student

The invariants in the table are linked through maximal compact subgroups. A connected Lie group has the homotopy type of its maximal compact subgroup, so $\pi_1$ of $SL_n(\R)$ is read off from $SO(n)$ and $\pi_1$ of $Sp(2n, \R)$ from $U(n)$ . Many rows pair a noncompact group with a compact form of the same complexified Lie algebra, such as $SL_n(\R)$ and $SU(n)$ . This explains why such pairs, like $Sp(2n, \R)$ and $Sp(n)$ , have the same dimension.

Reading path

To understand this page from the ground up, read these 41 pages in order. The foundations come first.

Binary operation definition
Clifford algebra concept
Compact space concept
Covering space concept
Cross product concept
Determinant concept
Fundamental group concept
General linear group object
Group concept
Identity matrix object
Lie group concept
Matrix definition
Real projective space object
Symplectic group object
Vector space definition
Commutativity definition
Ring definition
Square matrix definition
Transpose definition
Linear map definition
Unit of a ring definition
Conjugate transpose definition
Abelian group definition
Multilinear map definition
Orthogonal matrix definition
Division ring definition
Circle group object
Alternating multilinear map definition
Orthogonal group object
Unitary matrix definition
The quaternions example
Unitary group object
Determinant definition
Special orthogonal group object
Special linear group object
Special unitary group SU(n) example
Rotation group SO(3) example
SU(2) example
Spin group example
Compact symplectic group example
Table of Lie groups example

Tag	`720FQEBT68`
URI	`m2://720FQEBT68`
Kind	example
Writing	W0 stub
MSC	22E15, 22E10, 22C05
Illustrates	Lie group
Uses	General linear group, Special linear group, Orthogonal group, Special orthogonal group, Unitary group, Special unitary group SU(n), Compact symplectic group, Symplectic group, Spin group, Circle group, Fundamental group, Compact space
Same as	Wikidata Q3981612 Wikipedia Table of Lie groups