---
tag: 720FQEBT68
kind: example
name: Table of Lie groups
msc: ["22E15", "22E10", "22C05"]
illustrates: [7JBRGCVJRB]
uses: [0FCHE902D7, 0V9PMN0RBH, 0TKYAEJ8N7, 0CGYNDCWRR, 0R9R0Y1HDK, CZCWFPNS4N, 5NNPBFQA02, X12FS2JAP0, JJSXXJQGEH, 04EAFVFTF7, S1XK3HXHFX, 0TVJZYB6SH]
sources:
  - url: https://en.wikipedia.org/wiki/Table_of_Lie_groups
same_as:
  wikidata: Q3981612
  wikipedia: Table of Lie groups
---

## Example

The table lists common real [[Lie groups|7JBRGCVJRB]] with a few invariants.
These are the real dimension, [[compactness|0TVJZYB6SH]], the number of connected components, and the [[fundamental group|S1XK3HXHFX]] $\pi_1$ 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|04EAFVFTF7]] $U(1)$, the [[general linear group|0FCHE902D7]] and the [[special linear group|0V9PMN0RBH]].
Then come the [[orthogonal group|0TKYAEJ8N7]], the [[special orthogonal group|0CGYNDCWRR]], the [[spin group|JJSXXJQGEH]], the [[unitary group|0R9R0Y1HDK]] and the [[special unitary group|CZCWFPNS4N]].
The last rows are the real [[symplectic group|X12FS2JAP0]] $Sp(2n, \R)$ of $2n \times 2n$ real matrices.
They also include the [[compact symplectic group|5NNPBFQA02]] $Sp(n)$ 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.

## Explanations

### L1 human

Most Lie groups met in practice are groups of matrices: invertible matrices, matrices of determinant one, rotations, unitary matrices and symplectic matrices.
The table records a few basic numbers for each one.
The dimension counts how many independent parameters describe an element, as the three angles of a rotation in space.
Compactness says whether the group is bounded, and the components and the fundamental group describe its shape as a space.

### L2 human

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.