Intuitive

The Lie group $SO(3)$ describes all possible rotations in 3-dimensional Euclidean space. It thus describes an important symmetry of the physical space we live in. (Other symmetries of our space are translations and boosts.)

$SO(3)$ is closely related to the groups $SU(2)$ and $Sp(1)$. They are all locally isomorphic, that is, they have the same Lie algebra.

Concrete

Representations

Diagram by Eduard Sackinger

The diagram below shows the defining 3-dimensional representation of $SO(3)$ in its upper branch. A 5-dimensional representations of the same group is shown in the lower branch. For an explanation of this diagram and more representations of $SO(3)$ see Fun with Symmetry.

Abstract

Why is it interesting?