The group of three-dimensional rotations \({SO(3)}\) can “act” on various objects, for example the space \({\mathbb{R}^{3}}\) or the unit sphere \({S^{2}}\). These example actions are “symmetries” of the objects, in that the rotated objects are isomorphic to the original, with each element of \({SO(3)}\) thus “represented” by an automorphism of the object. These ideas are formalized by actions and representations, which are homomorphisms from each element of an algebraic object to a morphism from a space to itself. Group representations in particular are heavily used in physics.
Note that a matrix group (or algebra) has a defining representation (AKA standard representation) on the space of its entries, e.g. \({\mathbb{R}^{n}}\) or \({\mathbb{C}^{n}}\), but in a given situation one may be working with a different representation. The defining representation of a Lie group is also often called the fundamental representation, but this term has a different meaning when used in the classification of Lie algebras.
| Δ It is important to keep in mind which vector space is meant in a given situation; e.g. in the context of a representation of \({SO(n)}\) on an object in \({\mathbb{R}^{m}}\), there is the space \({\mathbb{R}^{n}}\) used to define the group, the possibly different Euclidean space \({\mathbb{R}^{m}}\) the representation is acting on, and the space \({\mathbb{R}^{n\left(n-1\right)/2}}\) that the charts of the manifold of the Lie group map to. |