Representations

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.