# Affine representation

An **affine representation** of a topological (Lie) group *G* on an affine space *A* is a continuous (smooth) group homomorphism from *G* to the automorphism group of *A*, the affine group Aff(*A*). Similarly, an affine representation of a Lie algebra **g** on *A* is a Lie algebra homomorphism from **g** to the Lie algebra **aff**(*A*) of the affine group of *A*.

An example is the action of the Euclidean group E(*n*) upon the Euclidean space E^{n}.

Since the affine group in dimension *n* is a matrix group in dimension *n* + 1, an affine representation may be thought of as a particular kind of linear representation. We may ask whether a given affine representation has a fixed point in the given affine space *A*. If it does, we may take that as origin and regard A as a vector space: in that case, we actually have a linear representation in dimension *n*. This reduction depends on a group cohomology question, in general.

## See alsoEdit

## ReferencesEdit

- Remm, Elisabeth; Goze, Michel (2003), "Affine Structures on abelian Lie Groups",
*Linear Algebra and its Applications*,**360**: 215–230, arXiv:math/0105023, doi:10.1016/S0024-3795(02)00452-4.

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