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.
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.
- 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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