mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, an affine representation of a topological

Lie group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...

''G'' on an

affine space In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relat ...

''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 In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...

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'') on the

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...

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 mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...

; in that case, we actually have a linear representation in dimension ''n''. This reduction depends on a group cohomology question, in general.

References

* . Homological algebra Representation theory of Lie algebras Representation theory of Lie groups {{algebra-stub

See also

References