Affine space

over a field $ k $

A set $ A $( the elements of which are called the points of the affine space) to which corresponds a vector space $ L $ over $ k $( which is called the space associated to $ A $) and a mapping of the set $ A \times A $ into the space $ L $( the image of an element $ (a, b) \in A \times A $ is denoted by $ \vec{ab} $ and is called the vector with beginning in $ a $ and end in $ b $), which has the following properties:

a) for any fixed point $ a $ the mapping $ x \rightarrow \vec{ax} $, $ x \in A $, is a bijection of $ A $ on $ L $;

b) for any points $ a, b, c \in A $ the relationship

$$ \vec{ab} + \vec{bc} + \vec{ca} = \vec{0} , $$

where $ \vec{0} $ denotes the zero vector, is valid. The dimension of $ L $ is taken for the dimension of the affine space $ A $. A point $ a \in A $ and a vector $ l \in L $ define another point, which is denoted by $ a + l $, i.e. the additive group of vectors of the space $ L $ acts freely and transitively on the affine space corresponding to $ L $.

Examples.

1) The set of the vectors of the space $ L $ is the affine space $ A(L) $; the space associated to it coincides with $ L $. In particular, the field of scalars is an affine space of dimension 1. If $ L = k ^ {n} $, then $ A( k ^ {n} ) $ is called the $ n $- dimensional affine space over the field $ k $, and its points $ a = ( a _ {1} \dots a _ {n} ) $ and $ b = (b _ {1} \dots b _ {n} ) $ determine the vector $ \vec{ab} = (b _ {1} - a _ {1} \dots b _ {n} - a _ {n} ) $.

2) The complement of any hyperplane in a projective space over the field $ k $ is an affine space.

3) The set of solutions of a system of linear (algebraic or differential) equations is an affine space the associated space of which is the space of solutions of the corresponding homogeneous set of equations.

A subset $ A ^ \prime $ of an affine space $ A $ is called an affine subspace (or a linear manifold) in $ A $ if the set of vectors $ \vec{ab} $, $ a, b \in A ^ \prime $, forms a subspace of $ L $. Each affine subspace $ A ^ \prime \subset A $ has the form $ a + L ^ \prime = \{ {a + l } : {l \in L ^ \prime } \} $, where $ L ^ \prime $ is some subspace in $ L $, while $ a $ is an arbitrary element of $ A ^ \prime $.

A mapping $ f: A _ {1} \rightarrow A _ {2} $ between affine spaces $ A _ {1} $ and $ A _ {2} $ is called affine if there exists a linear mapping of the associated vector spaces $ \phi : L _ {1} \rightarrow L _ {2} $ such that $ f(a + l) = f(a) + \phi (l) $ for all $ a \in A _ {1} $, $ l \in L _ {1} $. A bijective affine mapping is called an affine isomorphism. All affine spaces of the same dimension are mutually isomorphic.

The affine isomorphisms of an affine space $ A $ into itself form a group, called the affine group of the affine space $ A $ and denoted by $ { \mathop{\rm Aff} } (A) $. The affine group of the affine space $ A( k ^ {n} ) $ is denoted by $ { \mathop{\rm Aff} } _ {n} (k) $. Each element $ f \in { \mathop{\rm Aff} } _ {n} (k) $ is given by a formula

$$ f ( ( a _ {1} \dots a _ {n} ) ) = ( b _ {1} \dots b _ {n} ), $$

where

$$ b _ {i} = \sum _ { j } a _ {i} ^ {j} a _ {j} + c _ {i} , $$

$ ( a _ {i} ^ {j} ) $ being an invertible matrix. The affine group $ { \mathop{\rm Aff} } (A) $ contains an invariant subgroup, called the subgroup of (parallel) translations, consisting of the mappings $ f: A\rightarrow A $ for which $ \phi : L \rightarrow L $ is the identity. This group is isomorphic to the additive group of the vector space $ L $. The mapping $ f \rightarrow \phi $ defines a surjective homomorphism of $ { \mathop{\rm Aff} } (A) $ into the general linear group GL, with the subgroup of parallel translations as kernel. If $ L $ is a Euclidean space, the pre-image of the orthogonal group is called the subgroup of Euclidean motions. The pre-image of the special linear group SGL is called the equi-affine subgroup (cf. Affine unimodular group). The subgroup $ G _ {a} \subset { \mathop{\rm Aff} } _ {n} (A) $ consisting of the mappings $ f : A \rightarrow A $ such that $ f(a+l) = a + \phi (l) $ for a given $ a \in A $ and arbitrary $ l \in L $ is called the centro-affine subgroup; it is isomorphic to the general linear group GL of the space $ L $.

In algebraic geometry an affine algebraic set is sometimes called an affine space. A finite-dimensional affine space can be provided with the structure of an affine variety with the Zariski topology (cf. also Affine scheme).

Affine spaces associated with a vector space over a skew-field $ k $ are constructed in a similar manner.

References

[1]	N. Bourbaki, "Elements of mathematics. Algebra: Algebraic structures. Linear algebra" , 1 , Addison-Wesley (1974) pp. Chapt.1;2 (Translated from French) MR0354207

Comments

An affine isomorphism is also called an affine collineation. An equi-affine group is also called a Euclidean group.

References

[a1]	M. Berger, "Geometry" , I , Springer (1987) pp. Chapt. 2 MR0903026 MR0895392 MR0882916 MR0882541 Zbl 0619.53001 Zbl 0606.51001 Zbl 0606.00020