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Affinity in Generality
Let V be a vector space. Consider the set of all affine transformations of V: An affine transformation of V is a map from V to itself which can be expressed as
for some invertible linear map (automorphism) L and some vector b in V. The set of all affine transformations of V forms a group under composition and is called the affine group of V. Note that the set of automorphisms of V is a subgroup of the affine group and also that the set of translations of V is also subgroup of the affine group. Note that every element of the affine group can be expressed as a composition of an invertible linear map followed by a translation. Note also that the only affine transformation which is both an automorphism and a translation is the identity map. It is also quite easy to see that the translation group is a normal subgroup of the affine group. However, the automorphism group of V, which, since V is a vector space is known as the general linear group of V, is not.
Is the affine group the internal direct product of the translation group and the automorphism group? No, but it comes close. In the previous post, we saw that if G is the internal direct product of H and K, then not only can every element of G can be expressed (uniquely) in the form g = hk, but also the elements of H commute with the elements of K. This is the sense in which the two groups H and K do not interact with each other inside G. In the case of the affine group, there is interaction: The translation group of a vector space does not commute with its general linear group. This breakdown is evidenced by the fact that one of the two groups fails to be normal.
Recently, I have been reading some algebra. This has been immensely enjoyable; I had forgotten how much I used to like algebra. The material here is by no means advanced, but relies on some basic definitions from group theory.
Suppose I have two groups H and K. Can we combine them together? I learnt some years ago now that there is a straightforward way of forming a ‘sum’ of these two objects: One takes the cartesian product HK, and bestows it with a group operation in the most basic and obvious way possible: (h,k)(h’,k’) = (hh’,kk’). The group that results is known as the direct product of H and K.
The same can be done with vector spaces: Given two vector spaces V and W, I can form the direct sum of them, which is thought of as the vector space consisting of ordered pairs of the form (v,w) with v in V and w in W. It is easy to guess what the rules for addition and scalar multiplication must be. However, soon after learning these two definitions, I began to realise that at least a small amount of wool was obscuring my eyes: Suppose I have a vector space X and two subspaces Y and Z such that every vector x in X can be written uniquely in the form x = y + z, for y in Y and z in Z. I would then be invited to observe how this means that X was the direct sum of Y and Z. This didn’t sit well with me. It clearly wasn’t quite the same thing: There was no cartesian product; there was no way that x = y + z was the same as x = (y,z).