Torsors and affine spaces, or: I keep forgetting where I started

The topic of this post is torsors, which occur naturally throughout mathematics and physics whenever we have natural notions of relative — but not absolute — sizes, positions, temperatures and so forth. This post owes a lot to this 2009 post by John Baez, and so I’ll shamelessly steal borrow some (but not all) examples from him:

  • Even after choosing a unit of voltage (e.g. the SI unit, the volt), it makes no sense to say that the voltage at some point p in a circuit is, say, 7V. It does, however, make sense to say that the voltage at p, relative to that at another point q, is 7V. Relative to that chosen reference value at q, voltages are real numbers — but we are free to change the reference point, and without a reference point, voltages are not real numbers, but they do live in a real torsor.
  • A good geographical example is longitude: we habitually describe longitude on Earth as longitude using units of degrees and relative to the Greenwich meridian. However, the choice of the Greenwich meridian is basically arbitrary, and if we were to change to the Cairo, Paris, or Washington meridian instead, it would not change the difference in longitude between any two points on Earth. Longitudes are not elements of the circle group S1 (angles); it is longitude differences that are angles in S1, whereas longitudes live in an S1-torsor.
  • Both the previous two examples indicate that, whatever a “torsor” is, it’s like a well-behaved algebraic structure (like the real line ℝ or circle group S1) in which the usual reference point, the origin (0 in ℝ and 1 in S1) has been “forgotten”. The usual setting of plane geometry going all the way back to ancient Greece is like this, too: there is no preferred origin for plane Euclidean geometry: you are free to work relative to one corner of your graph paper, or relative to some point in the ground in your Athenian sand-pit.

So… what’s going on here?

Group Torsors

The starting point for the study of torsors is to fix a group G, i.e. a non-empty set equipped with a binary operation that is associative and invertible. Below, the binary operation will usually be written as juxtaposition, but if the operation is commutative (i.e. the group is Abelian), then I’ll follow the usual convention of writing the operation as + and the inverse of g ∈ G as −g instead of g−1.

Given a group G, a G-torsor (or principal homogeneous space for G) is a set X equipped with a free and transitive action of G on X. In less technical terms, this means that

  1. to each g ∈ G and each x ∈ X there corresponds an element of X, denoted by gx, called the action of g on x;
  2. this action is consistent with the group operation of G, here written as multiplication: for every g1, g2G and every xX, g2 • (g1x) = (g2g1) • x); in particular, denoting the identity element of G by 1, it follows that 1 • x = x for every xX;
  3. for every x, yX, there exists a unique gG such that gx = y.

When X is a G-torsor, the third part of the definition means that one can sensibly define the ratio (or difference, if G is an additive group) of two elements x, yX to be the unique gG such that g·x = y; we would write this g as y / x if G is written as a multiplicative group, or as y − x if G is written as an additive group. In summary, in the group G one can

  • multiply two elements g, h of G to get another element hg of G;
  • take the ratio of two elements g, h of G to get another element hg−1 of G;
  • invert an element g of G to get another element g−1 of G;

whereas in the G-torsor X one can only

  • take the ratio of two elements x, y of X to get an element y / x of G.

What is the relationship between G and a G-torsor? In one direction, the group G can be turned into a G-torsor by “forgetting” which element is the identity and simply defining ratios by

\displaystyle h/g := h g^{-1} \quad \mbox{for each } g, h \in G.

The other direction is more subtle. Suppose that X is a G-torsor. If one fixes an element x ∈ X, then the third defining property of a G-torsor defines a bijection fxX → G by

\displaystyle f_{x}(y) := y / x.

Think of fx(y) is the group element that represents y relative to our chosen base point x. Having done this, X can be turned into a group with operation ∗x defined by

\displaystyle y_{1} \ast_{x} y_{2} := f_{x}^{-1} \bigl( f_{x}(y_{1}) f_{x}(y_{2}) \bigr),

for which x is the identity element (note that fx(x) = 1, the identity element of G). Moreover, the bijection fx is a group isomorphism from (X, ∗x) to G. However, each element x of X defines a different (though isomorphic) group structure on X and a different isomorphism fx. So, a G-torsor really is like a copy of the group G that has “forgotten” its identity element, and there is no single preferred or “right” way to get that lost structure back.

Vector and Affine Spaces

If the group G above is the additive group of a vector space V, then we obtain as torsors the useful class of affine spaces. Just as a G-torsor is a group G that has “forgotten” its identity element, an affine space is a vector space that has “forgotten” its additive identity, i.e. the zero vector. What we keep is the notion of a straight line (and higher-dimensional concepts like planes, 3-spaces, &c.), but without requiring that these objects pass through the origin — because we no longer know where the origin is! So, for example, the two sets

\displaystyle L_{0} := \{ (x, y) \in \mathbb{R}^{2} \mid x = 0 \},
\displaystyle L_{1} := \{ (x, y) \in \mathbb{R}^{2} \mid x = 1 \},

are both lines in the affine plane ℝ2, but only L0 is a line in (i.e. a linear subspace of) the vector space ℝ2.

More formally, an affine space is a set A equipped with a free and transitive action of the additive group of a vector space V on A. The vector space V can be thought of as the space of differences of elements of A. Just as a group G can be turned into a G-torsor by defining

\displaystyle h/g := h g^{-1} \quad \mbox{for each } g, h \in G,

an vector space V can be turned into an affine space A by defining

\displaystyle \underbrace{v - u}_{\substack{\mbox{formal} \\ \mbox{difference} \\ \mbox{in } A}} := \underbrace{v - u}_{\substack{\mbox{vector} \\ \mbox{difference} \\ \mbox{in } V}} \quad \mbox{for each } u, v \in A = V

The classic example is n-dimensional affine space An, which is obtained from the vector space ℝn by exactly this “forget the origin” construction. To reiterate, lines in ℝn are linear subspaces and are required to pass through the origin (zero vector); lines in An are translations of lines in ℝn — they are still straight, but don’t have to pass through the origin; similarly for higher-dimensional objects like planes.

Going in the other direction, choosing any point o of an affine space A to be the origin turns A into a vector space that is an isomorphic copy of the difference space V: simply identify each affine point x in A with the displacement vector x − o in V and add points x and y of A using the formula

\displaystyle x + y := (x - o) + (y - o) + o,

which should be read as “work out the displacement vectors of x and y relative to o, add them together, and then add that new displacement vector to the original base point o”. Similarly, a scalar multiple c of an affine point x, relative to o, is defined to be c(x − o).

But now here’s an interesting fact: suppose that x1, …, xn are points of an affine space A and that c1, …, cn are scalars that sum to 1 (the multiplicative identity of the field). Then

\displaystyle \sum_{j = 1}^{n} c_{j} x_{j} := o + \sum_{j = 1}^{n} c_{j} (x_{j} - o) = \sum_{j = 1}^{n} c_{j} x_{j}.

Note that the right hand side is independent of the choice of origin o! In fact, this is an if-and-only-if: we can unambiguously take a linear combination of points of an affine space and get and origin-independent answer if and only if the coefficients in the linear combination sum to 1. Among other things, this means that we an unambiguously talk about the affine line spanned by two points x and y of A as being

\displaystyle \{ c x + (1 - c) y \mid c \in \mathbb{R} \}

and not have to worry about which point is taken as the temporary origin of A; similarly for planes spanned by three points, and so on in higher dimensions.

Forgetting Even More: Projective Spaces

In a forthcoming post, I’ll cover a little material on projective spaces: loosely, what happens to a vector space when you forget how long lines are, and just care about their directions, and how this relates to adding a “line at infinity” to an affine space.

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