There are only 3 real unital algebras of dimension 2. Awesome

This is such a trivial result that it hardly merits a post but it’s quite elegant and I was really happy to have come across yet another application of the involution-idempotent-relation which I mentioned briefly at the end of my second blog post as an example of the sort of general things and techniques which I always strive to find.

So just a brief summary of concepts to start us off: An algebra is just a natural extension of the concept of a vector space where the algebra itself is a vector space which you have endowed with some form of multiplication. Think matrix algebras where you can add and multiply but not necessarily divide and you have essentially captured the whole concept as all algebras can in a sense be represented as matrix algebras. An algebra is said to be real if the corresponding vector space is real and finally it is said to be unital if it has a multiplicative unit, an element 1 which leaves everything invariant 1a = a. (Mathematical purists might stress the properties of the multiplication operation but as I said its just supposed to have the same properties as with matrix  multiplication where we essentially only need to remember that it need not be commutative ab \neq ba).

The prototypical real algebra is the set of complex numbers \mathbf C  where the natural basis elements are 1 and i and where every element is written as a linear combination of the two; a + ib The complex numbers is a real 2 dimensional unital algebra and as it turns out its one of only 3 possible such algebras. The classification is fairly simple and can be done in terms of matrices but it’s not really necessary. Here goes:

Essentially if A is such an algebra then the fact that it is 2 dimensional means every element can be written as a (real) linear combination of 2 elements  a1 + b x = a + bx where 1 and x are the basis elements. Now being inspired by the complex numbers we consider the square of x, x^2, which naturally is also an element of A and so there are real numbers a,b such that

x^2 = a + bx

But similarly to completing the square when solving a quadratic equation we may construct a new element y from x whose square is a multiple of 1, that is  ‘real’.

y^2 =(x - b / 2)^2 = a - b^2 / 4

 If need be we could further rescale y so that its square is either -1, 0, or 1 and lets call this normalized element i. The case where i^2 = -1 is already familiar so we need not dwell on it but we are nevertheless left with two other potential algebras. Lets see what they are.

Lets begin with the case where i^2 = 1. Turns out that this is also a very trivial and/of familiar kind of algebra. Now what is i? It’s an involution! So we can construct from it and the unit 1 two idempotent elements

p_1 = \frac{1}{2}(1 + i), \qquad p_2 = \frac{1}{2}(1 - i)

which have the properties

p_1^2 = p_1, \qquad p_2^2 = p_2, \qquad p_1 p_2 = p_2 p_1 = 0

And since they are linearly independent we may take them as a basis for the algebra and we immediately identify the algebraic structure

(a p_1 + bp_2)(cp_1 + dp_2) = ac p_1 + bd p_2

which is just ‘term-wise’ multiplication, the trivial algebra in \mathbf{R}^2. Another name one can use for this algebra are the ‘Double numbers’.

The third type of algebra, the one with i^2 = 0 is also interesting and you probably have seen it before but it does not permit an illuminating rewrite but instead simply is what it is. Thanks to a friend at the forums I can attribute to them the name the Dual numbers

So there we are; There are the Complex numbers, the Double numbers and the Dual numbers. Funny enough, and I might return to this in a future post, each of these system links to one of the 3 types rotations and geometries; trigonometric, hyperbolic and parabolic. A summary full of color can be found in this arxiv article:


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