The Spin Chronicles (Part 8): Clifford Algebra Universal Property

 In The Spin Chronicles (Part 7): Whispers of the Cosmos Beneath the Fig Tree we have met quaternions and biquaternions living peacefully within the Clifford algebra of our 3D space. Today we will widen a little bit our horizons.

The Secret of Light

"God's imaginings extend from rest to rest in His three-dimensional radial universe of length, breadth and thickness--to become the stage of space for His imagined radial universe of matter, time, change and motion." (Walter Russell,  The Secret of Light, University of Science and Philosophy 1947). (Fig. 4)

Once we have space, it gets curved, but since All is One, the structure of the whole Cosmos is reflected in the structure of its smallest cubes, and what we are studying now is the algebra of such a primitive cube. We already know it's eight-dimensional. Three dimensions of space generated 2³ dimensions of its, first Grassmann Λ(V), then Clifford, algebra Cl(V). The Clifford algebra structure allows us to measure its substructure. And now we are ready to delve deeper into this issue. We will start with the natural automorphisms  and anti-automorphism of Cl(V). Recall that we have realized Cl(V) on the canvas of Λ(V):

Λ(V) = Λ⁰(V) ⊕ Λ¹(V) ⊕ Λ²(V) ⊕ Λ³(V)

by deforming its Grassmann product u∧v into the Clifford product uv

uv = u∧v+(u⋅v)1,

for u,v in V. Elements of each Λⁱ(V) are called homogeneous of degree i. We then write: i = deg(a).  Grassmann product of homogeneous elements is again homogeneous. Clifford product, on the other hand, does not preserve homogeneity.  We will return to this point soon, but first let us make some side observations

Observation 1. Our construction of the Grassmann and Clifford algebras can be repeated for any n-dimensional (real or complex) vector space V (there is nothing special about n=3 in this respect) and for any symmetric bilinear form B(u,v) (rather than the Euclidean product (u⋅v)). We just write

uv = u∧v+B(u,v)1,

istead of uv = u∧v+(u⋅v)1.

Sometimes certain properties of the Clifford algebra are even easier to prove for a general Cl(V,B) than for just our particular case.

Observation 2. Every Clifford algebra Cl(V,B) has a very important universal property. I will now state this property, without proving it. Some authors are even using this property to define Clifford algebra. 

Universal Property: Let φ be a linear map from V into an associative algebra A (with unit 1), such that φ(x)² = B(x,x)1 for all x in V. Then φ can be extended to a unique algebra homomorphism  φ^~ from Cl(V,B) to A.

We will now show two simple applications of this universal property. Since it is a nice exercise in a logical and precise thinking, we will do it in details. First take for A the algebra Cl(V,B) itself. We can do it since Cl(V,B) is an associative algebra with unit 1. For φ we take the map φ: x ⟶ -x. We consider it as a map from V to Cl(V,B). The x on the left hand side is considered as a vector in V, the -x on the right hand side is considered as an element of Cl(V,B). We can do it, since V is (identified with)  a subspace of Cl(V,B) - the subspace of elements of degree 1. We take φ(x) = -x. This is -x in Cl(V). We take its square in Cl(V). We get: (-x)(-x) = x²  = xx = B(x,x)1. Therefore φ satisfies the assumptions in the Universal Property. Thus it extends to a unique homomorphism φ^~ from Cl(V,B) to Cl(V,B). The fact that  φ^~  extends φ means  that φ^~ (x) = φ (x) = - x for x in V. But  it is a homomorphism of Cl(V,B). A composition of two homomorphisms is a homomorphism. Thus φ^~ ∘ φ^~ is also a homomorphism. But (φ^~ ∘ φ^~) (x) = x for x in V, and V generates Cl(V). Therefore (φ^~ ∘ φ^~) is the identity on Cl(V,B). Therefore φ^~  is not just a homomorphism, it has an inverse, thus it is an automorphism of Cl(V,B). It is called the canonical (or principal ) automorphism, and we denote it by Π. It is clear that for homogeneous elements of Cl(V) we have

Π(u) = (-1)^deg(u) u.

The elements u of Cl(V,B) on which Π(u) = u form the subalgebra of Cl(V,B) - it is called the even subalgebra, and denoted Cl(V,B)⁺.  The elements u of Cl(V,B) for which Π(u) = -u form a vector subspace of Cl(V,B), denoted Cl(V,B)^-. We have

Cl(V) = Cl(V,B)⁺ ⊕ Cl(V,B)^-.

We now move to the canonical (or principal ) anti-automorphism. This is another instructive exercise in semi-precise logico-algebraic thinking. This time we will take for A the algebra Cl(V,B)^T opposite to Cl(V,B). That means Cl(V,B)^T is the same as Cl(V,B) as a vector space, but multiplication in Cl(V,B)^T, which we denote as u*v is defined as opposite to the product uv in CL(V,B):

u*v = vu.

The algebra Cl(V,B)^T is also associative algebra with the same unit 1. The identity map id: Cl(V,B) → Cl(V,B)T has the property:

id(a)*id(b) = id(ba).

  Let now φ stand for the identity map φ: x ⟶ x, from V to Cl(V,B)^T. Then φ (x)*φ (x) = x*x = xx = B(x,x)1, and again we can apply the universal property to deduce that   φ extends to a unique algebra homomorphism  φ^~ from Cl(V,B) to Cl(V,B)^T . Since 1 and V generate (by sums of products) all CL(V,B) and all Cl(V,B)^T, and since φ^~  is the identity both on 1 and on vectors of V, it follows that φ^~  is a vector space isomorphism, therefore (φ^~ )^-1 exists. We define  τ: Cl(V,B)⟶Cl(V,B) by

τ = id^-1∘φ^~.

Then , for any u,v in Cl(V,B) we have

τ(uv) = vu,

and for any x in V we have τ(x)=x. We also have τ(1)=1. Thus τ is an anti-automorphism of Cl(V,B) - the canonical anti-authomorphism. It is, in fact, a unique anti-automorphism with these properties. Sometimes we will write u^T instead of τ(u), and call it a transposition.

Exercise 1. Show that Π∘τ = τ∘Π.

In the following chronicles we will use Π and τ rather often. They are the main characters of the Clifford algebras and spinors story.