prev

next

out of 31

View

0Download

0

Embed Size (px)

¿Octonions? A non-associative geometric algebra

Benjamin Prather

Florida State University Department of Mathematics

October 19, 2017

¿Octonions?

Prather

Scalar Product Spaces

Clifford Algebras

Composistion Algebras

Hurwitz’ Theorem

Summary

Scalar Product Spaces

Let K be a field with 1 6= −1 Let V be a vector space over K . Let 〈·, ·〉 : V × V → K .

Definition

V is a scalar product space if:

Symmetry

〈x , y〉 = 〈y , x〉 Linearity

〈ax , y〉 = a 〈x , y〉 〈x + y , z〉 = 〈x , z〉+ 〈y , z〉

for all x , y , z ∈ V and a, b ∈ K

¿Octonions?

Prather

Scalar Product Spaces

Clifford Algebras

Composistion Algebras

Hurwitz’ Theorem

Summary

Scalar Product Spaces

Let N(x) = −〈x , x〉 be a modulus on V .

N(x + y) = −〈x , x〉 − 2 〈x , y〉 − 〈y , y〉 〈x , y〉 = (N(x) + N(y)− N(x + y))/2 N(ax) = −〈ax , ax〉 = a2N(x)

〈x , y〉 can be recovered from N(x). N(x) is homogeneous of degree 2. Thus 〈x , y〉 is a quadratic form.

¿Octonions?

Prather

Scalar Product Spaces

Clifford Algebras

Composistion Algebras

Hurwitz’ Theorem

Summary

Scalar Product Spaces

(James Joseph) Sylvester’s Rigidity Theorem: (1852)

A scalar product space V over R, by an appropriate change of basis, can be made diagonal, with each term in {−1, 0, 1}. Further, the count of each sign is an invariant of V .

¿Octonions?

Prather

Scalar Product Spaces

Clifford Algebras

Composistion Algebras

Hurwitz’ Theorem

Summary

Scalar Product Spaces

The signature of V , (p, n, z), is the number of 1’s, −1’s and 0’s in such a basis.

The proof uses a modified Gram-Shmidt process to find an orthogonal basis. Any change of basis preserving orthogonality preserves the signs of the resulting basis. This basis is then scaled by 1/

√ |N(x)|.

¿Octonions?

Prather

Scalar Product Spaces

Clifford Algebras

Composistion Algebras

Hurwitz’ Theorem

Summary

Scalar Product Spaces

This can be generalized to any field K . Let a ∼ b if a = k2b for some k ∈ K . This forms an equivalence relation.

The signature of V over K is then unique, up to an ordering of equivalence classes.

A standard basis for V is an orthogonal basis ordered by signature.

¿Octonions?

Prather

Scalar Product Spaces

Clifford Algebras

Composistion Algebras

Hurwitz’ Theorem

Summary

Scalar Product Spaces

Over Q the signature is a multi-set of products of finite subsets of prime numbers, plus −1.

Any other q ∈ Q can be put in reduced form then multiplied by the square of its denominator to get a number of this form.

¿Octonions?

Prather

Scalar Product Spaces

Clifford Algebras

Composistion Algebras

Hurwitz’ Theorem

Summary

Scalar Product Spaces

For any finite field there are exactly 3 classes.

Over F5, for example, −1 ∼ 1 so we need another element for the third class. The choices are ±2. Thus the signature uses {−1, 2, 0}. For quadratically closed fields, like C, the signature is entirely 1’s and 0’s.

¿Octonions?

Prather

Scalar Product Spaces

Clifford Algebras

Composistion Algebras

Hurwitz’ Theorem

Summary

Scalar Product Spaces

A scalar product space is degenerate if its signature contains 0s.

x ∈ V is degenerate if 〈x , y〉 = 0 for all y ∈ V . V is degenerate iff it contains a degenerate vector.

Scaler product spaces will be assumed to be non-degenerate, unless stated otherwise.

¿Octonions?

Prather

Scalar Product Spaces

Clifford Algebras

Composistion Algebras

Hurwitz’ Theorem

Summary

Inner Product Spaces

V is an inner product space if additionally:

Positive definite

N(x) ≥ 0 N(x) = 0 iff x = 0

In particular this restricts K to ordered fields. Note: If K is a subset of C symmetry is typically replaced by conjugate symmetry. This forces N(x) ∈ R.

¿Octonions?

Prather

Scalar Product Spaces

Clifford Algebras

Composistion Algebras

Hurwitz’ Theorem

Summary

Clifford Algebras

William Kingdon Clifford

Let V , N(x) be a scalar product space. Let T (V ) be the tensor space over V .

Let I be the ideal 〈x2 + N(x)〉, for all x ∈ V .

Then the Clifford algebra over V is Cl(V ,N) = T (V )/I

¿Octonions?

Prather

Scalar Product Spaces

Clifford Algebras

Composistion Algebras

Hurwitz’ Theorem

Summary

Clifford Algebras

This shows Clifford algebras are initial, or the freest, among algebras containing V with x2 = −N(x). Thus they satisfy a universal property.

Further, if N(x) = 0 for all x this reduces to the exterior algebra over V .

Scalar product spaces over R relate to geometry. Thus if V is over R or C these are geometric algebras.

¿Octonions?

Prather

Scalar Product Spaces

Clifford Algebras

Composistion Algebras

Hurwitz’ Theorem

Summary

Clifford Algebras

Let ei be a standard basis for V . Let E be an ordered subset of basis vectors. Since I removes all squares of V ,∏

E ei forms a basis for Cl(V ,N) Let the product over the empty set be identified with 1.

Thus dim(Cl(V )) = 2d , where d = dim(V ).

¿Octonions?

Prather

Scalar Product Spaces

Clifford Algebras

Composistion Algebras

Hurwitz’ Theorem

Summary

Clifford Algebras

By construction: e2i = 〈ei , ei〉 = −N(ei).

Since this basis is orthogonal: 〈ei , ej〉 = 0 for i 6= j , and eiej = −ejei .

Since Clifford algebras are associative, this uniquely defines the product.

¿Octonions?

Prather

Scalar Product Spaces

Clifford Algebras

Composistion Algebras

Hurwitz’ Theorem

Summary

Clifford Algebras

Consider a vector space over Q with N(ae1 + be2 + ce3) = a

2 − 2b2 − 3c2. Thus e21 = −1, e22 = 2 and e23 = 3.

Let p = e1e2 + e2e3 and q = e2 + e1e2e3.

pq = e1e2e2 + e2e3e2 + e1e2e1e2e3 + e2e3e1e2e3

= 2e1 − e2e2e3 − e1e1e2e2e3 + e2e1e3e3e2 = 2e1 − 2e3 + 2e3 + 3e2e1e2 = 2e1 − 3e1e2e2 = 2e1 − 6e1 = −4e1

¿Octonions?

Prather

Scalar Product Spaces

Clifford Algebras

Composistion Algebras

Hurwitz’ Theorem

Summary

Clifford Algebras

Let Clp,n denote the geometric algebra with signature (p, n, 0). Let Cln represent Cl0,n.

Cl1 = C, Cl2 = H Cl1,0 = C− ∼= R⊕ R, Cl2,0 = Cl1,1 = H− ∼= M2(R). The algebras in second line, and all larger Clifford algebras have zero divisors.

¿Octonions?

Prather

Scalar Product Spaces

Clifford Algebras

Composistion Algebras

Hurwitz’ Theorem

Summary

Clifford Analysis

Cl3,1 and Cl1,3 are used to create algebras over Minkowski space-time. Cl0,3 is sometimes used, with the scalar treated as a time-coordinate.

Physicists like to use differential operators. This motivates active research in Clifford analysis. Clifford analysis gives us well behaved differential forms on the underlying space.

One difficulty in Clifford analysis is the lack of the composition property.

¿Octonions?

Prather

Scalar Product Spaces

Clifford Algebras

Composistion Algebras

Hurwitz’ Theorem

Summary

Unital Algebras

Definition

A unital algebra A is a vector space with:

bilinear product A× A→ A (x + y)z = xz + yz x(y + z) = xy + xz (ax)(by) = (ab)(xy)

Multiplicative identity 1 ∈ A 1x = x1 = x

For all x , y , z ∈ A and a, b ∈ K .

In particular, associativity is not required. a ∈ K is associated with a1 in A, in particular 1.

¿Octonions?

Prather

Scalar Product Spaces

Clifford Algebras

Composistion Algebras

Hurwitz’ Theorem

Summary

Composition Algebras

Definition

A composition algebra is a unital algebra A that is a scalar product space with:

Multiplicative modulus

N(xy) = N(x)N(y)

Clifford algebras only give N(xy) ≤ CN(x)N(y). C, H, C−, and H− do have this property. Any associative algebra over a scalar product space will be a Clifford algebra.

Can we get more by relaxing associativity?

¿Octonions?

Prather

Scalar Product Spaces

Clifford Algebras

Composistion Algebras

Hurwitz’ Theorem

Summary

Hurwitz’ Theorem

Hurwitz Theorem (1923)

The only positive definite composition algebras over R are R, C, H and O.

Adolf Hurwitz

Yes! We get precisely the octonions, O.

This can be generalized as follows: A non-degenerate compos