Algebras generalizing the Octonions

Algebras generalizing the Octonions

Marie Kreusch

Poster presented at the Topical Workshop "Integrability and Cluster Algebras: Geometry and Combinatorics" at ICERM, US, 25 – 29 August, 2014

Overview

R

C

H

O

. . .

Cl

_p,q

O

_p,q

Cayley-Dickson Algebras Clifford Algebras Algebras Op,q

Applications to the classical Hurwitz problem of "Sums of Squares".

Main Definition

The algebra O_p,q (p + q = n ≥ 3) is the 2n-dimensional vector space on R with the basis {u_x : x ∈ Zn₂}, and equipped with the product

u_x · u_y = (−1)fOp,q(x ,y )u_{x +y} for all x , y ∈ Zn₂, where

f_Op,q (x , y ) = X 1≤i<j<k ≤n (x_ix_jy_k + x_iy_jx_k + y_ix_jx_k) + X 1≤i≤j≤n x_iy_j + X 1≤i≤p x_i.

In particular, the algebra O_p,q is a twisted group algebra, noted (R[Zn₂], f_Op,q).

Generating Cubic Form

The defect of commutativity and associativity of a twisted algebra (R[Zn₂], f ) is measured by a symmetric function β : Zn₂ × Zn₂ → Z₂, and a function φ : Zn₂ × Zn₂ × Zn₂ → Z₂, respectively

u_x · u_y = (−1)β(x ,y ) u_y · u_x,

u_x · (u_y · u_z) = (−1)φ(x ,y ,z) (u_x · u_y) · u_z.

A function α : Zn₂ −→ Z₂ is called a generating function for (R[Zn₂], f ), if for x , y and z in Zn₂, (i) f (x , x ) = α(x ),

(ii) β(x , y ) = α(x + y ) + α(x ) + α(y ),

(iii) φ(x , y , z) = α(x + y + z) + α(x + y ) + α(x + z) + α(y + z) + α(x ) + α(y ) + α(z).

Theorem [M-G & O] :

I A twisted algebra (R[Zn₂], f ) has a generating function iff φ is symmetric. I The generating function α is a polynomial on Zn₂ of degree ≤ 3.

I Given any polynomial α on Zn₂ degree ≤ 3, there exists a unique (up to isomorphism)

twisted group algebra (R[Zn₂], f ) having α as generating function.

Algebras

O

_p,q

and Clifford Algebras

The algebras O_p,q (as well as the Clifford algebras) have generating cubic forms (res. quadratic forms). They are given by

α_p,q(x ) = f_Op,q(x , x ) = α_n(x ) + X 1≤i≤p x_i, where α_n(x ) = X 1≤i<j<k ≤n x_ix_jx_k + X 1≤i<j≤n x_ix_j + X 1≤i≤n x_i, for all x = (x₁, . . . , x_n) ∈ Zn₂.

The Hamming weight of x is |x | := #{x_i 6= 0}, and one has

α_n(x ) = (

0, if |x | ≡ 0 mod 4, 1, otherwise.

For q 6= 0, O_p,q contains a Clifford subalgebra Cl_p,q−1 ⊂ O_p,q which basis is given by {u_x : x ∈ (Z₂)n, IxI ≡ 0 mod2}

Classification

Theorem [1] :

I If pq 6= 0, then one has

Op,q ' Oq,p

Op,q+4 ' Op+4,q

)

⇔ Cl_p,q−1 ' Cl_p0_,q0₋₁

I For n ≥ 5, the algebras O_n,0 and O_0,n are exceptional.

Bott Periodicity

Theorem [2] : If n = p + q ≥ 3 and pq 6= 0, then

I O_0,n+4 ' O_0,n ⊗_C O_0,5 Cl_p,q+2 ' Cl_q,p ⊗_R Cl_0,2 Cl_p+2,q ' Cl_q,p ⊗_R Cl_2,0 Cl_p+1,q+1 ' Cl_p,q ⊗_R Cl_1,1 On+4,0 ' On,0 ⊗R⊕R O5,0 I O_p+2,q+2 ' O_p,q ⊗_C O_2,3 ' O_p,q ⊗_R⊕R O_3,2

Summary Table of Classification of Algebras

O

_p,q

O

9,0

O

8,1

O

7,2

O

6,3

O

5,4

O

4,5

O

3,6

O

2,7

O

1,8

O

0,9

O

8,0

O

7,1

O

6,2

O

5,3

O

4,4

O

3,5

O

2,6

O

1,7

O

0,8

O

7,0

O

6,1

O

5,2

O

4,3

O

3,4

O

2,5

O

1,6

O

0,7

O

6,0

O

5,1

O

4,2

O

3,3

O

2,4

O

1,5

O

0,6

O

5,0

O

4,1

O

3,2

O

2,3

O

1,4

O

0,5

O

4,0

O

3,1

O

2,2

O

1,3

O

0,4

O

3,0

O

2,1

O

1,2

O

0,3

M.Kreusch@ulg.ac.be

[1] Kreusch & Morier Genoud, Classification of the algebras

O

_p,q