Algebras generalizing the OctonionsMarie 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 . . .

Clp,q Op,q

Cayley-DicksonAlgebras

CliffordAlgebras

AlgebrasOp,q

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

Main DefinitionThe algebra Op,q (p + q = n ≥ 3) is the 2n-dimensional vector space on R withthe basis {ux : x ∈ Zn

2}, and equipped with the product

ux · uy = (−1)fOp,q(x ,y)ux+y

for all x , y ∈ Zn2, where

fOp,q (x , y) =∑

1≤i<j<k≤n

(xixjyk + xiyjxk + yixjxk) +∑

1≤i≤j≤n

xiyj +∑

1≤i≤p

xi.

In particular, the algebra Op,q is a twisted group algebra, noted (R[Zn2], fOp,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 × Zn

2 → Z2, and a function φ : Zn2 × Zn

2 × Zn2 → Z2, respectively

ux · uy = (−1)β(x ,y) uy · ux ,

ux · (uy · uz) = (−1)φ(x ,y ,z) (ux · uy) · uz.

A function α : Zn2 −→ Z2 is called a generating function for (R[Zn

2], 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 Op,q and Clifford Algebras

The algebras Op,q (as well as the Clifford algebras) have generatingcubic forms (res. quadratic forms). They are given by

αp,q(x) = fOp,q(x , x) = αn(x) +∑

1≤i≤p

xi,

whereαn(x) =

∑1≤i<j<k≤n

xixjxk +∑

1≤i<j≤n

xixj +∑

1≤i≤n

xi,

for all x = (x1, . . . , xn) ∈ Zn2.

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

αn(x) =

{0, if |x | ≡ 0 mod 4,1, otherwise.

For q 6= 0, Op,q contains a Clifford subalgebra Clp,q−1 ⊂ Op,q whichbasis is given by {ux : x ∈ (Z2)

n, IxI ≡ 0 mod2}

ClassificationTheorem [1] :

I If pq 6= 0, then one has

Op,q ' Oq,p

Op,q+4 ' Op+4,q

}⇔ Clp,q−1 ' Clp′,q′−1

I For n ≥ 5, the algebras On,0 and O0,n are exceptional.

Bott Periodicity

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

I O0,n+4 ' O0,n ⊗C O0,5 Clp,q+2 ' Clq,p ⊗R Cl0,2Clp+2,q ' Clq,p ⊗R Cl2,0

Clp+1,q+1 ' Clp,q ⊗R Cl1,1

On+4,0 ' On,0 ⊗R⊕R O5,0

I Op+2,q+2 ' Op,q ⊗C O2,3

' Op,q ⊗R⊕R O3,2

Summary Table of Classification of Algebras Op,q

O9,0 O8,1 O7,2 O6,3 O5,4 O4,5 O3,6 O2,7 O1,8 O0,9

O8,0 O7,1 O6,2 O5,3 O4,4 O3,5 O2,6 O1,7 O0,8

O7,0 O6,1 O5,2 O4,3 O3,4 O2,5 O1,6 O0,7

O6,0 O5,1 O4,2 O3,3 O2,4 O1,5 O0,6

O5,0 O4,1 O3,2 O2,3 O1,4 O0,5

O4,0 O3,1 O2,2 O1,3 O0,4

O3,0 O2,1 O1,2 O0,3

M.Kreusch@ulg.ac.be [1] Kreusch & Morier Genoud, Classification of the algebras Op,qUniversité de Liège (Belgique) [2] Kreusch, Bott type periodicity for higher octonions