This is a good start, as one of the defining properties of Jordan algebras is
that they are commutative. The other property is that the identity
(x
So the algebra is in fact a Jordan algebra (which was
clear by construction).
We finally have a look at the structure constants.
We construct the real Cayley algebra, which is a non-associative algebra of
dimension 8, containing 7 quaternion algebras. If the basis elements are
labelled 1, ..., 8 and 1 corresponds to the identity, these quaternion
algebras are spanned by
{ 1, (n + 1) mod 7 + 2, (n + 2) mod 7 + 2, (n + 4) mod 7 + 4 }, where
0 ≤n ≤6.
We first define a function, which, given three indices i, j, k constructs a
sequence with the structure constants for the quaternion algebra spanned by
1, i, j, k in the quadruple notation.
> quat := func<i,j,k | [<1,1,1, 1>, <i,i,1, -1>, <j,j,1, -1>, <k,k,1, -1>,
> <1,i,i, 1>, <i,1,i, 1>, <1,j,j, 1>, <j,1,j, 1>, <1,k,k, 1>, <k,1,k, 1>,
> <i,j,k, 1>, <j,i,k, -1>, <j,k,i, 1>, <k,j,i, -1>, <k,i,j, 1>, <i,k,j, -1>]>;
We now define the sequence of non-zero structure constants for the Cayley
algebra using the function
quat. Some structure constants are defined
more than once and we have to get rid of these when defining the algebra.
> con := &cat[quat((n+1) mod 7 +2, (n+2) mod 7 +2, (n+4) mod 7 +2):n in [0..6]];
> C := Algebra< Rationals(), 8 | Setseq(Set(con)) >;
> C;
Algebra of dimension 8 with base ring Rational Field
> IsAssociative(C);
false
> IsAssociative( sub< C | C.1, C.2, C.3, C.5 > );
true
The integral elements in this algebra are those where either all coefficients
are integral or exactly 4 coefficients lie in 1/2 + Z in positions
i
1, i
2, i
3, i
4, such that i
1, i
2, i
3, i
4 are a basis of one of the
7 quaternion algebras or a complement of such a basis. These elements are
called the integral Cayley numbers and form a Z-algebra. The units in this
algebra are the elements with either one entry ∓ 1 and the others 0 or
with 4 entries ∓ 1/2 and 4 entries 0, where the non-zero entries are in
the positions as described above. This gives 240 units and they form
(after rescaling with Sqrt(2)) the roots in the root lattice of type E
8.
> a := (C.1 - C.2 + C.3 - C.5) / 2;
> MinimalPolynomial(a);
$.1^2 - $.1 + 1
> MinimalPolynomial(a^-1);
$.1^2 - $.1 + 1
> MinimalPolynomial(C.2+C.3);
$.1^2 + 2
> MinimalPolynomial((C.2+C.3)^-1);
$.1^2 + 1/2
Tensoring the integral Cayley algebra with a finite field gives a finite
Cayley algebra. As the Z-algebra generated by the chosen basis for C has
index 2
4 in the full integral Cayley algebra, we can get the finite
Cayley algebras by applying the
ChangeRing function for finite fields
of odd characteristic.
The Cayley algebra over GF(q) has the simple group G
2(q) as its
automorphism group. Since the identity has to be fixed, every automorphism
is determined by its image on the remaining 7 basis elements. Each of
these has minimal polynomial x
2 + 1, hence one obtains a permutation
representation of G
2(q) on the elements with this minimal polynomial.
As ∓-pairs have to be preserved, this number can be divided by 2.
> C3 := ChangeRing( C, GF(3) );
> f := MinimalPolynomial(C3.2);
> f;
$.1^2 + 1
> #C3;
6561
> time Im := [ c : c in C3 | MinimalPolynomial(c) eq f ];
Time: 3.099
> #Im;
702
> C5 := ChangeRing( C, GF(5) );
> f := MinimalPolynomial(C5.2);
> f;
$.1^2 + 1
> #C5;
390625
> time Im := [ c : c in C5 | MinimalPolynomial(c) eq f ];
Time: 238.620
> #Im;
15750
In the case of the Cayley algebra over GF(3) we obtain a permutation
representation of degree 351, which is in fact the smallest possible degree
(corresponding to the representation on the cosets of the largest maximal
subgroup U
3(3):2).
Over GF(5), the permutation representation is of degree 7875, corresponding
to the maximal subgroup L
3(5):2, the smallest possible degree being 3906.
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