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Clifford algebras are associative structure constant algebras and
therefore the intrinsics in chapters on structure constant algebras
and associative algebras may be used with Clifford algebra arguments.
The homogeneous component of degree k of the Clifford algebra
element v.
> F := GF(5);
> C,V,f := CliffordAlgebra(IdentityMatrix(F,4));
> v := (f(V.1)*f(V.2)+3*f(V.2))*(f(V.3)+f(V.4));
> AsPolynomial(HomogeneousComponent(v,2));
3*e1*e3 + 3*e2*e4
The even subalgebra C_ + of the Clifford algebra C. This is
the algebra of fixed points of the main involution. The second
return value of this function is the canonical embedding of C_ + in C.
A (generalised) quaternion algebra can also be realised as the even
subalgebra of a Clifford algebra.
> F<a,b> := RationalFunctionField(Rationals(),2);
> Q := DiagonalMatrix(F,[1,-a,-b]);
> C,V,f := CliffordAlgebra(Q);
> E, h := EvenSubalgebra(C);
> i := E.2;
> j := E.3;
> i^2;
(a 0 0 0)
> j^2;
(b 0 0 0)
> i*j eq -j*i;
true
Center(C) : AlgClff -> AlgAss, Map
The centre of the Clifford algebra C. The second return value
is the embedding in C.
The following examples illustrate the fact that over a finite field F
a Clifford algebra C of a non-degenerate quadratic form Q is either a
simple algebra or the direct sum of two simple algebras. Furthermore,
the same is true of its even subalgebra E. Let V denote the quadratic
space of Q.
If the dimension of V is 2m and the Witt index of Q is m,
then C is a central simple algebra.
> F := GF(3);
> Q := StandardQuadraticForm(6,F);
> C,V,f := CliffordAlgebra(Q);
> WittIndex(V);
3
> IsSimple(C);
true
> #Centre(C);
3
The even subalgebra E of C is the direct sum of two simple ideals
E(1 - z) and E(1 + z), where z2 = 1 and z anticommutes with every
element of V.
> E,h := EvenSubalgebra(C);
> IsSimple(E);
false
> #MinimalIdeals(E);
2
> Z := Centre(E); Z;
Associative Algebra of dimension 2 with base ring GF(3)
> #{ z : z in Z | IsUnit(z) };
4
> exists(z){ z : z in Z | z^2 eq One(E) and
> forall{ v : v in V | f(v)*h(z) eq - h(z)*f(v) } };
true
> E1 := ideal< E | 1-z >;
> IsSimple(E1);
true
> E2 := ideal< E | 1+z >;
> IsSimple(E2);
true
If dim V = 2m and the Witt index of Q is m - 1, the even subalgebra of
C is a simple algebra whose centre is a quadratic extension of F.
> F := GF(3);
> Q := StandardQuadraticForm(6,F : Minus);
> C,V,f := CliffordAlgebra(Q);
> WittIndex(V);
2
> IsSimple(C);
true
> #Centre(C);
3
> E := EvenSubalgebra(C);
> IsSimple(E);
true
> Z := Centre(E); Z;
Associative Algebra of dimension 2 with base ring GF(3)
> #{ z : z in Z | IsUnit(z) };
8
If dim V = 2m + 1 and the Witt index of Q is m, the even subalgebra
is central simple and C is the either simple or the direct sum of two simple
algebras.
> F := GF(3);
> Q := StandardQuadraticForm(5,F);
> C,V,f := CliffordAlgebra(Q);
> WittIndex(V);
2
> IsSimple(C);
false
> Z := Centre(C); Z;
Associative Algebra of dimension 2 with base ring GF(3)
> #{ z : z in Z | IsUnit(z) };
4
> E := EvenSubalgebra(C);
> IsSimple(E);
true
In this example the Clifford algebra is the direct sum of two simple
ideals. But it is possible that the Clifford algebra of a scalar multiple
of Q is a simple algebra over a quadratic extension of the base field.
> C,V,f := CliffordAlgebra(2*Q);
> IsSimple(C);
true
> #Z,#{ z : z in Centre(C) | IsUnit(z) };
9 8
The non-zero elements of Z are invertible and hence Z is a field, namely
GF(9).
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