Clifford Algebra Structure#
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.
- HomogeneousComponent(v, k): AlgClffElt, RngIntElt -> AlgClffElt#
The homogeneous component of degree \(k\) of the Clifford algebra element \(v\).
- Example: Hom Comp (ex-40ad8c)#
> 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
- EvenSubalgebra(C : parameters): AlgClff -> AlgAss, Map#
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 \(C\).
- Example: Cliff Gen Quat Ex (ex-521a9d)#
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
- Centre(C): AlgClff -> AlgAss, Map#
- 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 \(Q\).
- Example: Even Dim Plus (ex-d39588)#
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 \(z^2 = 1\) 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
- Example: Even Dim Minus (ex-04b7d8)#
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
- Example: Odd Dim (ex-a8b683)#
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 \({\bf F}_{9}\).