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Clifford Algebras and their Elements

Clifford algebras are represented in Magma as structure constant algebras and so many of the functions described in Chapter ALGEBRAS apply to Clifford algebras. The Magma type of a Clifford algebra is AlgClff and all Clifford algebras have the attributes

space: the quadratic space from which the Clifford is derived;
embedding: the standard embedding of the quadratic space into the Clifford algebra;
mainInvolutionMatrix: the matrix of the antiautomorphism of the Clifford algebra that reverses the multiplication.

Let C be the Clifford algebra of the quadratic form Q defined on the vector space V. If e1, e2, ..., en is a basis for V, a basis for C is the set of all products e1i1e2i2 ... vnin, where ik is 0 or 1 for all k. The function k |-> ik is the characteristic function of a subset of {1, 2, ..., n}, namely S = {k | ik eq 1 }. The map S |-> 1 + ∑k∈S 2k - 1 is a bijection between the subsets of {1, 2, ..., n} and the integers in the interval [1 ... 2n].

Thus the elements of C can be represented by a sequence of pairs < S, a > where S is a subset of {1, 2, ..., n} and a is a field element. Multiplication is determined by the fact that for all u, v∈V we have

v2 = Q(v).1quadand uv + vu = β(u, v).1,

where βis the polar form of Q.

Subsections
CliffordAlgebra(Q) : AlgMatElt -> AlgClff, ModTupFld, Map
This function returns a triple C, V, f, where C is the Clifford algebra of the quadratic form Q, V is the quadratic space of Q, and f is the standard embedding of V into C.
CliffordAlgebra(V) : ModTupFld -> AlgClff, Map
If V is a quadratic space with quadratic form Q, this function returns the pair C, f, where C is the Clifford algebra of Q and f is the standard embedding of V into C.

Elements of a Clifford Algebra

elt< C | r1, r2, ..., rm > : AlgClff, RngElt, RngElt, ..., RngElt -> AlgClffElt
Given a Clifford algebra C of dimension m = 2n over a field F, and field elements r1, r2, ..., rm ∈F construct the element r1 * C.1 + r2 * C.2 + ... + rm * C.m of C.
C ! L : AlgClff, SeqEnum[RngElt] -> AlgGenElt
Given a Clifford algebra C of dimension m = 2n and a sequence L = [r1, r2, ..., rm] of elements of the base ring R of C, construct the element r1 * C.1 + r2 * C.2 + ... + rm * C.m of C.
BasisProduct(A, i, j) : AlgClff, RngIntElt, RngIntElt -> AlgGenElt
Return the product of the i-th and j-th basis element of the Clifford algebra C.
BasisElement(C, L) : QlgClff, SetEnum -> AlgClffElt
The basis element C.j of the Clifford algebra C corresponding to the subset L of {1, 2, ..., n} where j = 1 + ∑k∈L2k - 1. If e1, e2, ..., en is the standard basis for the vector space on which C is based, this corresponds to the product ei1 * ei2 * ... * eih, where L = {i1, i2, ..., ih} and i1 < i2 < ... < ih.
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Version: V2.19 of Wed Apr 24 15:09:57 EST 2013