Clifford algebras and geometric algebra
Let \(R^{p,q}\) be the universal Clifford algebra associated to a real vector space \(\mathbb{R}^n\), \(n=p+q\), equipped with a nondegenerated symmetric bilinear form \(B\) of signature \((p-q)\). Let \({\mathfrak G}\) be the infinite dimensional geometric algebra as introduced by \textit{D. Hestenes} and \textit{G. Sobczyk} [Clifford algebra to geometrical calculus. A unified language for mathematics and physics (1984; Zbl 0541.53059)]. Let \({\mathfrak G} (A_n)\) be the subalgebra generated by an \(n\)-blade \(A_n\), i.e., the algebra generated by all products and sums of vectors in \({\mathfrak G}^1 (A_n)= \{a\in {\mathfrak G}^1 \mid a\wedge A_n=0\}\), where \({\mathfrak G}^1\) is the 1-vector subspace of \({\mathfrak G}\). Then, by assuming (i) axiom 7 as: for every non-zero vector \(a\in {\mathfrak G}^1\) we have \(a^2= \langle a^2 \rangle_0 \in R\) and (ii) that there exists \(n\) linearly independent vectors \(e_1, \dots, e_p\), \(e_{p+1}, \dots, e_q\), \(e_i\in {\mathfrak G}^1 (A_n)\), \(e^1_i=+1\) for \(i=1, \dots, p\), \(e^2_i=-1\) for \(i=p+1, \dots, q\), the authors prove that \(R^{p,q} \simeq {\mathfrak G} (A_n)\). Here we would like to recall that a proof that \({\mathfrak G}^1 (A_n) \simeq R^{p,q}\) can also be given by using axiom 7 as: for every \(a\in {\mathfrak G}^1\) we have \(a^2= \langle a^2 \rangle_0>0\). In this case the bilinear form of signature \((p-q)\) can be constructed in \({\mathfrak G} (A_n)\) by using the relations between metric tensors and symmetries as discussed in section 3.7 of Hestenes and Sobczyk's book [loc. cit.].
scientific article; zbMATH DE number 1156795
scientific article