A Classification of Clifford Algebras as Images of Group Algebras of Salingaros Vee Groups

Abstract

The main objective of this work is to prove that every Clifford algebra $$C \ell _{p,q}$$ is $$\mathbb {R}$$ -isomorphic to a quotient of a group algebra $$\mathbb {R}[G_{p,q}]$$ modulo an ideal $$\mathcal {J}=(1+\tau )$$ where $$\tau $$ is a central element of order 2. Here, $$G_{p,q}$$ is a 2-group of order $$2^{p+q+1}$$ belonging to one of Salingaros isomorphism classes $$N_{2k-1},$$ $$N_{2k},$$ $$\Omega _{2k-1},$$ $$\Omega _{2k}$$ or $$S_k$$ . Thus, Clifford algebras $$C \ell _{p,q}$$ can be classified by Salingaros classes. Since the group algebras $$\mathbb {R}[G_{p,q}]$$ are $$\mathbb {Z}_2$$ -graded and the ideal $$\mathcal {J}$$ is homogeneous, the quotient algebras $$\mathbb {R}[G]/\mathcal {J}$$ are $$\mathbb {Z}_2$$ -graded. In some instances, the isomorphism $$\mathbb {R}[G]/\mathcal {J}\cong C \ell _{p,q}$$ is also $$\mathbb {Z}_2$$ -graded. By Salingaros’ Theorem, the groups $$G_{p,q}$$ in the class $$N_{2k-1}$$ are iterative central products of k copies of the dihedral group $$D_8$$ while the groups in the class $$N_{2k}$$ are iterative central products of $$k-1$$ copies of the dihedral group $$D_8$$ and one copy of the quaternion group $$Q_8$$ , and so they are extra-special. The groups $$G_{p,q}$$ in the classes $$\Omega _{2k-1}$$ and $$\Omega _{2k}$$ are central products of $$N_{2k-1}$$ and $$N_{2k}$$ with $$C_2 \times C_2$$ , respectively, while the groups in the class $$S_k$$ are central products of $$N_{2k-1}$$ or $$N_{2k}$$ with $$C_4$$ . Two algorithms to factor any $$G_{p,q}$$ into an internal central product, depending on the class, are given. A complete table of central factorizations for groups of order up to 1, 024 is presented.