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.