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

  • Rafał Abłamowicz
  • Manisha Varahagiri
  • Anne Marie Walley
Article
Part of the following topical collections:
  1. Proceedings ICCA 11, Ghent, 2017

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.

Keywords

2-group Central product Clifford algebra Cyclic group Elementary abelian group Extra-special group \(\mathbb {Z}_2\)-graded algebra Graded algebra morphism Group algebra Homogeneous ideal Quotient algebra Salingaros vee group 

Mathematics Subject Classification

Primary 15A66 16W50 20C05 20C40 20D15 Secondary 68W30 

Notes

Acknowledgements

The authors are very grateful to Dr. Padmini Veerapen (TTU) and two anonymous reviewers for their very helpful comments on the mathematical content of this paper as well as for their suggestions on how to improve our presentation.

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsTennessee Technological UniversityCookevilleUSA

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