On Ternary Clifford Algebras on Two Generators Defined by Extra-Special 3-Groups of Order 27

The main objective of this work is to show how to construct a ternary Z3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {Z}}_3$$\end{document}-graded Clifford algebra on two generators by using a group algebra of an extra-special 3-group G of order 27. The approach used is an extension of the method implemented to classify Z2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {Z}}_2$$\end{document}-graded Clifford algebras as images of group algebras of Salingaros 2-groups [2]. We will show how non-equivalent irreducible representations of the Z3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {Z}}_3$$\end{document}-graded Clifford algebra are determined by two distinct irreducible characters of G of degree 3. We comment on applying this approach to defining p-ary Clifford-like algebras on two generators and finding their irreducible representations on the basis of extra-special p-groups of order p3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p^3$$\end{document} for p>3.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p > 3.$$\end{document} Finally, we will comment on possibly using this approach to define p-ary Clifford-like algebras on three and more generators by using group central products and their group algebras.