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 Z(3)-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 Z(2)-graded Clifford algebras as images of group algebras of Salingaros 2-groups [2]. We will show how non-equivalent irreducible representations of the Z(3)-graded Clifford algebra are determined by two distinct irreducible characters of G of degree 3. We comment on applying this approach to defining pary Clifford-like algebras on two generators and finding their irreducible representations on the basis of extra-special p-groups of order p(3) for p > 3. 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.