Representations of Clifford algebras of binary and ternary cubic forms (Q1579070)

\textit{D. E. Haile} [Am. J. Math. 106, 1269-1280 (1984; Zbl 0585.13001)], \textit{Ph. Revoy} [Commun. Algebra 11, 1877-1891 (1983; Zbl 0526.15019)] and \textit{Ph. Revoy} and \textit{S. Tesser} [C.R. Acad. Sci., Paris, Sér. I. 318, No. 6, 509-514 (1994; Zbl 0818.11021)] have studied binary and ternary cubic forms which generalize Clifford algebras generated by quadratic forms. Clifford algebras of polynomial forms of degree \(d>2\) were introduced by N. Roby. In this article the author considers special binary cubic forms such as \(f=xy(x+y)\) and \(f=xy^2\). He describes explicitly a corresponding irreducible family of representations in the matrix algebra \(M_3(K)\) \((K\) is a base field) which only depend on one parameter. Associated Weierstrass functions help to represent the results in case of \(K=\mathbb{C}\). It is shown in a direct way that the number of 3D-representations of a cubic form is finite and different from zero.