Composition of quadratic forms and tensor product of quaternion algebras
Let C be the Clifford algebra of a quadratic space (U,\(\sigma)\) and let J be the involution of C such that \(x^ J=-x\) for \(x\in U\). The existence of composition laws for quadratic forms is related to the existence of C- modules W with nonsingular bilinear forms b such that \(b(cx,y)=b(x,c^ Jy).\) Such forms are called admissible. In this paper, the author describes all such modules W for a given quadratic space (U,\(\sigma)\). After studying simple C-modules, he discusses the existence of admissible forms b. In particular it is shown that the existence of symmetric admissible forms on a C-module W depends only on the dimensions of U and W and on the signed determinant of \(\sigma\) (if dim U is odd). Conditions on \(\sigma\) are given which imply that all symmetric admissible forms are hyperbolic. This paper is related to the work of \textit{D. B. Shapiro} [J. Algebra 46, 148-170, 171-181 (1977; Zbl 0358.15024, Zbl 0358.15025) and \textit{A. R. Wadsworth} and \textit{D. B. Shapiro} [ibid. 46, 182-188 (1977; Zbl 0358.15026)]. As a last result, the author proves a special case of a conjecture of Shapiro, stating that symmetric admissible forms are similar to Pfister forms.
scientific article; zbMATH DE number 3920573
scientific article