This paper is of interest for experts in noncommutative higher-dimensional algebra. It treats various aspects of the Clifford algebra bundle of the space of quadratic forms. The introduction provides an overview of existing literature, the general setup and the definition of tensor algebras, with special emphasis on N. Bourbaki’s work on sesquilinear and quadratic forms [Éléments de mathématique. XXIV. Part. 1: Les structures fondamentales de l’analyse. Livre II: Algèbre. Chap. 9: Formes sesquilinéaires et formes quadratiques. Paris: Hermann & Cie (1959; Zbl 0102.25503)].
Section 2 on Clifford algebras defines quadratic forms, bilinear forms in characteristic 2, and the action of bilinear forms on quadratic forms. After the definition of Clifford algebras, their universality property is established. Section 3 focuses on operations within and between Clifford algebras, and comments on the representation of Clifford algebras on exterior algebras (zero quadratic form). It further clarifies the roles of automorphisms and deformations in the Clifford algebra bundle. Bilinear forms \(F\), \(Q'(x) = Q(x)+F(x,x)\) linearly map fibers onto fibers \(\mathrm{Cl}(Q) \rightarrow \mathrm{Cl}(Q')\), and alternating forms define vertical automorphisms, mapping every fiber into itself. They are also called gauge transformations.
Finally, with field characteristic \(\neq 2\) the map \(\mathrm{Cl}(Q) \rightarrow \bigwedge (V)\) from a Clifford algebra with quadratic form to an exterior algebra over the same vector space \(V\) is called symbol map and its inverse is the quantization map. The work contains many helpful footnotes and detailed references to the related literature.