Kanzaki’s Generalized Quadratic Spaces and Graded Salingaros Groups

Abstract

In 1973, Kanzaki introduced a category $\text {Quad}_2(K)$ made of generalized quadratic spaces (E, f, g) where E was a vector space of finite dimension over a field K, f a linear form on E, and g a quadratic form on E such that the bilinear form $(a,b)\longmapsto f(a)f(b)+g(a+b)-g(a)-g(b)$ was non-degenerate on E. There are direct sums and tensor products in $\text {Quad}_2(K)$, and its objects are classified by a Witt ring $\text {W}_2(K)$. On another side, Salingaros groups have been studied as subsets of real Clifford algebras; and the parity gradation of those algebras leads to the concept of graded Salingaros group. This article shows that, up to isomorphy, the graded Salingaros groups are in bijection with the objects (E, f, g) of $\text {Quad}_2({\mathbb {Z}}/2{\mathbb {Z}})$ such that $f\not = 0$; therefore, they are classified by the Witt ring $\text {W}_2({\mathbb {Z}}/2{\mathbb {Z}})$ isomorphic to ${\mathbb {Z}}/8{\mathbb {Z}}$. As subsets of real Clifford algebras, they are also classified by the Brauer–Wall group $\text {BW}({\mathbb {R}})$; and thus they account for the isomorphism $\text {W}_2({\mathbb {Z}}/2{\mathbb {Z}})\cong \text {BW}({\mathbb {R}})$. The disclosure of those facts is made easier by the solution of an abstract enumeration problem that emerges in all contexts that are here under consideration.