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.

中文翻译：

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神崎的广义二次空间和分级Salingaros群

1973年，Kanzaki引入了一个类别\（\ text {Quad} _2（K）\），该类别由广义二次空间（E，  f，  g）构成，其中E是一个在域K上有限维的矢量空间，f为线性形式上Ë，和克上的二次形式Ë使得双线性形式\（（A，b）\ longmapsto F（A）F（b）+ G（A + b）-g（一）-g（b）\ ）在E上未退化。\（\ text {Quad} _2（K）\）中有直接和和张量积，其对象由Witt环\（\ text {W} _2（K）\）进行分类。另一方面，已将Salingaros群作为真实Clifford代数的子集进行了研究。这些代数的奇偶级配导致了Salingaros分组的概念。本文示出的是，高达isomorphy，渐变Salingaros基团是在双射与所述对象（ê，  ˚F， 克的）\（\文本{四} _2（{\ mathbb {Z}} / 2 {\ mathbb {Z }}）\）这样\（f \ not = 0 \） ; 因此，它们通过Witt环\（\ text {W} _2（{\ mathbb {Z}} / 2 {\ mathbb {Z}}）\）分类为\（{\ mathbb {Z}} / 8 {\ mathbb {Z}} \）。作为真实Clifford代数的子集，它们也按Brauer–Wall组\（\ text {BW}（{\ mathbb {R}}）\）分类。; 因此它们解释了同构\（\ text {W} _2（{\ mathbb {Z}} / 2 {\ mathbb {Z}}）\ cong \ text {BW}（{\ mathbb {R}}）\ ）。通过解决在此处考虑的所有情况下出现的抽象枚举问题，可以更轻松地公开这些事实。