The canonical quadratic pair on a Clifford algebra and triality (Q2040188)

Let \((A,\sigma)\) be a central simple algebra with involution over a field of characteristic not \(2\). Then one can construct the so-called Clifford algebra \(C(A,\sigma)\) of \(A\) that is itself equipped with a canonical involution that depends on \(\sigma\). This Clifford algebra with involution proves useful in many ways. For example, the Clifford algebra with its canonical involution is a complete invariant for the classification of orthogonal involutions with trivial discriminant on degree \(8\) algebras, and it can also be used to express the important trialitarian properties of such degree \(8\) algebras. In characteristic \(2\), the situation is more complicated. It turns out that the notion of involution is not subtle enough to capture many important properties that hold in characteristic not \(2\), one reason being that the notions of symmetric bilinear forms and of quadratic forms are no longer equivalent. So one considers algebras equipped with a so-called quadratic pair, the latter being, roughly speaking, a refined version of an involution. Again, one has a Clifford algebra of such an algebra with quadratic pair, and one obtains a canonical involution on the Clifford algebra, but hitherto there has not been an adequate notion of canonical quadratic pair of the Clifford algebra. In the present paper, the authors fill this gap. This allows them to get characteristic \(2\) versions of many notions and results that have been known before only in characteristic not \(2\). For example, they develop the concept of a trialitarian triple of degree \(8\) central simple algebras with quadratic pairs and isomorphism classes of such triples over a field \(F\), the latter being in canonical bijection with \(H^1(F, \mathrm{PGO}_8^+)\). Also, there is a natural action of \(A_3\) on such triples. Using this action, they deduce that a degree \(8\) algebra with quadratic pair is totally decomposable if and only if it has trivial discriminant and its Clifford algebra has a split factor, and they obtain structure results for trialitarian triples in the case where at least two of the algebras have Schur index at most \(2\). In an appendix, the authors provide a construction of a canonical semi-trace on the Clifford algebra of a nonsingular quadratic form in characteristic \(2\).