Abstract :
The Grassmann envelope is a useful tool for deciding varietal questions about supersystems . It is not useful in deciding simplicity, since the envelope is always fraught with trivial ideals coming from trivial superscalars. At first glance it would not seem useful in deciding superspeciality either, but we will show that it is a more sensitive tool than it seems. We say a Jordan supersystem Js (algebra, triple, or pair) is Grassmann special if its Grassmann envelope is special as an ordinary Jordan system over . Certainly superspeciality Js As+ over Φ implies Grassmann speciality Γ(Js) Γ(As)+ over , but it is not obvious how Grassmann speciality influences superspeciality of Js. Nevertheless, we show how to transform obstacles to speciality of Js over Φ (anti-superspecial elements) into obstacles to speciality of Γ(Js) over (anti-special elements) using Grassmann boosters, so that non-superspeciality Js over Φ implies non-speciality of Γ(Js) over . Thus Grassmann speciality is the same as superspeciality: a supersystem J is a superspecial Jordan supersystem over Φ iff Γ(Js) is a special Jordan system over . (It is not clear this is the same as speciality of Γ(Js) over Φ, since in general speciality of quadratic Jordan systems depends on the scalars: we give an example of a Jordan Ω-algebra which is not special, but is special over a scalar subring Φ.) As a corollary, any Jordan superalgebra with zero linear extreme radical which is viably evenly 4-interconnected is superspecial; thus exceptional superalgebras must be built over even parts of degree 3.
