Grassmann speciality of Jordan supersystems
The Grassmann envelope G(S) = S0tensor G0 +
S1tensor G1 is a useful tool for
deciding varietal questions about supersystems S = S0 + S1. 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 (algebra, triple, or
pair) is Grassmann special if its Grassmann envelope is special as an
ordinary Jordan system over G0. Certainly superspeciality, Js a subset of
As+ over Phi implies Grassmann speciality, G(Js) a subset of G(As+)
over G0, but it is not obvious how Grassmann speciality influences
superspeciality of Js. Nevertheless, we show how to transform obstacles to
speciality of Js over Phi (anti-superspecial elements) into obstacles to
speciality of G(Js) over G0 (anti-special elements) using Grassmann boosters,
so that non-superspeciality of Js over Phi implies non-speciality of G(S)
over G0. Thus Grassmann speciality is the same as superspeciality: a
supersystem Js is a superspecial Jordan supersystem over Phi iff G(Js) is a
special Jordan system over G0. [It is not clear this is the same as speciality
of G(Js) over Phi, since in general speciality of quadratic Jordan systems
depends on the scalars: we give an example of a Jordan Omega-algebra which is
not special, but is special over a scalar subring Phi.] 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 at most 3.
(This paper has appeared in J. Alg. 218 (2004), 3 -- 31)
K. McCrimmon < kmm4m@virginia.edu >