The splittest Kac superalgebra K10
We present an isotope of the usual version of the Kac 10-dimensional Jordan
superalgebra over a general ring of scalars (isomorphic to the original
version when 1/2 and -1 have square roots, but not in characteristic 2),
which we take as the ``correct'' split model for the simple superalgebra in
all characteristics. This has unit the sum of three reduced orthogonal
idempotents, with even part the direct sum of a 1-dimensional ideal and the
Jordan algebra of a split quadratic form. In characteristic 2 the usual
version has the property that in the quadratic form part all squares are
scalar multiples of 1 and there are no proper idempotents. In the structure
theory for quadratic Jordan algebras this is considered an aberrant case:
the ``standard'' degree-2 algebra has unit a sum of two reduced orthogonal
idempotents, and the traceless form arises as a (non-isomorphic) isotope of
this standard form.
We exhibit a "quaternionic'' model of the bimodule structure of the split
version, as well as an "exterior'' model for both the bimodule structure
and the odd product. We give a reference table for all quadratic and triple
products, and use this to explicitly describe all inner super-derivations.
In a subsequent article we will use these tables to investigate the
structure of the Grassmann envelope.
(12 Sep 2005: This is an improved version of the paper uploaded 1 Sep 2005)
(6 Oct 2005: This is now yet another improvement of the previous version)
Kevin McCrimmon < kmm4m@virginia.edu >