The Structure of Quadratic Jordan Systems of Clifford Type
Zel'manov's classification of Jordan triple systems and pairs states
that prime nondegenerate quadratic Jordan systems are either i-special,
or exceptional finite-dimensional over their centroid, and it further
establishes that whenever the characteristic is not 2 or 3, the
i-special triples (resp. pairs) are forms of either one of 5 simple
triples (resp. 2 pairs) of hermitian type or one of 4 simple triples
(resp. 2 pairs) of Clifford type. (As it turns out, he inadvertently
omitted two pairs - hence four triples - of the latter type.) The part
of the argument that pertains to systems of hermitian type has already
been extended to quadratic Jordan systems over an arbitrary ring of
scalars; in the present work, we finalize the general classification by
handling the case of quadratic systems of Clifford type. [By system of
Clifford type (triple or pair), we mean a system all the homotopes of
which strictly satisfy some Jordan Clifford identity, a Jordan
polynomial that does not vanish on the split 3 by 3 hermitian matrices.]
Essentially, a simple Jordan pair of Clifford type is either a "small"
(degree at most 2) pair of rectangular matrices, or a pair of
alternating 5 by 5 matrices, or a pair obtained by doubling a simple
Jordan algebra of a quadratic form of dimension different from 2 over
some algebraically closed field extension of the centroid. The structure
of simple triples then emerges from their categorical equivalence with
Jordan pairs with involution.
Finally, by combining the structure of locally finite-dimensional
idempotent-finite locally-nilpotent-free Jordan systems with a standard
ultrafilter argument, we conclude that a prime nondegenerate Jordan
system sits inside a simple one as a scalar form.
A. D'Amour
<damour@cc.denison.edu>
K. McCrimmon
<kmm4m@weyl.math.Virginia.edu>