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>