Generic Jordan Polynomials

Abstract

The universal multiplication envelope 𝒰 ℳ ℰ(J) of a Jordan system J (algebra, triple, or pair) encodes information about its linear actions—all of its possible actions by linear transformations on outer modules M (equivalently, on all larger split null extensions J ⊕ M). In this article, we study all possible actions, linear and nonlinear, on larger systems. This is encoded in the universal polynomial envelope 𝒰 𝒫 ℰ(J), which is a system containing J and a set X of indeterminates. Its elements are generic polynomials in X with coefficients in the system J, and it encodes information about all possible multiplications by J on extensions . The universal multiplication envelope is recovered as the “linear part,” the elements homogeneous of degree 1 in some variable x. We are especially interested in generic polynomial identities, free Jordan polynomials p(x 1,…, x n ; y 1,…, y m ) which vanish for particular a j ∈ J and all possible x i in all , i.e., such that the generic polynomial p(x 1,…, x n ; a 1,…, a m ) vanishes in 𝒰 𝒫 ℰ(J). These represent “generic” multiplication relations among elements a i , which will hold no matter where J is imbedded. This will play a role in the problem of imbedding J in a system of “fractions” (McCrimmon, McCrimmon Submitted, McCrimmon To appear). The natural domain for a fraction is the dominion K s ≻ n = Φ n + Φ s + U s (J) where the denominator s dominates the numerator n in the sense that U n , U n,s are divisible by U s on the left and right. We show that by passing to subdomains we can increase the “fractional” properties of the domain, especially if s generically dominates n in 𝒰 𝒫 ℰ(J).