Generic Jordan polynomials

The universal multiplication envelope UME(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 bimodules M (equivalently, on all larger split null extensions J+M ). In this paper we study all possible actions, linear and nonlinear, on larger systems. This is encoded in the universal polynomial envelope UPE(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 J^∼. 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(x₁, ... , x_n; y₁, ... , y_m) which vanish for particular a_j in J and all possible x_i in all J^∼, i.e., such that the generic polynomial p(x₁, ... , x_n; a₁, ... , a_m) vanishes in UPE(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'' J^∼. The natural domain for a fraction Q_s^-1n is the dominion K_{s > n} = Φ n + Φ s + Q_sV where the denominator s dominates the numerator n in the sense that Q_n, Q_n,s are divisible by Q_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 UPE(V).

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K. McCrimmon < kmm4m@virginia.edu >