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).

Key Words:

• Jordan fraction
• Multiplication and polynomial envelopes
• Quadratic Jordan algebras
• triples
• pairs

2000 Mathematics Subject Classification:

• Primary 17C05
• Secondary 17C10

ACKNOWLEDGMENT

This research was partially supported by the Spanish Ministerio de Educació y Ciencia MTM2004-06580-C02-01 and Fondos FEDER.

Notes

¹We throw (JP2)′ in for future reference, though it holds automatically.

²It was with great difficulty that my colleagues, mentioning old dogs and new tricks, dissuaded me from adopting the general notation J _x for the quadratic Jordan operator. This notation was suggested by that of the Koecher school (Braun and Koecher, 1965) (where A _x denotes multiplication by x and the commutative algebra A) and recently the Tits school of buildings, where the relevant quadratic operator was denoted by J. Perhaps a future generation will feel less constrained by tradition to mind their P's and Q's and U's.

³For algebras these were called quadratic specializations (McCrimmon, 1971), and for pairs were called associative representations (Loos, 1975, 2.4), but we now adopt the adjective multiplication; linear and quadratic specializations suggest specializations of linear and quadratic Jordan systems, wheras the real distinction is between representing the elements of J in an associative algebra, and representing their multiplication operators in an associative algebra. We will preserve the distinction between specialization in (map into an associative algebra) and representation on (map into an associative algebra of linear transformations acting on a space).

⁴Loos (1975, 2.4, pp. 16–17) leaves out (JS2) since it follows from (JS4, 4*) via . (JS4) in turn usually follows by applying (JS5) with y, a replaced by m, b, acting on a, and reading the result as an operator on m. But due to the asymmetry between the pair elements x, y and a, b we cannot derive (JS4) this way and must assume it as an axiom. This contrasts with the Jordan algebra case (McCrimmon, 1971), where u ₁ = 1, u _{U x (y)} = u _x u _y u _x , u _{U w (y),x}  = u _x v _y,x  = v _x,y u _x suffice. The Outer Module Theorem below shows that (JS8), (JS9) are more directly involved than (JS4), (JS5) in capturing outer module structure, but we prefer (JS5) as a basic result (d(x, a) is a diffeotopy), and (JS4)* since it is the dual of (JS4).

⁵As yet there is no definition or theory of abstract inner modules or inner representations. A regular inner module I ⊆ J is a submodule (I⁺, I⁻) invariant under inner multiplication by J, ∩_J(I) ⊆ I (i.e., Q _I^σ (J^−σ) ⊆ I^σ, which just means I is an inner ideal of J).

⁶It is not true that 𝒫ℳ (J) is generated by q(x _i ), q(x _i , x _j ), d(x _i , x _j ), ∩ _{x i} for generators {x _k } of J since ∩ _{Q x i (x j )}: a → Q _a Q _{x i} (x _j ) is not directly expressible in terms of these.

⁷It would suffice to consider only these relations and all their linearizations where x, y, a are themselves monomials. The full list of linearizations would involve further linearizing all the quadratic relations: replacing x → x + λ₁ x ₁ + λ₂ x ₂ + λ₃ x ₃, a → a + λ₄ a ₁, we would have to add for (JP1) the coefficients of λ₁ λ₂, for (JP2) those of λ₁, λ₄, λ₁ λ₄, and for (JP3) those of But that's too steep a price to pay!

⁸While it is not true that every scalar extension J_Ω is a homomorphic image of every finite set of elements of J_Ω lies in such a homomorphic image, and J_Ω itself is such an image if Ω is countably generated as Φ-algebra.

⁹As pair theorists, we can blithely ignore the complications in the category of unital algebras, where we would want 1 ∈ A to remain the unit in A⟨X⟩ and therefore must face the sort of collapse am ₁1m ₂ b − a(m ₁ m ₂)b familiar from the case of free groups. Indeed, associative ring theory wants the entire center C of A to remain the center of A⟨X⟩, forming the free product over C instead of Φ.

¹⁰Note that we have not required in our definition of Power Shifting, primarily because we have been unable to establish it here (not even ). The missing ingredient is a formula S′^σ N ^σ = N ^σ D _c,n , which holds in the case of fractions.

¹¹These serve much the same function as ear-tags to track migrating wildlife.

¹²A careful examination of the proof reveals that in (S13a), we only need that Δ vanish on 2J^−σ and Q _s (J^σ); the vanishing of D _s,Δ1(c) Q _s  = Q _{Q s (Δ1(c)),s} is automatic since Q _s (Δ₁) = {Q _s N(Q _c Q _s (w)),c,s} − {Q _s S′(w),Q _c (n),s} + {Q _s (w),Q _c Q _n (c),s} [by (JP1)] = [D _{Q n Q c (y),c}  − D _{D n,c (y),Q c (n)} + D _{y,Q c Q n (c)}](s) [by (S15) with y := Q _s (w)] vanishes by Inner Triality (H6) with m = 2, x,a → n,c. However, we couldn't derive 2Δ₁(c) = 0 and more easily than Δ₁(c) = 0. Similarly, we only need the vanishing (S13b) of Δ* on the space Φ c + Q _c (J^−σ), but that didn't simplify matters either.

Communicated by I. E. Zelmanov.