Diophantine sets of representations

Let k be a field of characteristic 0 and L   the special linear Lie algebra sl(2,k)sl(2,k). Denote by L(n)⊆k[x,y]L(n)⊆k[x,y] the L-representation of homogeneous polynomials of total degree n  . It is proved that if ⊢ψ(v)→φ(v)⊢ψ(v)→φ(v) is a true implication of positive-primitive formulae in the language L(U)L(U) of representations of the universal enveloping algebra U=U(L)U=U(L), then the function n↦dimk[φ(L(n))/ψ(L(n))]n↦dimk[φ(L(n))/ψ(L(n))] is primitive recursive. A special case of this result is that if M is a finitely generated representation of U  , then the function n↦dimkHomU(M,L(n)) is primitive recursive. The main consequence of the result is that the subset of natural numbers {n∈N|φ(L(n))/ψ(L(n))≠0}{n∈N|φ(L(n))/ψ(L(n))≠0}, associated with a basic open subset of the Ziegler spectrum of U, is computable, and therefore Diophantine.