Diophantine sets of representations (Q2445923)

Let \(U\) denote the universal enveloping algebra of \(\text{sl}(2,k)\) where \(\text{char}(k)=0\). Given a pp-pair \(\phi/\psi\) in the language of \(U\)-modules, the support of \(\phi/\psi\) on the finite-dimensional \(U\)-representations is the set of natural numbers \(n\) such that \(F(L(n))\neq 0\). It is shown that this is a primitive recursive, hence diophantine, set. This supports the conjecture of Herzog that the theory of the finite-dimensional \(\text{sl}(2,k)\) representations is decidable; the theory of all \(\text{sl}(2,k)\) representations is known [\textit{M. Prest} and \textit{G. Puninski}, J. Algebra 211, No. 1, 268--297 (1999; Zbl 0942.16003)] to be undecidable.