Diophantine sets of representations

作者：Herzog Ivo^*; L' Innocente Sonia

来源：Advances in Mathematics, 2014, 255: 338-351.

DOI：10.1016/j.aim.2014.01.005

• Affine plane
• Special linear Lie algebra
• Diophantine set
• Primitive recursive
• Morley rank
• Computably presented field

Let k be a field of characteristic 0 and L the special linear Lie algebra sl(2, k). Denote by L(n) subset of k[x, y] the L-representation of homogeneous polynomials of total degree n. It is proved that if H (V) (p(v) is a true implication of positive-primitive formulae in the language L(U) of representations of the universal enveloping algebra U = U(L), then the function n -%26gt; dim(k)[phi(L(n))/psi(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 dim k Homu(M,L(n)) is primitive recursive. The main consequence of the result is that the subset of natural numbers {n is an element of N vertical bar phi(L(n))/psi(L(n)) not equal 0}, associated with a basic open subset of the Ziegler spectrum of U, is computable, and therefore Diophantine.

• 出版日期2014-4-1