Let k be a field of characteristic 0 and L the special linear Lie algebra sl(2,k) . Denote by L(n)⊆k[x,y] the L-representation of homogeneous polynomials of total degree n. It is proved that if ⊢ψ(v)→φ(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↦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 View the MathML source 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} , associated with a basic open subset of the Ziegler spectrum of U, is computable, and therefore Diophantine.
Diophantine sets of representations
L'INNOCENTE, Sonia
2014-01-01
Abstract
Let k be a field of characteristic 0 and L the special linear Lie algebra sl(2,k) . Denote by L(n)⊆k[x,y] the L-representation of homogeneous polynomials of total degree n. It is proved that if ⊢ψ(v)→φ(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↦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 View the MathML source 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} , associated with a basic open subset of the Ziegler spectrum of U, is computable, and therefore Diophantine.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
