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.
2014
262
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11581/304984
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