Abstract
For an associative algebra A and a class C of representations of A the following question (related to nullstellensatz) makes sense: Characterize all tuples of elements a1,…,an∈A such that vectors π(a1)v,…,π(an)v are linearly dependent for every π∈C and every v in the representation space of π. We answer this question in the following cases:(1)A=U(L) is the enveloping algebra of a finite-dimensional complex Lie algebra L and C is the class of all finite-dimensional representations of A.(2)A=U(sl2(C)) and C is the class of all finite-dimensional irreducible representations of A.(3)A=U(sl3(C)) and C is the class of all finite-dimensional irreducible representations of A with sufficiently high weights. In case (1) the answer is: tuples that are linearly dependent over C while in cases (2) and (3) the answer is: tuples that are linearly dependent over the center of A. Similar results have been proved before for free algebras and Weyl algebras.
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