A local-global principle for linear dependence in enveloping algebras of Lie algebras (Q2309689)

Let \(A\) be an associative algebra and let \(\mathcal C\) be a class of representations of \(A\). The elements \(p_1,\ldots,p_k\in A\) are \(\mathcal C\)-locally linearly dependent if for any representation \(\pi:A\to\text{End}(V_{\pi})\) in \(\mathcal C\) the endomorphisms \(\pi(p_1),\ldots,\pi(p_k)\in\text{End}(V_{\pi})\) are linearly dependent; \(p_1,\ldots,p_k\) are \(\mathcal C\)-locally directionally linearly dependent if for any \(v\in V_{\pi}\) the vectors \(\pi(p_1)v,\ldots,\pi(p_k)v\in V_{\pi}\) are linearly dependent. By results of [\textit{J. F. Camino} et al., Integral Equations Oper. Theory 46, No. 4, 399--454 (2003; Zbl 1046.68139)] and [\textit{M. Brešar} and \textit{I. Klep}, Isr. J. Math. 193, 71--82 (2013; Zbl 1293.16021)] when \(A\) is the free associative algebra \({\mathbb F}\langle X_1,\ldots,X_n\rangle\) over a field \(\mathbb F\) and \(\mathcal C\) is the class of all finite dimensional representations, then the local linear dependence and the local directional linear dependence of \(p_1,\ldots,p_k\) are equivalent to the linear dependence of the elements \(p_1,\ldots,p_k\) in \({\mathbb F}\langle X_1,\ldots,X_n\rangle\). A similar result was established by the first author of the paper under review [Integral Equations Oper. Theory 90, No. 3, Paper No. 38, 8 p. (2018; Zbl 06909748)] when \(A=A_n({\mathbb C})\) is the \(n\)-th Weyl algebra and \({\mathcal C}=\{\pi_0\}\) is its Schrödinger representation. We shall mention that such kind of results are related with the Nullstellensatz. In the paper under review, the authors consider similar problems when \(A=U(L)\) is the universal enveloping algebra of a finite dimensional complex Lie algebra \(L\) and \(\mathcal C\) consists of finite dimensional representations of \(U(L)\). The first main result gives that the linear dependence, the local linear dependence and the local directional linear dependence are equivalent when \(L\) is an arbitrary and \(\mathcal C\) consists of all finite dimensional representations. The second main result handles the case when \(L=\mathfrak{sl}_2\) is the complex Lie algebra of \(2\times 2\) traceless matrices and \(\mathcal C\) consists of all finite dimensional irreducible representations. Then the local linear dependence and the local directional linear dependence of \(p_1,\ldots,p_k\in\mathfrak{sl}_2\) are equivalent to the existence of elements \(z_1,\ldots,z_k\) in the center of \(U(\mathfrak{sl}_2)\), not all equal to zero, such that \(z_1p_1+\cdots+z_kp_k=0\). The third main result is for \(U(\mathfrak{sl}_3)\) and the class \({\mathcal C}={\mathcal I}_d\) consisting of all irreducible representations of \(\mathfrak{sl}_3\) with highest weights \((m_1,m_2)\), \(m_1,m_2\geq d\). Then the condition that \(z_1p_1+\cdots+z_kp_k=0\), \(z_1,\ldots,z_k\) in the center of \(U(\mathfrak{sl}_3)\), is equivalent to the \({\mathcal I}_d\)-local linear dependence and the \({\mathcal I}_d\)-local directional linear dependence of \(p_1,\ldots,p_k\) for some sufficiently large \(d\).