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

TL;DR

This paper characterizes when tuples of elements in enveloping algebras of Lie algebras lead to linearly dependent vectors across all relevant representations, revealing a local-global principle in these algebraic structures.

Contribution

It provides a new characterization of linear dependence in enveloping algebras of Lie algebras, extending previous results to these specific algebraic contexts.

Findings

Tuples linearly dependent over fc in the enveloping algebra of a Lie algebra.

Tuples linearly dependent over the center of the algebra for certain Lie algebra representations.

Extension of known results from free and Weyl algebras to enveloping algebras.

Abstract

For every associative algebra $A$ and every class $\mathcal{C}$ of representations of $A$ the following question (related to nullstellensatz) makes sense: Characterize all tuples of elements $a_1,\ldots,a_n \in A$ such that vectors $\pi(a_1)v,\ldots,\pi(a_n)v$ are linearly dependent for every $\pi \in \mathcal{C}$ and every $v$ from the representation space of $\pi$. 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 $\mathbb{C}$ is the class of all finite-dimensional representations of $A$. (2) $A=U(\mathfrak{sl}_2(\mathbb{C}))$ and $\mathbb{C}$ is the class of all finite-dimensional irreducible representations of $A$. (3) $A=U(\mathfrak{sl}_3(\mathbb{C}))$ and $\mathbb{C}$ 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 $\mathbb{C}$ 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.