Abstract

We present a new perspective on the invariants of Lie algebras (Casimir functions). Our approach is based on the connection of a linear mapping \(F\in End(V)\) , which has a given eigenvector v, to a Lie algebra. We obtain a solvable Lie algebra by considering a single pair (F, v). However, by considering a set of such pairs \((F_i, v_i)\) , \(i=1,2,\ldots , s\) , we can obtain any finite-dimensional Lie algebra. We also describe the Casimir function equations in terms of pairs, since the eigenvalue problem of (F, v) yields a Lie bracket. We outline the criterion for the quantity of Casimirs and their formulas for any Lie algebra, which depends on the decomposability of the tensor built from the pairs \((F_i,v_i)\) . In addition, we present the meaning of lifting Lie algebras in this context and explain how to construct Casimir functions for the lifted Lie algebra based on Casimir functions for the initial Lie algebra. One of the main results of the paper is to present the method to identify all Casimirs for a lifted Lie algebra starting from the initial one.

中文翻译：

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李代数的特征值问题与卡西米尔函数

摘要

我们对李代数（卡西米尔函数）的不变量提出了新的视角。我们的方法基于线性映射\(F\in End(V)\) 与李代数的连接，该线性映射具有给定的特征向量v。我们通过考虑单个对 ( F ,  v ) 来获得可解的李代数。然而，通过考虑一组这样的对\((F_i, v_i)\) , \(i=1,2,\ldots , s\)，我们可以获得任何有限维李代数。我们还以对的形式描述卡西米尔函数方程，因为 ( F ,  v ) 的特征值问题产生李括号。我们概述了卡西米尔数量的标准及其对于任何李代数的公式，这取决于由对\((F_i,v_i)\)构建的张量的可分解性。此外，我们在此背景下介绍了提升李代数的含义，并解释了如何基于初始李代数的卡西米尔函数构造提升李代数的卡西米尔函数。本文的主要成果之一是提出了一种从初始李代数开始识别所有卡西米尔的方法。