Eigenvalue problem versus Casimir functions for Lie algebras

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.