Enveloping Algebras of Semi-Simple Lie Algebras

Abstract

In a recent paper we studied systems of equations of the form

$$ \left[ {\left[ {{x_i},{x_j}} \right],{x_k}} \right] = {\delta_{ki}}{x_j} - {\delta_{kj}}{x_{i,}}\;i,j,k = 1,2, \ldots, n $$

(1)

$$ \phi \left( {{x_1}} \right) = 0 $$

(2)

where as usual [a,b] = ab−ba and ϕ(λ) is a polynomial.¹ Equations of this type have arisen in quantum mechanics. In our paper we gave a method of determining the matrix solutions of such equations. The starting point of our discussion was the observation that if the elements x _i satisfy (1) then the elements x _i , [x _j , x _k ] satisfy the multiplication table of a certain basis of the Lie algebra ${\mathfrak{S}_{n + 1}}$ of skew symmetric (n + 1) × (n + 1) matrices. We proved that if (2) is imposed as an added condition, then the algebra generated by the x’s has a finite basis, and we obtained the structure of the most general associative algebra that is generated in this way.