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Enveloping Algebras of Semi-Simple Lie Algebras

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Part of the Contemporary Mathematicians book series (CM)

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] = abba 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.

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References

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Copyright information

© Birkhäuser Boston 1989

Authors and Affiliations

  1. 1.Yale UniversityUSA

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