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Enveloping Algebras of Semi-SimpleLie Algebras

Published online by Cambridge University Press:  20 November 2018

N. Jacobson
N. Jacobson*
Affiliation:
Yale University
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In a recent paper we studied systems of equations of the form

(1)

(2)

where as usual [a,b] = ab — ba and ϕ(λ) is a polynomial. 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 xi satisfy (1) then the elements xi, [xj,xk] satisfy the multiplication table of a certain basis of the Lie algebra 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 has a finite basis, and we obtained the structure of the most general associative algebra that is generated in this way.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 2 , 1950 , pp. 257 - 266
Copyright
Copyright © Canadian Mathematical Society 1950

References

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