Lie Groups pp 61-66 | Cite as

The Universal Enveloping Algebra

  • Daniel Bump
Part of the Graduate Texts in Mathematics book series (GTM, volume 225)


We have seen that elements of the Lie algebra of a Lie group G are derivations of C (G). They are thus first-order differential operators that are left-invariant. The universal enveloping algebra is a purely algebraically defined ring that may be identified with the ring of all left-invariant differential operators, including higher-order ones.


  1. 92.
    Victor G. Kac. Infinite-dimensional Lie algebras. Cambridge University Press, Cambridge, third edition, 1990.Google Scholar

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© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Daniel Bump
    • 1
  1. 1.Department of MathematicsStanford UniversityStanfordUSA

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