Abstract
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.
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References
Victor G. Kac. Infinite-dimensional Lie algebras. Cambridge University Press, Cambridge, third edition, 1990.
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Bump, D. (2013). The Universal Enveloping Algebra. In: Lie Groups. Graduate Texts in Mathematics, vol 225. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-8024-2_10
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DOI: https://doi.org/10.1007/978-1-4614-8024-2_10
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