The Universal Enveloping Algebra
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.
- 92.Victor G. Kac. Infinite-dimensional Lie algebras. Cambridge University Press, Cambridge, third edition, 1990.Google Scholar