Comparison of two infinite dimensional Lie Algebras

Question

Is there an example of a real analytic (compact) manifold $M$ such that the following two lie algebras are isomorphic Lie algebras:

1. $\chi^{\infty}(M)$, the Lie algebra of all smooth vector fields on $M$.

2. $\chi^{\omega}(M)$, the Lie algebra of all analytic vector fields on $M$.

(Before any lie algebraic obstruction, Is there a trivial obstruction as cardinality,etc?)

Motivations: Lets reduce (and change) the question to a "ring" setting." Are the ring of smooth and analytic real functions on $M$ isomorphic?" This is a trivial question because the analytic ring is an integral domain but the smooth one is not. So in this trivial case the "zero divisor" is a ring theoretical obstruction for such ismorphism. Now it would be interesting to search for some Lie algebraic obstructions for isomorphicity of $\chi^{\infty}(M)$ and $\chi^{\omega}(M)$. This "ring" situation which I mentioned, shows that (perhaps) no $\mathbb{S}^{n}$ can be an example for my question. Because two isomorphic Lie algebras have isomorphic complexification. on the other hand the complexification of $T\mathbb{S}^{n}$ is the trivial bundle. Moreover sections of the trivial bundles corresponds to "functions".

As my final question:

What are the structures of the enveloping algebras of two Lie algebras under my questions? How can we compare them?

Answer 1

The Lie algebra of analytic vector fields on a compact real analytic manifold is simple —this was proved in [Grabowski, J. Isomorphisms and ideals of the Lie algebras of vector fields. Invent. Math. 50 (1978/79), no. 1, 13--33. MR0516602 (80g:57036)]—while the algebra of smooth vector fields has lots and lots of ideals. Indeed, in this last case the algebra has a maximal ideal per point of the manifold —see [Shanks, M. E.; Pursell, Lyle E. The Lie algebra of a smooth manifold. Proc. Amer. Math. Soc. 5, (1954). 468--472. MR0064764 (16,331a)]