# Comparison of two infinite dimensional Lie Algebras

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?

## 1 Answer

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)]

• Thank you very much for your answer and very interesting information. Mar 22, 2014 at 20:20