Given a smooth manifold $M$, one can consider the Lie algebra $\mathcal{X}(M)$ of vector fields equipped with the standard Lie bracket. This is a standard machinery of differential geometry. Gelfand and Fuchs defined the Lie algebra of formal vector fields at $0 \in \mathbb{R}^n$ as linear combinations $$\sum_{j=1}^np_j(x_1,...,x_n)e_j$$ where $e_j$ is a standard basis of $\mathbb{R}^n$ and $p_j$ are formal power series in variables $x_1,...,x_n$. This definition is at the given point $0$ in $\mathbb{R}^n$, so formal vector fields are not ,,globally'' defined.

Is it possible to define formal vector fields globally on a given manifold $M$?

Also I would like to know

What is the significance of formal vector fields? For example, do they naturally arise as a Lie algebra of some natural (infinite dimensional) group?

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    $\begingroup$ They arise as derivations on formal power series in $n$ variables; said another way they are the Lie algebra of the automorphism group of the formal completion at $0$. $\endgroup$ – Qiaochu Yuan Nov 4 '17 at 16:34
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    $\begingroup$ If you want to talk about them globally (for which reason?) you could consider fields of formal vector fields, where the coefficients of the formal series depend smoothly on the base points. This is basically the infinte order jet bundle of the bundle of vector fields. $\endgroup$ – Michael Bächtold Nov 5 '17 at 10:12
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    $\begingroup$ Or you could use the fact that every power series (convergent or not) arises as the Taylor series of some $C^\infty$ function. $\endgroup$ – Sebastian Goette Nov 5 '17 at 14:44

The Lie algebra of all formal vector fields of a real or complex manifold germ $(M,p)$ (where $p\in M$ is any point), does arise as the Lie algebra of the Lie group of all formal power series automorphisms of $(M,p)$, which is the same as the group of all invertible infinite jets of local diffeomorphisms of $(M,p)$, smooth or holomorphic respectively.

Also, formal vector fields of $(M,p)$ can be defined as derivations of the algebra of all formal power series at $p$.

Yet alternatively, the formal vector field algebra can be seen as the quotient of the algebra of all germs of smooth vector fields at $p$ modulo all germs of "flat" vector fields, i.e. ones vanishing of infinite order. For a complex manifold, that definition needs to be appropriately modified by allowing germs of smooth vector fields that are holomorphic of infinite order at $p$.

From that perspective, all these notions are fundamentally local. Of course, you can always consider subalgebras of formal vector fields extendible to global ones on $M$, which would be only interesting on a complex manifold. On a real manifold, of course, any formal vector field would arise that way.

  • $\begingroup$ Thank you very much for a great answer! Could you please just explain what do you mean by flat vector field? $\endgroup$ – truebaran Nov 6 '17 at 0:28
  • $\begingroup$ @truebaran You are welcome. I have added a clarification that was indeed needed. $\endgroup$ – Dmitri Zaitsev Nov 8 '17 at 10:53

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