# Is there a way to associate a Lie algebra to the group of diffeomorphisms?

Let $M$ a closed (smooth) manifold. The group $Diff(M)$ of all difeomorphisms of $M$ is infinite dimensional, therefore it is not a Lie group.

Is there a way to associate a Lie algebra to this? If so, is there some concrete descrpition of such Lie algebra?

EDIT: I would welcome some references. According to the question in the comments: the Lie algebra of the Lie group $G$ is defined as Lie algebra of left invariant vetor fields on $G$. Therefore for an infinite dimensional $G$ we would like to have a notion of a vector field. Vector fields are defined as section of the tangent bundle: the fibers of the tangent bundle are tangent spaces. We define the tangent space at a given point $x$ as the set of classes of smooth curves $\gamma:I \to G, \gamma(0)=x$ with equivalence relation $\gamma_1 \sim \gamma_2$ iff for any chart $\varphi$ we have $\frac{d}{dt}(\varphi \circ \gamma_1)(0)=\frac{d}{dt}(\varphi \circ \gamma_2)(0)$. This set is in one to one correspondence with $\mathbb{R}^n$ for $n$-dimensional manifolds and the linear structure is transported via this correspondence. In infinite dimension there are few delicate moments: instead $\mathbb{R}^n$ we need another ,,model space''. Which model space we choose for infinite dimensional Lie groups, in particular for $Diff(M)$?
Moreover while defining tangent space we need derivative: which notion do we use?

• The Lie Algebra section of the Wikipedia article says "The Lie algebra of the diffeomorphism group of M consists of all vector fields on M equipped with the Lie bracket of vector fields." en.wikipedia.org/wiki/Diffeomorphism#Lie_algebra Aug 12, 2017 at 17:55
• What is it that you don't like about infinite dimensional Lie groups? Aug 12, 2017 at 21:09
• I made an edit clarifying my doubts Aug 15, 2017 at 12:30
• This closely related question might be helpful.
– ಠ_ಠ
Oct 25, 2017 at 23:14

Having infinite dimensional geometry at hand, you can look at infinite dimensional Lie groups, and indeed the Lie algebra of a diffeomorphism group is the Lie algebra of vector fields with the negative of the usual Lie bracket. Intuitively, you can understand that as follows: A curve in $Diff(M)$ can be viewed as a 1-parameter family $\phi_t$ of diffeomorphisms. To get a tangent vector at the point $\phi=\phi_0$ you should differentiate this curve with respect to $t$ at $t=0$. If you look what happens in a point $x\in M$, then $t\mapsto\phi_t(x)$ is a smooth curve starting at $\phi(x)$, so differentiating this at $t=0$ determines a tangent vector at the point $\phi(x)\in M$. Now you check that this can be done smoothly, so you associate to each $x\in M$ a tangent vector at $\phi(x)$ and this defines a section of the pullback bundle $\phi^*(TM)$. With a bit more work you can indeed construct charts with values in a neighborhood of the zero section in the space of smooth sections of $\phi^*(TM)$ and thus identify that vector space as the modelling vector space. From $\phi=id$, you just get smooth sections of $TM$, i.e. vector fields on $M$. (And the exponential map associates to each vector field its local flow.) You can find a lot of information in that direction in the book by Kriegl and Michor mentioned above.