### Laubinger on Lie Algebras for Frölicher Groups

#### Posted by Urs Schreiber

[*guest post by Martin Laubinger* in the context of Smootheology: the study of generalized smooth spaces]

I have posted a preprint which contains the central new result I obtained in my
dissertation.

The result may not directly relate to higher category theory. Still, I would appreciate feedback and problems for further investigation:

Martin Laubinger
*A Lie algebra for Frölicher groups*

(arXiv)

Here is a short description: The category of Frölicher spaces is a cartesian closed category which contains the category of smooth finite-dimensional manifolds as a full subcategory. The same is true for the closely related category of diffeological spaces. However, it is easier to define tangent spaces to Frölicher spaces than to diffeological spaces. Many groups have natural Frölicher structures, including all Lie groups, but also certain groups of mappings such as $C^\infty(M,G)$ or $\Diff(M)$, which can not be given a manifold structure in general. A basic question is whether the tangent space (in the Frölicher sense) at the identity of these groups can be equipped with a Lie bracket. I have been able to construct such a Lie bracket, but there is an additional condition which has to be verified. This condition is very natural, but I have not found a general proof. In my thesis, I did not have an example for a group which satisfies the extra condition, but in the meantime I verified the condition for the additive group $\mathbb{R}^J$ (product of the reals) if J is not too big. This is explained in detail in the preprint.

Posted at July 1, 2009 10:19 AM UTC
## Re: Laubinger on Lie Algebras for Frölicher Groups

Welcome Martin!

I’ve made it through the first few pages and I like it very much. Very clear writing. Anyone who can write something that penetrates my skull has some amazing talent :)

I had a very basic question about Frölicher spaces.

When I look at $(X,C,F)$, I’m reminded of coordinate maps. If $X$ is a smooth $m$-dimensional manifold we have coordinate maps

$\phi:U\subset X\to\mathbb{R}^m$

such that for any other coordinate map $\psi$

$\phi\circ \psi^{-1}:\mathbb{R}^m\to\mathbb{R}^m$

is smooth (where defined). Comparing this to Frölicher spaces, I’m tempted to consider a

Frölicher $m$-space$(X,C_m,F_m)$, where$C_m = \{c:\mathbb{R}^m\to X | (\forall f \in F_m) f\circ c\in C^\infty(\mathbb{R}^m,\mathbb{R}^m)\}$

$F_m = \{f:X\to \mathbb{R}^m | (\forall c \in C_m) f\circ c\in C^\infty(\mathbb{R}^m,\mathbb{R}^m)\}.$

Then a Frölicher space is a Frölicher $m$-space for $m=1$.

Does that make any sense?

Then you can imagine maybe crossing dimensions. For example, $(X,\mathcal{C},\mathcal{F})$, where

$\mathcal{C} = \bigoplus_{0\lt m\le D} C_m,$

$\mathcal{F} = \bigoplus_{0\lt n\le D} F_n,$

$C_m = \{c:\mathbb{R}^m\to X | (0 \lt n \le D, \forall f \in F_n) f\circ c\in C^\infty(\mathbb{R}^m,\mathbb{R}^n)\},$

$F_n = \{f:X\to \mathbb{R}^n | (0 \lt m \le D, \forall c \in C_m) f\circ c\in C^\infty(\mathbb{R}^m,\mathbb{R}^n)\},$

and $D$ is to be thought of as the dimension of $X$.

Does that make any sense?