TITLE: Are there F_un Lie algebras ?  
QUESTION [16 upvotes]: Background See WP-article on F_1 = F_{un} = Field with one element (and also this MO question). Paraphrasing someone:
we do not know what is it, but it is not a field :). For this question it is enough to keep in mind J. Tits idea (1957) that Weyl groups should be thought as semisimple groups over F_1.
E.g. symmetric group S_n = GL_n(F_1). 
Q1 What might be Lie algebras over F_1 ? In particular for GL_n ?
What numerology should correspond to gl_n(F_1) ? I.e. are there some numbers related to
gl_n(F_q) which have a limit when q->1 (may be renormalized like with GL_n(F_q)) ? 

Comments on further questions are also welcome:
Q2 To what extent representation theory of S_n can be thought as limit q->1 of representation theory of GL_n(F_q) ? (There is some paper "Translating the Irreducible Representations of S_n into GL_n(F_q)", but I do not quite understand it).
Q3 What might be "orbit method" to construct representations of S_n = GL_n(F_1) ?
Q4 What might be Langlands correspondence over F_1 ? Should it be related to bijection between
irreps of S_n and its conj. classes (keep in mind that GL^L=GL).

REPLY [7 votes]: I will only attempt to answer the first question.
$n$-dimensional vector space over $\mathbf{F}_1$ is the same as a pointed set with $n+1$ elements. It is natural to call $GL_n(\mathbf{F}_1)$ the group of automorphisms of $\mathbf{F}_1^n$ and $\mathfrak{gl}_n(\mathbf{F}_1)$ the monoid of endomorphisms. There are at least two notions of morphisms:

Plain morphisms of pointed sets. The monoid of endomorphisms has cardinality $(n+1)^n$.
Maps of pointed sets which are injective if you throw away the basepoints (see http://arxiv.org/abs/1006.0912). It is not too hard to see that the cardinality of $End(\mathbf{F}_1)$ is
$$\sum_{k=0}^n\left(\begin{array}{c}n\\ k\end{array}\right)\frac{n!}{(n-k)!}=\sum_{k=0}^n\frac{(n!)^2}{k!(n-k)!^2}.$$
Here $k$ is the number of elements that don't go to the basepoint.

Note, that in both cases the group of automorphisms is the same ($S_n$).