# A Question About Lie(n) (Paul Selick and Jie
Wu)

Let **k** be a field and let V be the **k**-module generated by
x_1,...,x_n. Recall that **Lie(n)** is the **k**-submodule of the n-fold tensor
product of V generated by the iterated commutators
[...[x_{s(1)},x_{s(2)}],...],x_{s(n)}]
for any element s in the symmetric group S_{n}. Note that
Lie(n) is a module over the symmetric group S_{n}, where
S_{n} acts Lie(n) by permuting letters.

**Problems:**
- Determine the dimension of maximal projective
**k**(S_{n})-submodules
of Lie(n).
- Determine one of maximal projective
**k**(S_{n})-submodules
of Lie(n).

**Remarks:**
An application of this problem to Homotopy Theory is to determine natural
splittings of certain loop spaces.
The maximal projective **k**(\Sigma_n)-submodules
of Lie(n) is unique up to isomorphism.
Let Lie^{max}(n) denote a maximal projective
**k**(S_{n})-submodules
of Lie(n).
- If
**k** is of characteristic 0, then
Lie^{max}(n)=Lie(n).
- If
**k** is of characteristic p>0, then

(1) Lie^{max}(n)=Lie(n) if and only if n is not divisible by
p.
(2) dim(Lie^{max}(p))=(p-1)!-p+1.
(3) Lie^{max}(n) is UNKNOWN for general n.