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
[...[xs(1),xs(2)],...],xs(n)]
for any element s in the symmetric group Sn. Note that Lie(n) is a module over the symmetric group Sn, where Sn acts Lie(n) by permuting letters.

Problems:
  1. Determine the dimension of maximal projective k(Sn)-submodules of Lie(n).
  2. Determine one of maximal projective k(Sn)-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 Liemax(n) denote a maximal projective k(Sn)-submodules of Lie(n).
    1. If k is of characteristic 0, then Liemax(n)=Lie(n).
    2. If k is of characteristic p>0, then
    (1) Liemax(n)=Lie(n) if and only if n is not divisible by p.
    (2) dim(Liemax(p))=(p-1)!-p+1.
    (3) Liemax(n) is UNKNOWN for general n.