  We define, for each subset $S$ of primes, an $S_n$-module $Lie_n^S$ with
interesting properties. When $S=\emptyset,$ this is the well-known
representation $Lie_n$ of $S_n$ afforded by the free Lie algebra.
  The most intriguing case is $S=\{2\},$ giving a decomposition of the regular
representation as a sum of {exterior} powers of modules $Lie_n^{(2)}.$ This is
in contrast to the theorems of Poincar\'e-Birkhoff-Witt and Thrall which
decompose the regular representation into a sum of symmetrised $Lie$ modules.
We show that nearly every known property of $Lie_n$ has a counterpart for the
module $Lie_n^{(2)},$ suggesting connections to the cohomology of configuration
spaces via the character formulas of Sundaram and Welker, to the Eulerian
idempotents of Gerstenhaber and Schack, and to the Hodge decomposition of the
complex of injective words arising from Hochschild homology, due to Hanlon and
Hersh.
  For arbitrary $S,$ the symmetric and exterior powers of the module $Lie_n^S$
allow us to deduce Schur positivity for a new class of multiplicity-free sums
of power sums.
