  The peak algebra is a unital subalgebra of the symmetric group algebra,
linearly spanned by sums of permutations with a common set of peaks. By
exploiting the combinatorics of sparse subsets of [n-1] (and of certain classes
of compositions of n called almost-odd and thin), we construct three new linear
bases of this algebra. We discuss two peak analogs of the first Eulerian
idempotent and construct a basis of semi-idempotent elements.
  We use these bases to describe the Jacobson radical of the peak algebra and
to characterize the elements of this algebra in terms of the canonical action
of the symmetric groups on the tensor algebra of a vector space. We define a
chain of ideals such that the ideal at the bottom of the chain is the linear
span of sums of permutations with a common set of interior peaks and the ideal
at the top is the whole algebra. We extend the above results to these ideals,
generalizing results of Schocker (the case of the bottom ideal).
