The linear span P n of the sums of all permutations in the symmetric group S n with a given set of peaks is a sub-algebra of the symmetric group algebra, due to Nyman. This peak algebra is a left ideal of the descent algebra D n ; and the direct sum P of all P n is a Hopf sub-algebra of the direct sum D of all D n , dual to the Stembridge algebra of peak functions. In our self-contained approach, peak counterparts of several results on the descent algebra are established, including a simple combinatorial characterization of the algebra P n ; an algebraic characterization of P n based on the action on the Poincaré–Birkhoff–Witt basis of the free associative algebra; the display of peak variants of the classical Lie idempotents; an Eulerian-type sub-algebra of P n ; a description of the Jacobson radical of P n and its nil-potency index, of the principal indecomposable and irreducible P n -modules, and of the Cartan matrix of P n . Furthermore, it is shown that the primitive Lie algebra of P is free, and that P is its enveloping algebra.