In mathematics, the peak algebra is a (non-unital) subalgebra of the group algebra of the symmetric group S<sub>n</sub>, studied by . It consists of the elements of the group algebra of the symmetric group whose coefficients are the same for permutations with the same peaks. (Here a peak of a permutation àon {1,2,...,n} is an index i such that ÃÂ(iâÂÂ1)<ÃÂ(i)>ÃÂ(i+1).) It is a left ideal of the descent algebra. The direct sum of the peak algebras for all n has a natural structure of a Hopf algebra.