In mathematics, the peak algebra is a (non-unital)

subalgebra In mathematics, a subalgebra is a subset of an algebra, closed under all its operations, and carrying the induced operations. "Algebra", when referring to a structure, often means a vector space or module equipped with an additional bilinear opera ...

of the group algebra of the

symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric grou ...

''S''_''n'', studied by . It consists of the elements of the group algebra of the symmetric group whose coefficients are the same for

permutation In mathematics, a permutation of a set can mean one of two different things: * an arrangement of its members in a sequence or linear order, or * the act or process of changing the linear order of an ordered set. An example of the first mean ...

s with the same peaks. (Here a peak of a permutation σ on is an index ''i'' such that σ(''i''–1)<σ(''i'')>σ(''i''+1).) It is a left ideal of the descent algebra. The

direct sum The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently but analogously for different kinds of structures. As an example, the direct sum of two abelian groups A and B is anothe ...

of the peak algebras for all ''n'' has a natural structure of a

Hopf algebra In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously a ( unital associative) algebra and a (counital coassociative) coalgebra, with these structures' compatibility making it a bialgebra, and that moreover ...

References

*{{citation, mr=2001673 , last=Nyman, first= Kathryn L. , title=The peak algebra of the symmetric group , journal=J. Algebraic Combin., volume= 17 , year=2003, issue= 3, pages= 309–322 , doi=10.1023/A:1025000905826, doi-access=free Algebras