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 oper ...

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 group ...

''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 is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or p ...

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 In algebra, Solomon's descent algebra of a Coxeter group is a subalgebra of the integral group ring of the Coxeter group, introduced by . The descent algebra of the symmetric group In the special case of the symmetric group ''S'n'', the desc ...

. 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. To see how the direct sum is used in abstract algebra, consider a mo ...

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

Hopf algebra Hopf is a German surname. Notable people with the surname include: * Eberhard Hopf (1902–1983), Austrian mathematician * Hans Hopf (1916–1993), German tenor * Heinz Hopf (1894–1971), German mathematician * Heinz Hopf (actor) (1934–2001), Sw ...

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