Hopf algebra of permutations

In algebra, the Malvenuto–Poirier–Reutenauer Hopf algebra of permutations or MPR Hopf algebra is a Hopf algebra with a basis of all elements of all the finite symmetric groups S_n. It is both free as an algebra and graded-cofree as a graded coalgebra, so in in some sense as for as possible from being either commutative or cocommutative. It was introduced by Malvenuto & Reutenauer (1994) and studied by Poirier & Reutenauer (1995).

References

• Malvenuto, Clauda; Reutenauer, Christophe (1995), "Duality between quasi-symmetric functions and the Solomon descent algebra", J. Algebra, 177 (3): 967–982, MR 1358493
• Poirier, Stéphane; Reutenauer, Christophe (1995), "Algèbres de Hopf de tableaux", Ann. Sci. Math. Québec, 19 (1): 79–90, MR 1334836