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 Sn, and is a non-commutative analogue of the Hopf algebra of symmetric functions. 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).
Definition
References
- Hazewinkel, Michiel; Gubareni, Nadiya; Kirichenko, V. V. (2010), Algebras, rings and modules. Lie algebras and Hopf algebras, Mathematical Surveys and Monographs, vol. 168, Providence, RI: American Mathematical Society, ISBN 978-0-8218-5262-0, MR 2724822, Zbl 1211.16023
{{citation}}: line feed character in|title=at position 30 (help) - 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