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Hopf algebras

  • Kurt Luoto
  • Stefan Mykytiuk
  • Stephanie van Willigenburg
Chapter
Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)

Abstract

We give the basic theory of graded Hopf algebras, and then illustrate the theory in detail with three examples: the Hopf algebra of symmetric functions, Sym, the Hopf algebra of quasisymmetric functions, QSym, and the Hopf algebra of noncommutative symmetric functions, NSym. In each case we describe pertinent bases, the product, the coproduct and the antipode. Once defined we see how Sym is a subalgebra of QSym, and a quotient of NSym. We also discuss the duality of QSym and NSym and a variety of automorphisms on each. We end by defining combinatorial Hopf algebras and discussing the role QSym plays as the terminal object in the category of all combinatorial Hopf algebras.

Key words

combinatorial Hopf algebras coproducts duality Hopf algebras noncommutative symmetric functions P-partitions products quasisymmetric functions symmetric functions 

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Copyright information

© Kurt Luoto, Stefan Mykytiuk, Stephanie van Willigenburg 2013

Authors and Affiliations

  • Kurt Luoto
    • 1
  • Stefan Mykytiuk
    • 2
  • Stephanie van Willigenburg
    • 1
  1. 1.Department of MathematicsUniversity of British ColumbiaVancouverCanada
  2. 2.Department of Mathematics and StatisticsYork UniversityTorontoCanada

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