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On Hopf algebras of symmetric and quasisymmetric functionsDahlgren, Isabel January 2024 (has links)
This bachelor thesis aims to give an introduction to various Hopf algebras that arise in combinatorics, with a view towards symmetric functions. We begin by covering the algebraic background needed to define Hopf algebras, including a discussion of the algebra-coalgebra duality. Takeuchi's formula for the antipode is stated and proved. It is then generalised to incidence Hopf algebras. This is followed by a discussion of the Hopf algebra of symmetric functions. It is shown that the Hopf algebra of symmetric functions is self-dual. We also show that the graded dual of the Hopf algebra of quasisymmetric functions is the Hopf algebra of non-commutative symmetric functions. Relations to the Hopf algebra of symmetric functions in non-commuting variables are emphasised. Finally, we state and prove the Aguiar-Bergeron-Sottile universality theorem.
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