# The Structure of Hopf Algebras – University of Copenhagen

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# The Structure of Hopf Algebras

Specialeforsvar ved Bjarke Østergård Nielsen

Titel: The Structure of Hopf Algebra

Abstract: In this project we introduce the notion of algebras and coalgebras over a commutative ring, in order to define an algebraic structure called a Hopf algebra. We develop some (co)module-theory over these (co)algebras, and discuss the structure and duality between algebras and coalgebras. We use this to show that a connected, graded-commutative and associative quasi Hopf algebra H of finite type over a perfect field, is isomorphic as an algebra to the tensor product of monogenic Hopf algebras. Furthermore, we show that if we impose the condition of "being primitively generated" on H, then the previous mentioned isomorphism is an isomorphism of Hopf algebras. We introduce the notion of (restricted) Lie algebras and their universal enveloping algebra in order to show that a (not necessarily connected) graded-commutative and primitively generated Hopf algebra H of finite type over a perfect field is isomorphic as a Hopf algebra to the tensor product of monogenic Hopf algebras. All of this is done by working from the perspective of the Milnor-Moore paper "On the Structure of Hopf Algebras", and the article "Some Remarks on the Structure of Hopf Algebras" by J. P. May.

Vejleder:  David Sprehn
Censor:    Kasper K.S. Andersen, Lund Universitet