Fundamentals of Hopf Algebras by Robert G. Underwood

Author:Robert G. Underwood
Language: eng
Format: epub, pdf
Publisher: Springer International Publishing, Cham

3.1 Introduction to Hopf Algebras

In this section we define a K-Hopf algebra H as a K-bialgebra with an additional map called the coinverse (or antipode) satisfying the coinverse (or antipode) property. We give some examples of Hopf algebras including the group ring KG which in many respects is the prototype Hopf algebra that others are modelled on. We define convolution, a binary operation on linear transformations, and use it to prove that the coinverse is an algebra anti-homomorphism, and consequently, that the coinverse has order 2 under composition whenever H is cocommutative. Using convolution, we also show that the coinverse is a coalgebra anti-homomorphism, a property that can be used to show that the coinverse has order 2 whenever H is commutative. (We will again employ the coalgebra anti-homomorphism property in §3.2.) Next, we consider Hopf ideals, quotient Hopf algebras, and homomorphisms of Hopf algebras. Finally we show that H is a finite dimensional Hopf algebra if and only if H ∗ is a finite dimensional Hopf algebra.

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Definition 3.1.1.

A K-Hopf algebra is a bialgebra over a field K

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