Last update: 2 June 2014
This is a typed copy of the LOMI preprint Quantized universal enveloping algebras, the Yang-Baxter equation and invariants of links, I by N.Yu. Reshetikhin.
Recommended for publication by the Scientific Council of Steklov Mathematical Institute, Leningrad Department (LOMI) 6, June, 1987.
Here we give some elementary facts about Hopf algebras. The details are given in [Abe1980].
Definition 1. An associative algebra with unit and multiplication is a bialgebra if there is a homomorphism of algebras satisfying the coassociativity condition. The coassociativity implies the commutativity of the following diagram:
The homomorphism is called a comultiplication.
Proposition A.1. On the dual space of a bialgebra the structure of bialgebra with multiplication comultiplication and with counit is also defined.
Definition 2. Bialgebra is a Hopf algebra if on there is an antiautomorphism such that the following diagram is commutative This antiautomorphism is called the antipode.