## Quantized universal enveloping algebras, the Yang-Baxter equation and invariants of links, I

Last update: 2 June 2014

## Notes and References

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.

## Appendix

Here we give some elementary facts about Hopf algebras. The details are given in [Abe1980].

Definition 1. An associative algebra $A$ with unit $e$ and multiplication $m:A\otimes A\to A$ is a bialgebra if there is a homomorphism of algebras $\mathrm{\Delta }:A\to A\otimes A$ satisfying the coassociativity condition. The coassociativity implies the commutativity of the following diagram: $A A\otimes A A\otimes A A\otimes A\otimes A \mathrm{\Delta } \mathrm{\Delta } \mathrm{\Delta }\otimes \text{id} \text{id}\otimes \mathrm{\Delta }$

The homomorphism $\mathrm{\Delta }$ is called a comultiplication.

Proposition A.1. On the dual space of a bialgebra $A$ the structure of bialgebra with multiplication $\mathrm{\Delta }:{A}^{*}\otimes {A}^{*}\to {A}^{*}$ comultiplication $m:{A}^{*}\to {A}^{*}\otimes {A}^{*}$ and with counit $\epsilon \text{:}$ $\epsilon \left(a\right)={\delta }_{a,e}$ is also defined.

Definition 2. Bialgebra $A$ is a Hopf algebra if on there is an antiautomorphism $\gamma$ such that the following diagram is commutative $A A\otimes A\underset{\gamma \otimes \text{id}}{\overset{\text{id}\otimes \gamma }{⇉}}A\otimes A ℂ A \mathrm{\Delta } \epsilon m e$ This antiautomorphism is called the antipode.