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

Arun Ram

Department of Mathematics and Statistics

University of Melbourne

Parkville, VIC 3010 Australia

aram@unimelb.edu.au

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:
$$\begin{array}{c}\n\n\nA\n\nA\u2297A\nA\u2297A\n\nA\u2297A\u2297A\n\n\Delta \n\Delta \n\n\Delta \u2297id\nid\u2297\Delta \n\n\n\n\n\n\n\end{array}$$

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
$$\begin{array}{c}\n\n\nA\n\nA\u2297A\u21c9\gamma \u2297idid\u2297\gamma A\u2297A\n\u2102\n\nA\n\n\Delta \n\epsilon \n\nm\ne\n\n\n\n\n\n\n\end{array}$$
This antiautomorphism is called the antipode.

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