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: $$\begin{array}{c} A A⊗A A⊗A A⊗A⊗A Δ Δ Δ⊗id id⊗Δ \end{array}$$

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 This antiautomorphism is called the antipode.

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