# QuantumGroups`

## Contents

For now, QuantumGroups` is only available from its SVN repository (about SVN), or as a subpackage of KnotTheory`.

## Installation

If you're planning on using QuantumGroup` from inside KnotTheory`, you just have to execute `<<QuantumGroups`` after loading the KnotTheory` package. Otherwise, you'll first need to add it to the path (that is, add the directory containing QuantumGroups.m to \$Path), and then execute `<<QuantumGroups``.

## Examples of use

### Quantum knot invariants

If you've installed the KnotTheory` package, there's a very simple interface for computing quantum knot invariants. (Be warned however, that without precomputed data files installed, this will recompute everything from scratch, for each knot, and thus be extremely slow.)

 `In[]:=` `QuantumKnotInvariant[A2, Irrep[A2][{1,1}]][Knot[8,19]][q]` `Out[]=` ``` 2 -68 2 2 -60 4 4 4 2 4 6 7 8 6 4 -36 2 4 6 7 --- - q - --- - --- + q + --- + --- + --- + --- - --- - --- - --- - --- - --- - --- - q + --- + --- + --- + --- + 72 66 64 58 56 54 52 48 46 44 42 40 38 34 32 30 28 q q q q q q q q q q q q q q q q q 6 4 2 -20 --- + --- + --- + q 26 24 22 q q q ```

## Things to do

• PBW basis for the quantum group
• Special bases for representations
• Gelfand-Tsetlin basis for type A
• Canonical bases
• Web bases for $A_1$, $A_2$, $B_2$, $G_2$