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<div class="ChapSects"><a href="chap1.html#X7DFB63A97E67C0A1">1 <span class="Heading">Introduction</span></a>
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<h3>1 <span class="Heading">Introduction</span></h3>

<p>This is the manual for the <strong class="pkg">GAP</strong> package <strong class="pkg">QuaGroup</strong>, for doing computations with quantized enveloping algebras of semisimple Lie algebras.</p>

<p>Apart from the chapter you are currently reading, this document consists of two chapters. In Chapter <a href="chap2.html#X84AF2F1D7D4E7284"><span class="RefLink">2</span></a> we give a short summary of parts of the theory of quantized enveloping algebras. This fixes the notations and definitions that we use. Then in Chapter <a href="chap3.html#X80C9BD6584865157"><span class="RefLink">3</span></a> we describe the functions that constitute the package.</p>

<p>The package can be obtained from <span class="URL"><a href="http://www.math.uu.nl/people/graaf/quagroup.html">http://www.math.uu.nl/people/graaf/quagroup.html</a></span> The directory <code class="file">quagroup/doc</code> contains the manual of the package in <code class="file">dvi</code>, <code class="file">ps</code>, <code class="file">pdf</code> and <code class="file">html</code> format. The manual was built with the <strong class="pkg">GAP</strong> share package <strong class="pkg">GAPDoc</strong>, <a href="chapBib.html#biBLN01">[LN01]</a>. This means that, in order to be able to use the on-line help of <strong class="pkg">QuaGroup</strong>, you have to install <strong class="pkg">GAPDoc</strong> before calling <var class="Arg">LoadPackage("quagroup");</var>.</p>

<p>The main algorithm of the package (on which virtually the whole functionality relies) is a method for computing with so-called PBW-type bases, analogous to Poincar\'{e}-Birkhoff-Witt bases in universal enveloping algebras. In both cases commutation relations between the generators are used. However, in the latter case all commutation relations are of the form <span class="SimpleMath">yx=xy+z</span>, where <span class="SimpleMath">x,y</span> are generators, and <span class="SimpleMath">z</span> is a linear combination of generators. In the case of quantized enveloping algebras the situation is generally much more complicated. For example, in the quantized enveloping algebra of type <span class="SimpleMath">E_7</span> we have the following relation:</p>


<div class="example"><pre>
F62*F26 = (q)*F26*F62+(1-q^2)*F28*F61+(-q+q^3)*F30*F60+(-q^4+q^2)*F31*F59+
          (-q^4+q^2)*F33*F58+(-q^3+q^5)*F34*F57+(q^4-q^6)*F35*F56+
          (-q+q^-1-q^5+q^7)*F36*F55+(q^6)*F54
</pre></div>

<p>Due to the complexity of these commutation relations, some computations (even with rather small input) may take quite some time.</p>

<p>Remark: The package can deal with quantized enveloping algebras corresponding to root systems of rank at least up to eight, except <span class="SimpleMath">E_8</span>. In that case the computation of the necessary commutation relations took more than 2 GB. I wish to thank Steve Linton for trying this computation on the machines in St Andrews.</p>

<p>The following example illustrates some of the features of the package.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput"># We define a root system by giving its type:</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">R:= RootSystem( "B", 2 );</span>
<root system of type B2>
<span class="GAPprompt">gap></span> <span class="GAPinput"># Corresponding to the root system we define a quantized enveloping algebra:</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">U:= QuantizedUEA( R );</span>
QuantumUEA( <root system of type B2>, Qpar = q )
<span class="GAPprompt">gap></span> <span class="GAPinput"># It is generated by the generators of a so-called PBW-type basis:</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsOfAlgebra( U );</span>
[ F1, F2, F3, F4, K1, (-q^2+q^-2)*[ K1 ; 1 ]+K1, K2, (-q+q^-1)*[ K2 ; 1 ]+K2, 
  E1, E2, E3, E4 ]
<span class="GAPprompt">gap></span> <span class="GAPinput"># We can construct highest-weight modules:</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">V:= HighestWeightModule( U, [1,1] );</span>
<16-dimensional left-module over QuantumUEA( <root system of type B
2>, Qpar = q )>
<span class="GAPprompt">gap></span> <span class="GAPinput"># For modules of small dimension we can compute the corresponding</span>
<span class="GAPprompt">gap></span> <span class="GAPinput"># R-matrix:</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">U:= QuantizedUEA( RootSystem("A",2) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">V:= HighestWeightModule( U, [1,0] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">RMatrix( V );</span>
[ [ q^2, 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, q^3, 0, -q^4+q^2, 0, 0, 0, 0, 0 ], 
  [ 0, 0, q^3, 0, 0, 0, -q^4+q^2, 0, 0 ], [ 0, 0, 0, q^3, 0, 0, 0, 0, 0 ], 
  [ 0, 0, 0, 0, q^2, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, q^3, 0, -q^4+q^2, 0 ], 
  [ 0, 0, 0, 0, 0, 0, q^3, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, q^3, 0 ], 
  [ 0, 0, 0, 0, 0, 0, 0, 0, q^2 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput"># We can compute elements of the canonical basis of the "negative" part</span>
<span class="GAPprompt">gap></span> <span class="GAPinput"># of a quantized enveloping algebra:</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">U:= QuantizedUEA( RootSystem("F",4) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">B:= CanonicalBasis( U );</span>
<canonical basis of QuantumUEA( <root system of type F4>, Qpar = q ) >
<span class="GAPprompt">gap></span> <span class="GAPinput">p:= PBWElements( B, [0,1,2,1] ); </span>
[ F3*F9^(2)*F24, F3*F9*F23+(q^2)*F3*F9^(2)*F24, 
  (q^3+q)*F3*F9^(2)*F24+F7*F9*F24, (q^2)*F3*F9*F23+(q^4+q^2)*F3*F9^(2)*F
    24+(q)*F7*F9*F24+F7*F23, (q^4)*F3*F9^(2)*F24+(q)*F7*F9*F24+F8*F24, 
  (q^4)*F3*F9*F23+(q^6)*F3*F9^(2)*F24+(q^3)*F7*F9*F24+(q^2)*F7*F23+(q^2)*F8*F
    24+F9*F21, (q^3+q)*F3*F9*F23+(q^5+q^3)*F3*F9^(2)*F24+(q^2)*F7*F9*F24+(q)*F
    7*F23+(q)*F9*F21+F16 ]
<span class="GAPprompt">gap></span> <span class="GAPinput"># We can construct (anti-) automorphisms of quantized enveloping</span>
<span class="GAPprompt">gap></span> <span class="GAPinput"># algebras:</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">t:= AntiAutomorphismTau( U );</span>
<anti-automorphism of QuantumUEA( <root system of type F4>, Qpar = q )>
<span class="GAPprompt">gap></span> <span class="GAPinput">Image( t, p[1] );</span>
(q^4)*F3*F9*F23+(q^6)*F3*F9^(2)*F24+(q^3)*F7*F9*F24+(q^2)*F7*F23+(q^2)*F8*F
24+F9*F21
<span class="GAPprompt">gap></span> <span class="GAPinput"># (This is the sixth element of p.)</span>
</pre></div>


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