<?xml version="1.0" encoding="ISO-8859-1"?>
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
"DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml">
<head>
<title>GAP (quagroup) - Chapter 1: Introduction</title>
<meta http-equiv="content-type" content="text/html; charset=iso-8859-1" />
<meta name="generator" content="GAPDoc2HTML" />
<link rel="stylesheet" type="text/css" href="manual.css" />
</head>
<body>
<div class="pcenter"><table class="chlink"><tr><td class="chlink1">Goto Chapter: </td><td><a href="chap0.html">Top</a></td><td><a href="chap1.html">1</a></td><td><a href="chap2.html">2</a></td><td><a href="chap3.html">3</a></td><td><a href="chapBib.html">Bib</a></td><td><a href="chapInd.html">Ind</a></td></tr></table><br /></div>
<p><a id="s0ss0" name="s0ss0"></a></p>
<h3>1. Introduction</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#s0ss0"><b>2.</b></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#s0ss0"><b>3.</b></a> we describe the functions that constitute the package.</p>
<p>The package can be obtained from <a href="http://www.math.uu.nl/people/graaf/quagroup.html">http://www.math.uu.nl/people/graaf/quagroup.html</a> 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>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 yx=xy+z, where x,y are generators, and z 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 E_7 we have the following relation:</p>
<table class="example">
<tr><td><pre>
F62*F26 = (q)*F26*F62+(1-q^2)*F28*F61+(-q+q^3)*F30*F60+(q^2-q^4)*F31*F59+
(q^2-q^4)*F33*F58+(-q^3+q^5)*F34*F57+(q^4-q^6)*F35*F56+
(q^-1-q-q^5+q^7)*F36*F55+(q^6)*F54
</pre></td></tr></table>
<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 E_8. 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>
<table class="example">
<tr><td><pre>
# We define a root system by giving its type:
gap> R:= RootSystem( "B", 2 );
<root system of type B2>
# Corresponding to the root system we define a quantized enveloping algebra:
gap> U:= QuantizedUEA( R );
QuantumUEA( <root system of type B2>, Qpar = q )
# It is generated by the generators of a so-called PBW-type basis:
gap> GeneratorsOfAlgebra( U );
[ F1, F2, F3, F4, K1, K1+(q^-2-q^2)*[ K1 ; 1 ], K2, K2+(q^-1-q)*[ K2 ; 1 ],
E1, E2, E3, E4 ]
# We can construct highest-weight modules:
gap> V:= HighestWeightModule( U, [1,1] );
<16-dimensional left-module over QuantumUEA( <root system of type B
2>, Qpar = q )>
# For modules of small dimension we can compute the corresponding
# R-matrix:
gap> U:= QuantizedUEA( RootSystem("A",2) );;
gap> V:= HighestWeightModule( U, [1,0] );;
gap> RMatrix( V );
[ [ q^2, 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, q^3, 0, q^2-q^4, 0, 0, 0, 0, 0 ],
[ 0, 0, q^3, 0, 0, 0, q^2-q^4, 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^2-q^4, 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 ] ]
# We can compute elements of the canonical basis of the "negative" part
# of a quantized enveloping algebra:
gap> U:= QuantizedUEA( RootSystem("F",4) );;
gap> B:= CanonicalBasis( U );
<canonical basis of QuantumUEA( <root system of type F4>, Qpar = q ) >
gap> p:= PBWElements( B, [0,1,2,1] );
[ F3*F9^(2)*F24, F3*F9*F23+(q^2)*F3*F9^(2)*F24,
(q+q^3)*F3*F9^(2)*F24+F7*F9*F24, (q^2)*F3*F9*F23+(q^2+q^4)*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)*F
8*F24+F9*F21, (q+q^3)*F3*F9*F23+(q^3+q^5)*F3*F9^(2)*F24+(q^2)*F7*F9*F
24+(q)*F7*F23+(q)*F9*F21+F16 ]
# We can construct (anti-) automorphisms of quantized enveloping
# algebras:
gap> t:= AntiAutomorphismTau( U );
<anti-automorphism of QuantumUEA( <root system of type F4>, Qpar = q )>
gap> Image( t, p[1] );
(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
# (This is the sixth element of p.)
</pre></td></tr></table>
<div class="pcenter">
<table class="chlink"><tr><td><a href="chap0.html">Top of Book</a></td><td><a href="chap0.html">Previous Chapter</a></td><td><a href="chap2.html">Next Chapter</a></td></tr></table>
<br />
<div class="pcenter"><table class="chlink"><tr><td class="chlink1">Goto Chapter: </td><td><a href="chap0.html">Top</a></td><td><a href="chap1.html">1</a></td><td><a href="chap2.html">2</a></td><td><a href="chap3.html">3</a></td><td><a href="chapBib.html">Bib</a></td><td><a href="chapInd.html">Ind</a></td></tr></table><br /></div>
</div>
<hr />
<p class="foot">generated by <a href="http://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc">GAPDoc2HTML</a></p>
</body>
</html>
distrib
>
Mandriva
>
2010.0
>
i586
>
media
>
contrib-release
>
by-pkgid
>
91213ddcfbe7f54821d42c2d9e091326
>
files
>
2373
