1. Introduction
  
  This is the manual for the GAP package QuaGroup, for doing computations with
  quantized enveloping algebras of semisimple Lie algebras.
  
  Apart  from the chapter you are currently reading, this document consists of
  two  chapters.  In Chapter 2. 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 3. we describe the functions that constitute
  the package.
  
  The          package          can          be          obtained         from
  http://www.math.uu.nl/people/graaf/quagroup.html  The directory quagroup/doc
  contains  the  manual  of  the  package in dvi, ps, pdf and html format. The
  manual was built with the GAP share package GAPDoc, [LN01]. This means that,
  in order to be able to use the on-line help of QuaGroup, you have to install
  GAPDoc before calling LoadPackage("quagroup");.
  
  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:
  
  ---------------------------  Example  ----------------------------
    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
  ------------------------------------------------------------------
  
  Due  to  the  complexity  of  these commutation relations, some computations
  (even with rather small input) may take quite some time.
  
  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.
  
  The following example illustrates some of the features of the package.
  
  ---------------------------  Example  ----------------------------
    # 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.)
  ------------------------------------------------------------------