  
  [1m[4m[31m1. Introduction[0m
  
  This is the manual for the [1mGAP[0m package [1mQuaGroup[0m, 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 [1m2.[0m 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 [1m3.[0m we describe the functions that constitute
  the package.
  
  The          package          can          be          obtained         from
  [34mhttp://www.math.uu.nl/people/graaf/quagroup.html[0m  The directory [1mquagroup/doc[0m
  contains  the  manual  of  the  package in [1mdvi[0m, [1mps[0m, [1mpdf[0m and [1mhtml[0m format. The
  manual was built with the [1mGAP[0m share package [1mGAPDoc[0m, [LN01]. This means that,
  in order to be able to use the on-line help of [1mQuaGroup[0m, you have to install
  [1mGAPDoc[0m before calling [22m[34mLoadPackage("quagroup");[0m.
  
  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:
  
  [22m[35m---------------------------  Example  ----------------------------[0m
    [22m[35mF62*F26 = (q)*F26*F62+(1-q^2)*F28*F61+(-q+q^3)*F30*F60+(q^2-q^4)*F31*F59+[0m
    [22m[35m          (q^2-q^4)*F33*F58+(-q^3+q^5)*F34*F57+(q^4-q^6)*F35*F56+[0m
    [22m[35m          (q^-1-q-q^5+q^7)*F36*F55+(q^6)*F54[0m
  [22m[35m------------------------------------------------------------------[0m
  
  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.
  
  [22m[35m---------------------------  Example  ----------------------------[0m
    [22m[35m# We define a root system by giving its type:[0m
    [22m[35mgap> R:= RootSystem( "B", 2 );[0m
    [22m[35m<root system of type B2>[0m
    [22m[35m# Corresponding to the root system we define a quantized enveloping algebra:[0m
    [22m[35mgap> U:= QuantizedUEA( R );[0m
    [22m[35mQuantumUEA( <root system of type B2>, Qpar = q )[0m
    [22m[35m# It is generated by the generators of a so-called PBW-type basis:[0m
    [22m[35mgap> GeneratorsOfAlgebra( U );[0m
    [22m[35m[ F1, F2, F3, F4, K1, K1+(q^-2-q^2)*[ K1 ; 1 ], K2, K2+(q^-1-q)*[ K2 ; 1 ],[0m
    [22m[35m  E1, E2, E3, E4 ][0m
    [22m[35m# We can construct highest-weight modules:[0m
    [22m[35mgap> V:= HighestWeightModule( U, [1,1] );[0m
    [22m[35m<16-dimensional left-module over QuantumUEA( <root system of type B[0m
    [22m[35m2>, Qpar = q )>[0m
    [22m[35m# For modules of small dimension we can compute the corresponding[0m
    [22m[35m# R-matrix:[0m
    [22m[35mgap> U:= QuantizedUEA( RootSystem("A",2) );;[0m
    [22m[35mgap> V:= HighestWeightModule( U, [1,0] );;[0m
    [22m[35mgap> RMatrix( V );[0m
    [22m[35m[ [ q^2, 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, q^3, 0, q^2-q^4, 0, 0, 0, 0, 0 ], [0m
    [22m[35m  [ 0, 0, q^3, 0, 0, 0, q^2-q^4, 0, 0 ], [ 0, 0, 0, q^3, 0, 0, 0, 0, 0 ], [0m
    [22m[35m  [ 0, 0, 0, 0, q^2, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, q^3, 0, q^2-q^4, 0 ], [0m
    [22m[35m  [ 0, 0, 0, 0, 0, 0, q^3, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, q^3, 0 ], [0m
    [22m[35m  [ 0, 0, 0, 0, 0, 0, 0, 0, q^2 ] ][0m
    [22m[35m# We can compute elements of the canonical basis of the "negative" part[0m
    [22m[35m# of a quantized enveloping algebra:[0m
    [22m[35mgap> U:= QuantizedUEA( RootSystem("F",4) );;[0m
    [22m[35mgap> B:= CanonicalBasis( U );[0m
    [22m[35m<canonical basis of QuantumUEA( <root system of type F4>, Qpar = q ) >[0m
    [22m[35mgap> p:= PBWElements( B, [0,1,2,1] ); [0m
    [22m[35m[ F3*F9^(2)*F24, F3*F9*F23+(q^2)*F3*F9^(2)*F24, [0m
    [22m[35m  (q+q^3)*F3*F9^(2)*F24+F7*F9*F24, (q^2)*F3*F9*F23+(q^2+q^4)*F3*F9^(2)*F[0m
    [22m[35m    24+(q)*F7*F9*F24+F7*F23, (q^4)*F3*F9^(2)*F24+(q)*F7*F9*F24+F8*F24, [0m
    [22m[35m  (q^4)*F3*F9*F23+(q^6)*F3*F9^(2)*F24+(q^3)*F7*F9*F24+(q^2)*F7*F23+(q^2)*F[0m
    [22m[35m    8*F24+F9*F21, (q+q^3)*F3*F9*F23+(q^3+q^5)*F3*F9^(2)*F24+(q^2)*F7*F9*F[0m
    [22m[35m    24+(q)*F7*F23+(q)*F9*F21+F16 ][0m
    [22m[35m# We can construct (anti-) automorphisms of quantized enveloping[0m
    [22m[35m# algebras:[0m
    [22m[35mgap> t:= AntiAutomorphismTau( U );[0m
    [22m[35m<anti-automorphism of QuantumUEA( <root system of type F4>, Qpar = q )>[0m
    [22m[35mgap> Image( t, p[1] );[0m
    [22m[35m(q^4)*F3*F9*F23+(q^6)*F3*F9^(2)*F24+(q^3)*F7*F9*F24+(q^2)*F7*F23+(q^2)*F8*F[0m
    [22m[35m24+F9*F21[0m
    [22m[35m# (This is the sixth element of p.)[0m
  [22m[35m------------------------------------------------------------------[0m
  
