  
  [1X1 [33X[0;0YIntroduction[133X[101X
  
  [33X[0;0YThis is the manual for the [5XGAP[105X package [5XQuaGroup[105X, for doing computations with
  quantized enveloping algebras of semisimple Lie algebras.[133X
  
  [33X[0;0YApart  from the chapter you are currently reading, this document consists of
  two chapters. In Chapter [14X2[114X 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  [14X3[114X we describe the functions that constitute the
  package.[133X
  
  [33X[0;0YThe          package          can          be          obtained         from
  [7Xhttp://www.math.uu.nl/people/graaf/quagroup.html[107X  The directory [11Xquagroup/doc[111X
  contains  the  manual  of  the  package in [11Xdvi[111X, [11Xps[111X, [11Xpdf[111X and [11Xhtml[111X format. The
  manual was built with the [5XGAP[105X share package [5XGAPDoc[105X, [LN01]. This means that,
  in order to be able to use the on-line help of [5XQuaGroup[105X, you have to install
  [5XGAPDoc[105X before calling [3XLoadPackage("quagroup");[103X.[133X
  
  [33X[0;0YThe   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
  [22Xyx=xy+z[122X,  where  [22Xx,y[122X  are  generators,  and  [22Xz[122X  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 [22XE_7[122X we have the following relation:[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28XF62*F26 = (q)*F26*F62+(1-q^2)*F28*F61+(-q+q^3)*F30*F60+(-q^4+q^2)*F31*F59+[128X[104X
    [4X[28X          (-q^4+q^2)*F33*F58+(-q^3+q^5)*F34*F57+(q^4-q^6)*F35*F56+[128X[104X
    [4X[28X          (-q+q^-1-q^5+q^7)*F36*F55+(q^6)*F54[128X[104X
  [4X[32X[104X
  
  [33X[0;0YDue  to  the  complexity  of  these commutation relations, some computations
  (even with rather small input) may take quite some time.[133X
  
  [33X[0;0YRemark:   The   package   can   deal   with  quantized  enveloping  algebras
  corresponding  to  root systems of rank at least up to eight, except [22XE_8[122X. 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.[133X
  
  [33X[0;0YThe following example illustrates some of the features of the package.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27X# We define a root system by giving its type:[127X[104X
    [4X[25Xgap>[125X [27XR:= RootSystem( "B", 2 );[127X[104X
    [4X[28X<root system of type B2>[128X[104X
    [4X[25Xgap>[125X [27X# Corresponding to the root system we define a quantized enveloping algebra:[127X[104X
    [4X[25Xgap>[125X [27XU:= QuantizedUEA( R );[127X[104X
    [4X[28XQuantumUEA( <root system of type B2>, Qpar = q )[128X[104X
    [4X[25Xgap>[125X [27X# It is generated by the generators of a so-called PBW-type basis:[127X[104X
    [4X[25Xgap>[125X [27XGeneratorsOfAlgebra( U );[127X[104X
    [4X[28X[ F1, F2, F3, F4, K1, (-q^2+q^-2)*[ K1 ; 1 ]+K1, K2, (-q+q^-1)*[ K2 ; 1 ]+K2, [128X[104X
    [4X[28X  E1, E2, E3, E4 ][128X[104X
    [4X[25Xgap>[125X [27X# We can construct highest-weight modules:[127X[104X
    [4X[25Xgap>[125X [27XV:= HighestWeightModule( U, [1,1] );[127X[104X
    [4X[28X<16-dimensional left-module over QuantumUEA( <root system of type B[128X[104X
    [4X[28X2>, Qpar = q )>[128X[104X
    [4X[25Xgap>[125X [27X# For modules of small dimension we can compute the corresponding[127X[104X
    [4X[25Xgap>[125X [27X# R-matrix:[127X[104X
    [4X[25Xgap>[125X [27XU:= QuantizedUEA( RootSystem("A",2) );;[127X[104X
    [4X[25Xgap>[125X [27XV:= HighestWeightModule( U, [1,0] );;[127X[104X
    [4X[25Xgap>[125X [27XRMatrix( V );[127X[104X
    [4X[28X[ [ q^2, 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, q^3, 0, -q^4+q^2, 0, 0, 0, 0, 0 ], [128X[104X
    [4X[28X  [ 0, 0, q^3, 0, 0, 0, -q^4+q^2, 0, 0 ], [ 0, 0, 0, q^3, 0, 0, 0, 0, 0 ], [128X[104X
    [4X[28X  [ 0, 0, 0, 0, q^2, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, q^3, 0, -q^4+q^2, 0 ], [128X[104X
    [4X[28X  [ 0, 0, 0, 0, 0, 0, q^3, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, q^3, 0 ], [128X[104X
    [4X[28X  [ 0, 0, 0, 0, 0, 0, 0, 0, q^2 ] ][128X[104X
    [4X[25Xgap>[125X [27X# We can compute elements of the canonical basis of the "negative" part[127X[104X
    [4X[25Xgap>[125X [27X# of a quantized enveloping algebra:[127X[104X
    [4X[25Xgap>[125X [27XU:= QuantizedUEA( RootSystem("F",4) );;[127X[104X
    [4X[25Xgap>[125X [27XB:= CanonicalBasis( U );[127X[104X
    [4X[28X<canonical basis of QuantumUEA( <root system of type F4>, Qpar = q ) >[128X[104X
    [4X[25Xgap>[125X [27Xp:= PBWElements( B, [0,1,2,1] ); [127X[104X
    [4X[28X[ F3*F9^(2)*F24, F3*F9*F23+(q^2)*F3*F9^(2)*F24, [128X[104X
    [4X[28X  (q^3+q)*F3*F9^(2)*F24+F7*F9*F24, (q^2)*F3*F9*F23+(q^4+q^2)*F3*F9^(2)*F[128X[104X
    [4X[28X    24+(q)*F7*F9*F24+F7*F23, (q^4)*F3*F9^(2)*F24+(q)*F7*F9*F24+F8*F24, [128X[104X
    [4X[28X  (q^4)*F3*F9*F23+(q^6)*F3*F9^(2)*F24+(q^3)*F7*F9*F24+(q^2)*F7*F23+(q^2)*F8*F[128X[104X
    [4X[28X    24+F9*F21, (q^3+q)*F3*F9*F23+(q^5+q^3)*F3*F9^(2)*F24+(q^2)*F7*F9*F24+(q)*F[128X[104X
    [4X[28X    7*F23+(q)*F9*F21+F16 ][128X[104X
    [4X[25Xgap>[125X [27X# We can construct (anti-) automorphisms of quantized enveloping[127X[104X
    [4X[25Xgap>[125X [27X# algebras:[127X[104X
    [4X[25Xgap>[125X [27Xt:= AntiAutomorphismTau( U );[127X[104X
    [4X[28X<anti-automorphism of QuantumUEA( <root system of type F4>, Qpar = q )>[128X[104X
    [4X[25Xgap>[125X [27XImage( t, p[1] );[127X[104X
    [4X[28X(q^4)*F3*F9*F23+(q^6)*F3*F9^(2)*F24+(q^3)*F7*F9*F24+(q^2)*F7*F23+(q^2)*F8*F[128X[104X
    [4X[28X24+F9*F21[128X[104X
    [4X[25Xgap>[125X [27X# (This is the sixth element of p.)[127X[104X
  [4X[32X[104X
  
