GAP (quagroup) - Chapter 1: Introduction

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1 Introduction

### 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:

    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

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.

    gap> # We define a root system by giving its type:
    gap> R:= RootSystem( "B", 2 );
    <root system of type B2>
    gap> # Corresponding to the root system we define a quantized enveloping algebra:
    gap> U:= QuantizedUEA( R );
    QuantumUEA( <root system of type B2>, Qpar = q )
    gap> # It is generated by the generators of a so-called PBW-type basis:
    gap> GeneratorsOfAlgebra( U );
    [ F1, F2, F3, F4, K1, (-q^2+q^-2)*[ K1 ; 1 ]+K1, K2, (-q+q^-1)*[ K2 ; 1 ]+K2,
      E1, E2, E3, E4 ]
    gap> # 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 )>
    gap> # For modules of small dimension we can compute the corresponding
    gap> # 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^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 ] ]
    gap> # We can compute elements of the canonical basis of the "negative" part
    gap> # 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^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 ]
    gap> # We can construct (anti-) automorphisms of quantized enveloping
    gap> # 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
    gap> # (This is the sixth element of p.)

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