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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