This is the manual for the &GAP; package <Package>QuaGroup</Package>, for
doing computations with quantized enveloping algebras of semisimple Lie
algebras.<P/>
Apart from the chapter you are currently reading, this document consists of
two chapters. In Chapter <Ref Chap="chap2"/> 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
<Ref Chap="chap3"/> we describe the functions that
constitute the package. <P/>
The package can be obtained from
<URL>http://www.math.uu.nl/people/graaf/quagroup.html</URL>
The directory <F>quagroup/doc</F> contains the manual of the package in
<F>dvi</F>, <F>ps</F>, <F>pdf</F> and <F>html</F> format.
The manual was built with the &GAP; share package
<Package>GAPDoc</Package>, <Cite Key="LN01"/>.
This means that, in order to be able to use the on-line help of
<Package>QuaGroup</Package>, you have to install
<Package>GAPDoc</Package> before calling
<A>LoadPackage("quagroup");</A>. <P/>
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 <M>yx=xy+z</M>,
where <M>x,y</M> are generators, and <M>z</M> 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 <M>E_7</M> we have the following relation:
Due to the complexity of these commutation relations, some computations
(even with rather small input) may take quite some time.<P/>
Remark: The package can deal with quantized enveloping algebras corresponding
to root systems of rank at least up to eight, except <M>E_8</M>. 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.<P/>
The following example illustrates some of the features of the package.<P/>
<Example><![CDATA[
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.)
]]></Example>
Let <Math>v</Math> be an indeterminate over <Math>\mathbb{Q}</Math>. For a positive integer <Math>n</Math>
we set
<Display> [n] = v^{n-1}+v^{n-3}+\cdots + v^{-n+3}+v^{-n+1}. </Display>
We say that <Math>[n]</Math> is the <E> Gaussian integer </E> corresponding to <Math>n</Math>.
The <E> Gaussian factorial </E> <Math>[n]!</Math> is defined by
<Display> [0]! = 1, ~ [n]! = [n][n-1]\cdots [1], \text{ for } n>0.</Display>
Finally, the <E> Gaussian binomial </E> is
<Display> \begin{bmatrix} n \\ k \end{bmatrix} = \frac{[n]!}{[k]![n-k]!}.</Display>
Let <Math>\mathfrak{g}</Math> be a semisimple Lie algebra with root system <Math>\Phi</Math>.
By <Math>\Delta=\{\alpha_1,\ldots, \alpha_l \}</Math> we denote a fixed simple system
of <Math>\Phi</Math>. Let <Math>C=(C_{ij})</Math> be the Cartan matrix of <Math>\Phi</Math> (with respect to
<Math>\Delta</Math>, i.e., <Math> C_{ij} = \langle \alpha_i, \alpha_j^{\vee} \rangle</Math>).
Let <Math>d_1,\ldots, d_l</Math> be the unique sequence of positive integers with
greatest common divisor <Math>1</Math>, such that <Math> d_i C_{ji} = d_j C_{ij} </Math>, and set
<Math> (\alpha_i,\alpha_j) = d_j C_{ij} </Math>. (We note that this implies that
<Math>(\alpha_i,\alpha_i)</Math> is divisible by <Math>2</Math>.) By <Math>P</Math> we denote the weight
lattice, and we extend the form <Math>(~,~)</Math> to <Math>P</Math> by bilinearity. <P/>
By <Math>W(\Phi)</Math> we denote the Weyl group of <Math>\Phi</Math>. It is generated by the
simple reflections <Math>s_i=s_{\alpha_i}</Math> for <Math>1\leq i\leq l</Math> (where
<Math>s_{\alpha}</Math> is defined by <Math>s_{\alpha}(\beta) = \beta - \langle\beta,
\alpha^{\vee}\rangle \alpha</Math>).<P/>
We work over the field <Math>\mathbb{Q}(q)</Math>. For <Math>\alpha\in\Phi </Math> we set
<Display> q_{\alpha} = q^{\frac{(\alpha,\alpha)}{2}},</Display>
and for a non-negative integer <Math>n</Math>, <Math>[n]_{\alpha}= [n]_{v=q_{\alpha}}</Math>;
<Math>[n]_{\alpha}!</Math> and <Math>\begin{bmatrix} n \\ k \end{bmatrix}_{\alpha}</Math> are
defined analogously.<P/>
The quantized enveloping algebra <Math>U_q(\mathfrak{g})</Math> is the associative
algebra (with one) over <Math>\mathbb{Q}(q)</Math> generated by <Math>F_{\alpha}</Math>,
<Math>K_{\alpha}</Math>, <Math>K_{\alpha}^{-1}</Math>, <Math>E_{\alpha}</Math> for <Math>\alpha\in\Delta</Math>,
subject to the following relations
<Display>
\begin{aligned}
K_{\alpha}K_{\alpha}^{-1} &= K_{\alpha}^{-1}K_{\alpha} = 1,~
K_{\alpha}K_{\beta} = K_{\beta}K_{\alpha}\\
E_{\beta} K_{\alpha} &= q^{-(\alpha,\beta)}K_{\alpha} E_{\beta}\\
K_{\alpha} F_{\beta} &= q^{-(\alpha,\beta)}F_{\beta}K_{\alpha}\\
E_{\alpha} F_{\beta} &= F_{\beta}E_{\alpha} +\delta_{\alpha,\beta}
\frac{K_{\alpha}-K_{\alpha}^{-1}}{q_{\alpha}-q_{\alpha}^{-1}}
\end{aligned}
</Display>
together with, for <Math>\alpha\neq \beta\in\Delta</Math>,
<Display>
\begin{aligned}
\sum_{k=0}^{1-\langle \beta,\alpha^{\vee}\rangle }
(-1)^k \begin{bmatrix} 1-\langle \beta,\alpha^{\vee}\rangle \\ k
\end{bmatrix}_{\alpha} E_{\alpha}^{1-\langle \beta,\alpha^{\vee}\rangle-k}
E_{\beta} E_{\alpha}^k =0 & \\
\sum_{k=0}^{1-\langle \beta,\alpha^{\vee}\rangle } (-1)^k \begin{bmatrix}
1-\langle \beta,\alpha^{\vee}\rangle \\ k \end{bmatrix}_{\alpha}
F_{\alpha}^{1-\langle \beta,\alpha^{\vee}\rangle-k}
F_{\beta} F_{\alpha}^k =0 &.
\end{aligned}
</Display>
The quantized enveloping algebra has an automorphism <Math>\omega</Math> defined by
<Math>\omega( F_{\alpha} ) = E_{\alpha}</Math>, <Math>\omega(E_{\alpha})= F_{\alpha}</Math> and
<Math>\omega(K_{\alpha})=K_{\alpha}^{-1}</Math>. Also there is an anti-automorphism
<Math>\tau</Math> defined by <Math>\tau(F_{\alpha})=F_{\alpha}</Math>, <Math>\tau(E_{\alpha})=
E_{\alpha}</Math> and <Math>\tau(K_{\alpha})=K_{\alpha}^{-1}</Math>. We have <Math>\omega^2=1</Math>
and <Math>\tau^2=1</Math>.<P/>
If the Dynkin diagram of <Math>\Phi</Math> admits a diagram automorphism <Math>\pi</Math>, then
<Math>\pi</Math> induces an automorphism of <Math>U_q(\mathfrak{g})</Math> in the obvious way
(<Math>\pi</Math> is a permutation of the simple roots; we permute the <Math>F_{\alpha}</Math>,
<Math>E_{\alpha}</Math>, <Math>K_{\alpha}^{\pm 1}</Math> accordingly).<P/>
Now we view <Math>U_q(\mathfrak{g})</Math> as an algebra over <Math>\mathbb{Q}</Math>, and we let
<Math>\overline{\phantom{A}} : U_q(\mathfrak{g})\to U_q(\mathfrak{g})</Math> be the
automorphism defined by <Math>\overline{F_{\alpha}}=F_{\alpha}</Math>,
<Math>\overline{K_{\alpha}}= K_{\alpha}^{-1}</Math>, <Math>\overline{E_{\alpha}}=E_{\alpha}</Math>,
<Math>\overline{q}=q^{-1}</Math>.
</Section>
<Section Label="sec2.3"> <Heading>Representations of <Math>U_q(\mathfrak{g})</Math> </Heading>
Let <Math>\lambda\in P</Math> be a dominant weight. Then there is a unique irreducible
highest-weight module over <Math>U_q(\mathfrak{g})</Math> with highest weight <Math>\lambda</Math>.
We denote it by <Math>V(\lambda)</Math>. It has the same character as the irreducible
highest-weight module over <Math>\mathfrak{g}</Math> with highest weight <Math>\lambda</Math>.
Furthermore, every finite-dimensional <Math>U_q(\mathfrak{g})</Math>-module is a direct
sum of irreducible highest-weight modules.<P/>
It is well-known that <Math>U_q(\mathfrak{g})</Math> is a Hopf algebra. The
comultiplication <Math>\Delta : U_q(\mathfrak{g})\to U_q(\mathfrak{g})
\otimes U_q(\mathfrak{g})</Math> is defined by
<Display> \begin{aligned}
\Delta(E_{\alpha}) &= E_{\alpha}\otimes 1 + K_{\alpha}\otimes E_{\alpha}\\
\Delta(F_{\alpha}) &= F_{\alpha}\otimes K_{\alpha}^{-1} +
1\otimes F_{\alpha}\\
\Delta(K_{\alpha}) &= K_{\alpha}\otimes K_{\alpha}.
\end{aligned} </Display>
(Note that we use the same symbol to denote a simple system of <Math>\Phi</Math>; of
course this does not cause confusion.) The counit <Math>\varepsilon :
U_q(\mathfrak{g}) \to \mathbb{Q}(q)</Math> is a homomorphism defined by
<Math>\varepsilon(E_{\alpha})=\varepsilon(F_{\alpha})=0</Math>,
<Math>\varepsilon( K_{\alpha}) =1</Math>. Finally, the antipode
<Math>S: U_q(\mathfrak{g})\to U_q(\mathfrak{g})</Math> is an anti-automorphism given
by <Math>S(E_{\alpha})=-K_{\alpha}^{-1}E_{\alpha}</Math>, <Math>S(F_{\alpha})=-F_{\alpha}
K_{\alpha}</Math>, <Math>S(K_{\alpha})=K_{\alpha}^{-1}</Math>.<P/>
Using <Math>\Delta</Math> we can make the tensor product <Math>V\otimes W</Math> of two
<Math>U_q(\mathfrak{g})</Math>-modules <Math>V,W</Math> into a <Math>U_q(\mathfrak{g})</Math>-module.
The counit <Math>\varepsilon</Math> yields a trivial <Math>1</Math>-dimensional
<Math>U_q(\mathfrak{g})</Math>-module. And with <Math>S</Math> we can define a
<Math>U_q(\mathfrak{g})</Math>-module structure on the dual <Math>V^*</Math> of a
<Math>U_q(\mathfrak{g})</Math>-module <Math>V</Math>, by <Math>(u\cdot f)(v) = f(S(u)\cdot v )</Math>.<P/>
The Hopf algebra structure given above is not the only one possible. For
example, we can twist <Math>\Delta,\varepsilon,S</Math> by an automorphism, or an
anti-automorphism <Math>f</Math>. The twisted comultiplication is given by
<Display>\Delta^f = f\otimes f \circ\Delta\circ f^{-1}.</Display>
The twisted antipode by
<Display> S^f = \begin{cases} f\circ S\circ f^{-1} & \text{ if }f\text{ is an
automorphism}\\ f\circ S^{-1}\circ f^{-1}
& \text{ if }f\text{ is an anti-automorphism.}\end{cases}</Display>
And the twisted counit by <Math>\varepsilon^f = \varepsilon\circ f^{-1}</Math>
(see <Cite Key="J96"/>, 3.8).
The first problem one has to deal with when working with <Math>U_q(\mathfrak{g})</Math>
is finding a basis of it, along with an algorithm for expressing the product
of two basis elements as a linear combination of basis elements.
First of all we have that <Math>U_q(\mathfrak{g})\cong U^-\otimes U^0\otimes U^+</Math>
(as vector spaces), where <Math>U^-</Math> is the subalgebra generated by the
<Math>F_{\alpha}</Math>, <Math>U^0</Math> is the subalgebra generated by the <Math>K_{\alpha}</Math>, and
<Math>U^+</Math> is generated by the <Math>E_{\alpha}</Math>. So a basis of <Math>U_q(\mathfrak{g})</Math> is
formed by all elements <Math>FKE</Math>, where <Math>F</Math>, <Math>K</Math>, <Math>E</Math> run through bases of <Math>U^-</Math>,
<Math>U^0</Math>, <Math>U^+</Math> respectively.<P/>
Finding a basis of <Math>U^0</Math> is easy: it is spanned by all <Math>K_{\alpha_1}^{r_1}
\cdots K_{\alpha_l}^{r_l}</Math>, where <Math>r_i\in\mathbb{Z}</Math>. For <Math>U^-</Math>, <Math>U^+</Math> we
use the so-called <E>PBW-type</E> bases. They are defined as follows.
For <Math>\alpha,\beta\in\Delta</Math> we set <Math>r_{\beta,\alpha} = -\langle \beta,
\alpha^{\vee}\rangle</Math>. Then for <Math>\alpha\in\Delta</Math> we have the automorphism
<Math>T_{\alpha} : U_q(\mathfrak{g})\to U_q(\mathfrak{g})</Math> defined by
<Display> \begin{aligned}
T_{\alpha}(E_{\alpha}) &= -F_{\alpha}K_{\alpha}\\
T_{\alpha}(E_{\beta}) &= \sum_{i=0}^{r_{\beta,\alpha}}
(-1)^i q_{\alpha}^{-i} E_{\alpha}^{(r_{\beta,\alpha}-i)}E_{\beta}
E_{\alpha}^{(i)}\text{ if } \alpha\neq\beta \\
T_{\alpha}(K_{\beta}) &= K_{\beta}K_{\alpha}^{r_{\beta,\alpha}}\\
T_{\alpha}(F_{\alpha}) &= -K_{\alpha}^{-1} E_{\alpha}\\
T_{\alpha}(F_{\beta}) &= \sum_{i=0}^{r_{\beta,\alpha}}
(-1)^i q_{\alpha}^{i} F_{\alpha}^{(i)}F_{\beta}F_{\alpha}^
{(r_{\beta,\alpha}-i)}\text{ if }\alpha\neq\beta,\\
\end{aligned} </Display>
(where <Math>E_{\alpha}^{(k)} = E_{\alpha}^k/[k]_{\alpha}!</Math>, and likewise for
<Math>F_{\alpha}^{(k)}</Math>). <P/>
Let <Math>w_0=s_{i_1}\cdots s_{i_t}</Math> be a reduced expression for the longest element in the Weyl group <Math>W(\Phi)</Math>. For <Math>1\leq k\leq t</Math> set
<Math>F_k = T_{\alpha_{i_1}}\cdots T_{\alpha_{i_{k-1}}}(F_{\alpha_{i_k}})</Math>,
and <Math>E_k = T_{\alpha_{i_1}}\cdots T_{\alpha_{i_{k-1}}}(E_{\alpha_{i_k}})</Math>.
Then <Math>F_k\in U^-</Math>, and <Math>E_k\in U^+</Math>. Furthermore, the elements <Math>F_1^{m_1}
\cdots F_t^{m_t}</Math>, <Math>E_1^{n_1}\cdots E_t^{n_t}</Math> (where the <Math>m_i</Math>, <Math>n_i</Math>
are non-negative integers) form bases of <Math>U^-</Math> and <Math>U^+</Math> respectively. <P/>
The elements <Math>F_{\alpha}</Math> and <Math>E_{\alpha}</Math> are said to have weight
<Math>-\alpha</Math> and <Math>\alpha</Math> respectively, where <Math>\alpha</Math> is a simple root.
Furthermore, the weight of a product <Math>ab</Math> is the sum of the weights of <Math>a</Math>
and <Math>b</Math>. Now elements of <Math>U^-</Math>, <Math>U^+</Math> that are linear combinations of elements
of the same weight are said to be homogeneous. It can be shown that the
elements <Math>F_k</Math>, and <Math>E_k</Math> are homogeneous of weight <Math>-\beta</Math> and <Math>\beta</Math>
respectively, where <Math>\beta=s_{i_1}\cdots s_{i_{k-1}}(\alpha_{i_k})</Math>. <P/>
In the sequel we use the notation <Math>F_k^{(m)} = F_k^m/[m]_{\alpha_{i_k}}!</Math>,
and <Math>E_k^{(n)} = E_k^n/[n]_{\alpha_{i_k}}!</Math>. <P/>
</Section>
<Section Label="sec2.5"> <Heading>The <Math>{\mathbb Z}</Math>-form of <Math>U_q(\mathfrak{g})</Math> </Heading>
For <Math>\alpha\in\Delta</Math> set
<Display>\begin{bmatrix} K_{\alpha} \\ n \end{bmatrix} = \prod_{i=1}^n
\frac{q_{\alpha}^{-i+1}K_{\alpha} - q_{\alpha}^{i-1} K_{\alpha}^{-1}}
{q_{\alpha}^i-q_{\alpha}^{-i}}.</Display>
Then according to <Cite Key="L90"/>, Theorem 6.7 the elements
<Display>F_1^{(k_1)}\cdots F_t^{(k_t)} K_{\alpha_1}^{\delta_1}
\begin{bmatrix} K_{\alpha_1} \\ m_1 \end{bmatrix}
\cdots K_{\alpha_l}^{\delta_l}
\begin{bmatrix} K_{\alpha_l} \\ m_l \end{bmatrix}
E_1^{(n_1)}\cdots E_t^{(n_t)},</Display>
(where <Math>k_i,m_i,n_i\geq 0</Math>, <Math>\delta_i=0,1</Math>) form a basis of
<Math>U_q(\mathfrak{g})</Math>, such that the product of any two basis elements is a
linear combination of basis elements with coefficients in
<Math>\mathbb{Z}[q,q^{-1}]</Math>. The quantized enveloping algebra over
<Math>\mathbb{Z}[q,q^{-1}]</Math> with this basis is called the <Math>\mathbb{Z}</Math>-form of
<Math>U_q(\mathfrak{g})</Math>, and denoted by <Math>U_{\mathbb{Z}}</Math>. Since <Math>U_{\mathbb{Z}}</Math>
is defined over <Math>\mathbb{Z}[q,q^{-1}]</Math> we can specialize <Math>q</Math> to any nonzero element <Math>\epsilon</Math> of a field <Math>F</Math>, and obtain an algebra <Math>U_{\epsilon}</Math> over
<Math>F</Math>. <P/>
We call <Math>q\in \mathbb{Q}(q)</Math>, and <Math>\epsilon \in F</Math> the quantum parameter of
<Math>U_q(\mathfrak{g})</Math> and <Math>U_{\epsilon}</Math> respectively. <P/>
Let <Math>\lambda</Math> be a dominant weight, and <Math>V(\lambda)</Math> the irreducible highest
weight module of highest weight <Math>\lambda</Math> over <Math>U_q(\mathfrak{g})</Math>. Let
<Math>v_{\lambda}\in V(\lambda)</Math> be a fixed highest weight vector. Then
<Math>U_{\mathbb{Z}}\cdot v_{\lambda}</Math> is a <Math>U_{\mathbb{Z}}</Math>-module. So by
specializing <Math>q</Math> to an element <Math>\epsilon</Math> of a field <Math>F</Math>, we get a
<Math>U_{\epsilon}</Math>-module. We call it the Weyl module of highest weight <Math>\lambda</Math>
over <Math>U_{\epsilon}</Math>. We note that it is not necessarily irreducible.
As in Section <Ref Sect="sec2.4"/> we let <Math>U^-</Math> be the subalgebra of
<Math>U_q(\mathfrak{g})</Math> generated by the <Math>F_{\alpha}</Math> for <Math>\alpha\in\Delta</Math>.
In <Cite Key="L0a"/> Lusztig
introduced a basis of <Math>U^-</Math> with very nice properties,
called the <E>canonical basis</E>. (Later this basis was also constructed
by Kashiwara, using a different method. For a brief overview on the history of
canonical bases we refer to <Cite Key="C06"/>.) <P/>
Let <Math>w_0=s_{i_1}\cdots s_{i_t}</Math>, and the elements <Math>F_k</Math> be as in Section
<Ref Sect="sec2.4"/>. Then, in order to stress the dependency of the monomial
<Display>
F_1^{(n_1)}\cdots F_t^{(n_t)}
</Display>
on the choice of reduced expression for the longest element in <Math>W(\Phi)</Math> we
say that it is a <Math>w_0</Math>-monomial.<P/>
Now we let <Math>\overline{\phantom{a}}</Math> be the automorphism of <Math>U^-</Math> defined in
Section <Ref Sect="sec2.2"/>. Elements that are invariant under
<Math>\overline{\phantom{a}}</Math> are said to be bar-invariant. <P/>
By results of Lusztig (<Cite Key="L93"/> Theorem 42.1.10,
<Cite Key="L96"/>, Proposition 8.2), there is a unique basis
<Math>{\bf B}</Math> of <Math>U^-</Math> with the following properties. Firstly, all elements of
<Math>{\bf B}</Math> are bar-invariant. Secondly, for any choice of reduced expression
<Math>w_0</Math> for the longest element in the Weyl group, and any element <Math>X\in{\bf B}</Math>
we have that <Math>X = x +\sum \zeta_i x_i</Math>, where <Math>x,x_i</Math> are <Math>w_0</Math>-monomials,
<Math>x\neq x_i</Math> for all <Math>i</Math>, and <Math>\zeta_i\in q\mathbb{Z}[q]</Math>. The basis <Math>{\bf B}</Math>
is called the canonical basis. If we work with a fixed reduced expression
for the longest element in <Math>W(\Phi)</Math>, and write <Math>X\in{\bf B}</Math> as above,
then we say that <Math>x</Math> is the <E>principal monomial</E> of <Math>X</Math>.<P/>
Let <Math>\mathcal{L}</Math> be the <Math>\mathbb{Z}[q]</Math>-lattice in <Math>U^-</Math> spanned by <Math>{\bf B}</Math>.
Then <Math>\mathcal{L}</Math> is also spanned by all <Math>w_0</Math>-monomials (where <Math>w_0</Math> is
a fixed reduced expression for the longest element in <Math>W(\Phi)</Math>). Now
let <Math>\widetilde{w}_0</Math> be a second reduced expression for the longest element
in <Math>W(\Phi)</Math>. Let <Math>x</Math> be a <Math>w_0</Math>-monomial, and let <Math>X</Math> be the element of
<Math>{\bf B}</Math> with principal monomial <Math>x</Math>. Write <Math>X</Math> as a linear combination of
<Math>\widetilde{w}_0</Math>-monomials, and let <Math>\widetilde{x}</Math> be the principal monomial
of that expression. Then we write <Math>\widetilde{x} = R_{w_0}^{\tilde{w}_0}(x)</Math>.
Note that <Math>x = \widetilde{x} \bmod q\mathcal{L}</Math>. <P/>
Now let <Math>\mathcal{B}</Math> be the set of all <Math>w_0</Math>-monomials <Math>\bmod q\mathcal{L}</Math>.
Then <Math>\mathcal{B}</Math> is a basis of the <Math>\mathbb{Z}</Math>-module
<Math>\mathcal{L}/q\mathcal{L}</Math>. Moreover, <Math>\mathcal{B}</Math> is independent of the
choice of <Math>w_0</Math>. Let <Math>\alpha\in\Delta</Math>, and let <Math>\widetilde{w}_0</Math> be a
reduced expression for the longest element in <Math>W(\Phi)</Math>, starting with
<Math>s_{\alpha}</Math>. The Kashiwara operators <Math>\widetilde{F}_{ \alpha} :
\mathcal{B}\to \mathcal{B}</Math> and <Math>\widetilde{E}_{\alpha} : \mathcal{B}\to
\mathcal{B}\cup\{0\}</Math> are defined as follows. Let <Math>b\in\mathcal{B}</Math> and
let <Math>x=</Math>
be the <Math>w_0</Math>-monomial such that <Math>b = x \bmod q\mathcal{L}</Math>. Set
<Math>\widetilde{x} = R_{w_0}^ {\tilde{w}_0}(x)</Math>. Then <Math>\widetilde{x}' is the
<Math>\widetilde{w}_0</Math>-monomial constructed from <Math>\widetilde{x}</Math> by increasing
its first exponent by <Math>1</Math> (the first exponent is <Math>n_1</Math> if we write
<Math>\widetilde{x}=F_1^{(n_1)}\cdots F_t^{(n_t)}</Math>).
Then <Math>\widetilde{F}_{ \alpha}(b) = R_{\tilde{w}_0}^{w_0}(\widetilde{x}')
\bmod q\mathcal{L}</Math>. For <Math>\widetilde{E}_{\alpha}</Math> we let <Math>\widetilde{x}'
be the <Math>\widetilde{w}_0</Math>-monomial constructed from <Math>\widetilde{x}</Math> by
decreasing its first exponent by <Math>1</Math>, if this exponent is <Math>\geq 1</Math>. Then
<Math>\widetilde{E}_{\alpha}(b) = R_{\tilde{w}_0}^{w_0}(\widetilde{x}')\bmod
q\mathcal{L}</Math>. Furthermore, <Math>\widetilde{E}_{\alpha}(b) =0</Math> if the first
exponent of <Math>\widetilde{x}</Math> is <Math>0</Math>. It can be shown that this definition does
not depend on the choice of <Math>w_0</Math>, <Math>\widetilde{w}_0</Math>. Furthermore we have
<Math>\widetilde{F}_{\alpha}\widetilde{E}_{\alpha}(b)=b</Math>, if
<Math>\widetilde{E}_{\alpha}(b)\neq 0</Math>, and <Math>\widetilde{E}_{\alpha}
\widetilde{F}_ {\alpha}(b)=b</Math> for all <Math>b\in \mathcal{B}</Math>.<P/>
Let <Math>w_0=s_{i_1}\cdots s_{i_t}</Math> be a fixed reduced expression for the
longest element in <Math>W(\Phi)</Math>. For <Math>b\in\mathcal{B}</Math> we define a sequence of
elements <Math>b_k\in\mathcal{B}</Math> for <Math>0\leq k\leq t</Math>, and a sequence of integers
<Math>n_k</Math> for <Math>1\leq k\leq t</Math> as follows. We set <Math>b_0=b</Math>, and if <Math>b_{k-1}</Math> is
defined we let <Math>n_k</Math> be maximal such that <Math>\widetilde{E}_{\alpha_{i_k}}^
{n_k}(b_{k-1})\neq 0</Math>. Also we set <Math>b_k = \widetilde{E}_{\alpha_{i_k}}^{n_k}
(b_{k-1})</Math>. Then the sequence <Math>(n_1,\ldots,n_t)</Math> is called the <E>string</E>
of <Math>b\in\mathcal{B}</Math> (relative to <Math>w_0</Math>). We note that
<Math>b=\widetilde{F}_ {\alpha_{i_1}}^{n_1}\cdots \widetilde{F}_{\alpha_{i_t}}^
{n_t}(1)</Math>. The set of all strings parametrizes the elements of <Math>\mathcal{B}</Math>,
and hence of <Math>{\bf B}</Math>.<P/>
Now let <Math>V(\lambda)</Math> be a highest-weight module over <Math>U_q(\mathfrak{g})</Math>,
with highest weight <Math>\lambda</Math>. Let <Math>v_{\lambda}</Math> be a fixed highest weight
vector. Then <Math>{\bf B}_{\lambda} = \{ X\cdot v_{\lambda}\mid X\in {\bf B}\}
\setminus \{0\}</Math> is a basis of <Math>V(\lambda)</Math>, called the <E>canonical basis</E>,
or <E>crystal basis</E> of <Math>V(\lambda)</Math>. Let <Math>\mathcal{L}(\lambda)</Math> be the
<Math>\mathbb{Z}[q]</Math>-lattice in <Math>V(\lambda)</Math> spanned by <Math>{\bf B}_{\lambda}</Math>. We
let <Math>\mathcal{B}({\lambda})</Math> be the set of all <Math>x\cdot v_{\lambda}\bmod
q\mathcal{L}(\lambda)</Math>, where <Math>x</Math> runs through all <Math>w_0</Math>-monomials, such
that <Math>X\cdot v_{\lambda} \neq 0</Math>, where <Math>X\in {\bf B}</Math> is the element with
principal monomial <Math>x</Math>. Then the Kashiwara operators are also viewed as
maps <Math>\mathcal{B}(\lambda)\to \mathcal{B}(\lambda)\cup\{0\}</Math>, in the following
way. Let <Math>b=x\cdot v_{\lambda}\bmod q\mathcal{L}(\lambda)</Math> be an element
of <Math>\mathcal{B}(\lambda)</Math>, and let <Math>b'=x\bmod q\mathcal{L} be the
corresponding element of <Math>\mathcal{B}</Math>. Let <Math>y</Math> be the <Math>w_0</Math>-monomial such
that <Math>\widetilde{F}_{\alpha}(b')=y\bmod q\mathcal{L}. Then
<Math>\widetilde{F}_{ \alpha}(b) = y\cdot v_{\lambda} \bmod q\mathcal{L}(\lambda)</Math>.
The description of <Math>\widetilde{E}_{\alpha}</Math> is analogous. (In
<Cite Key="J96"/>, Chapter 9 a different definition is given; however, by
<Cite Key="J96"/>, Proposition 10.9, Lemma 10.13, the two definitions
agree).<P/>
The set <Math>\mathcal{B}(\lambda)</Math> has <Math>\dim V(\lambda)</Math> elements. We let
<Math>\Gamma</Math> be the coloured directed graph defined as follows. The points of
<Math>\Gamma</Math> are the elements of <Math>\mathcal{B}(\lambda)</Math>, and there is an arrow
with colour <Math>\alpha\in\Delta</Math> connecting <Math>b,b'\in \mathcal{B}, if
<Math>\widetilde{F}_{\alpha}(b)=b'. The graph is called the
<E>crystal graph</E> of <Math>V(\lambda)</Math>.
</Section>
<Section Label="sec2.7"> <Heading> The path model </Heading>
In this section we recall some basic facts on Littelmann's path model.
From Section <Ref Sect="sec2.2"/> we recall that <Math>P</Math> denotes the weight
lattice. Let <Math>P_{\mathbb{R}}</Math> be the vector space over <Math>\mathbb{R}</Math> spanned
by <Math>P</Math>. Let <Math>\Pi</Math> be the set of all piecewise linear paths
<Math>\xi : [0,1]\to P_{\mathbb{R}} </Math>, such that <Math>\xi(0)=0</Math>. For <Math>\alpha\in\Delta</Math>
Littelmann defined operators <Math>f_{\alpha}, e_{\alpha} : \Pi \to \Pi\cup \{0\}</Math>.
Let <Math>\lambda</Math> be a dominant weight and let <Math>\xi_{\lambda}</Math> be the path joining
<Math>\lambda</Math> and the origin by a straight line. Let <Math>\Pi_{\lambda}</Math> be the set
of all nonzero <Math>f_{\alpha_{i_1}}\cdots f_{\alpha_{i_m}}(\xi_{\lambda})</Math> for
<Math>m\geq 0</Math>. Then <Math>\xi(1)\in P</Math> for all <Math>\xi\in \Pi_{\lambda}</Math>. Let <Math>\mu\in P</Math>
be a weight, and let <Math>V(\lambda)</Math> be the highest-weight module over
<Math>U_q(\mathfrak{g})</Math> of highest weight <Math>\lambda</Math>. A theorem of Littelmann
states that the number of paths <Math>\xi\in \Pi_{\lambda}</Math> such that <Math>\xi(1)=\mu</Math>
is equal to the dimension of the weight space of weight <Math>\mu</Math> in <Math>V(\lambda)</Math>
(<Cite Key="L95"/>, Theorem 9.1).<P/>
All paths appearing in <Math>\Pi_{\lambda}</Math> are so-called Lakshmibai-Seshadri
paths (LS-paths for short). They are defined as follows. Let <Math>\leq</Math> denote
the Bruhat order on <Math>W(\Phi)</Math>. For <Math>\mu,\nu\in W(\Phi)\cdot \lambda</Math> (the
orbit of <Math>\lambda</Math> under the action of <Math>W(\Phi)</Math>), write <Math>\mu\leq \nu</Math> if
<Math>\tau\leq\sigma</Math>, where <Math>\tau,\sigma\in W(\Phi)</Math> are the unique elements
of minimal length such that <Math>\tau(\lambda)=\mu</Math>, <Math>\sigma(\lambda)= \nu</Math>.
Now a rational path of shape <Math>\lambda</Math> is a pair <Math>\pi=(\nu,a)</Math>, where
<Math>\nu=(\nu_1,\ldots, \nu_s)</Math> is a sequence of elements of <Math>W(\Phi)\cdot
\lambda</Math>, such that <Math>\nu_i> \nu_{i+1}</Math> and <Math>a=(a_0=0, a_1, \cdots ,a_s=1)</Math>
is a sequence of rationals such that <Math>a_i <a_{i+1}</Math>. The path <Math>\pi</Math>
corresponding to these sequences is given by
<Display> \pi(t) =\sum_{j=1}^{r-1} (a_j-a_{j-1})\nu_j + \nu_r(t-a_{r-1})</Display>
for <Math>a_{r-1}\leq t\leq a_r</Math>. Now an LS-path of shape <Math>\lambda</Math> is a rational
path satisfying a certain integrality condition (see
<Cite Key="L94"/>, <Cite Key="L95"/>). We note that the path <Math>\xi_{\lambda}
= ( (\lambda), (0,1) )</Math> joining the origin and <Math>\lambda</Math> by a straight line
is an LS-path.<P/>
Now from <Cite Key="L94"/>, <Cite Key="L95"/> we transcribe the following:
<Enum>
<Item> Let <Math>\pi</Math> be an LS-path. Then <Math>f_{\alpha}\pi</Math> is an LS-path or <Math>0</Math>;
and the same holds for <Math>e_{\alpha}\pi</Math>. </Item>
<Item> The action of <Math>f_{\alpha},e_{\alpha}</Math> can easily be described
combinatorially (see <Cite Key="L94"/>). </Item>
<Item> The endpoint of an LS-path is an integral weight. </Item>
<Item> Let <Math>\pi=(\nu,a)</Math> be an LS-path. Then by <Math>\phi(\pi)</Math> we denote the
unique element <Math>\sigma</Math> of <Math>W(\Phi)</Math> of shortest length such that
<Math>\sigma(\lambda)=\nu_1</Math>. </Item>
</Enum>
Let <Math>\lambda</Math> be a dominant weight. Then we define a labeled directed graph
<Math>\Gamma</Math> as follows. The points of <Math>\Gamma</Math> are the paths in <Math>\Pi_{\lambda}</Math>.
There is an edge with label <Math>\alpha\in\Delta</Math> from <Math>\pi_1</Math> to <Math>\pi_2</Math> if
<Math>f_{\alpha}\pi_1 =\pi_2</Math>. Now by <Cite Key="K96"/> this graph <Math>\Gamma</Math> is
isomorphic to the crystal graph of the highest-weight module with highest
weight <Math>\lambda</Math>. So the path model provides an efficient way of computing
the crystal graph of a highest-weight module, without constructing the module
first. Also we see that <Math>f_{\alpha_{i_1}}\cdots f_{\alpha_{i_r}}\xi_{\lambda}
=0</Math> is equivalent to <Math>\widetilde{F}_{\alpha_{i_1}}\cdots \widetilde{F}_
{\alpha_{i_r}}v_{\lambda}=0</Math>, where <Math>v_{\lambda}\in V(\lambda)</Math> is a
highest weight vector (or rather the image of it in <Math>\mathcal{L}(\lambda)/
q\mathcal{L} (\lambda)</Math>), and the <Math>\widetilde{F}_{\alpha_k}</Math> are the
Kashiwara operators on <Math>\mathcal{B}(\lambda)</Math> (see Section
<Ref Sect="sec2.6"/>).
</Section>
<Section> <Heading> Notes</Heading>
I refer to <Cite Key="H90"/> for more information on Weyl groups, and to
<Cite Key="S01"/> for an overview of algorithms for computing with weights,
Weyl groups and their elements.<P/>
For general introductions into the theory of quantized enveloping algebras I
refer to <Cite Key="C98"/>, <Cite Key="J96"/> (from where most of the material
of this chapter is taken), <Cite Key="L92"/>, <Cite Key="L93"/>,
<Cite Key="R91"/>. I refer to the papers by Littelmann (<Cite Key="L94"/>,
<Cite Key="L95"/>, <Cite Key="L98"/>) for more information on the path model.
The paper by Kashiwara (<Cite Key="K96"/>) contains a proof of the connection
between path operators and Kashiwara operators.<P/>
Finally, I refer to <Cite Key="G01"/> (on computing with PBW-type bases),
<Cite Key="G02"/> (computation of elements of the canonical basis) for an
account of some of the algorithms used in <Package>QuaGroup</Package>.
<Var Name="QuantumField"/>
<Description>
This is the field <M>Q(q)</M> of rational functions in <M>q</M>,
over <M>Q</M>.
<Example><![CDATA[
gap> QuantumField;
QuantumField
]]></Example>
</Description>
</ManSection>
<ManSection>
<Var Name="_q"/>
<Description>
This is an indeterminate; <A>QuantumField</A> is the field of rational
functions in this indeterminate. The identifier <A>&uscore;q</A> is fixed
once the package <Package>QuaGroup</Package> is loaded. The symbol
<A>&uscore;q</A> is chosen (instead of <A>q</A>) in order to avoid potential
name clashes. We note that <A>&uscore;q</A> is printed as <A>q</A>.
<Example><![CDATA[
gap> _q;
q
gap> _q in QuantumField;
true
]]></Example>
</Description>
</ManSection>
</Section>
<Section> <Heading>Gaussian integers</Heading>
<ManSection>
<Oper Name="GaussNumber" Arg="n, par"/>
<Description>
This function computes for the integer <A>n</A> the Gaussian integer
<M>[n]_{v=<A>par</A>}</M> (cf. Section <Ref Sect="sec2.1"/>).
<Example><![CDATA[
gap> GaussNumber( 4, _q );
q^3+q+q^-1+q^-3
]]></Example>
</Description>
</ManSection>
<ManSection>
<Oper Name="GaussianFactorial" Arg="n, par"/>
<Description>
This function computes for the integer <A>n</A> the Gaussian factorial
<M>[n]!_{v=<A>par</A>}</M>.
<Example><![CDATA[
gap> GaussianFactorial( 3, _q );
q^3+2*q+2*q^-1+q^-3
gap> GaussianFactorial( 3, _q^2 );
q^6+2*q^2+2*q^-2+q^-6
]]></Example>
</Description>
</ManSection>
<ManSection>
<Oper Name="GaussianBinomial" Arg="n, k, par"/>
<Description>
This function computes for two integers <A>n</A> and <A>k</A> the
Gaussian binomial <A>n</A> choose <A>k</A>, where the parameter <M>v</M>
is replaced by <A>par</A>.
<Example><![CDATA[
gap> GaussianBinomial( 5, 2, _q^2 );
q^12+q^8+2*q^4+2+2*q^-4+q^-8+q^-12
]]></Example>
</Description>
</ManSection>
</Section>
<Section> <Heading>Roots and root systems</Heading>
In this section we describe some functions for dealing with root systems.
These functions supplement the ones already present in the &GAP; library.
<ManSection>
<Oper Name="RootSystem" Arg="type, rank"/>
<Oper Name="RootSystem" Arg="list"/>
<Description>
Here <A>type</A> is a capital letter between <A>"A"</A> and <A>"G"</A>,
and <A>rank</A> is a positive integer (<M>\geq 1</M> if <A>type="A"</A>,
<M>\geq 2</M> if <A>type="B"</A>, <A>"C"</A>, <M>\geq 4</M> if
<A>type="D"</A>, <M>6,7,8</M> if <A>type="E"</A>, <M>4</M> if
<A>type="F"</A>, and <M>2</M> if <A>type="G"</A>). This function returns
the root system of type <A>type</A> and rank <A>rank</A>. In the second
form <A>list</A> is a list of types and ranks, e.g.,
<A>[ "B", 2, "F", 4, "D", 7 ]</A>. <P/>
The root system constructed by this function comes with he attributes
<A>PositiveRoots</A>, <A>NegativeRoots</A>, <A>SimpleSystem</A>,
<A>CartanMatrix</A>, <A>BilinearFormMat</A>. Here the attribute
<A>SimpleSystem</A> contains a set of simple roots, written as
unit vectors. <A>PositiveRoots</A> is a list of the positive roots, written as
linear combinations of the simple roots, and likewise for <A>NegativeRoots</A>.
<A>CartanMatrix( R )</A> is the Cartan matrix of the root system <A>R</A>,
where the entry on position <M>( i, j )</M> is given by <M>\langle \alpha_i,
\alpha_j^{\vee}\rangle</M> where <M>\alpha_i</M> is the <M>i</M>-th simple root.
<A>BilinearFormMat( R )</A> is the matrix of the bilinear form,
where the entry on position <M>( i, j )</M> is given by <M>( \alpha_i,
\alpha_j )</M> (see Section <Ref Sect="sec2.2"/>).<P/>
<A>WeylGroup( R )</A> returns the Weyl group of the root system <A>R</A>.
We refer to the &GAP; reference manual for an overview of the functions for
Weyl groups in the &GAP; library. We mention the functions
<A>ConjugateDominantWeight( W, wt )</A> (returns the dominant weight in the
<A>W</A>-orbit of the weight <A>wt</A>), and <A>WeylOrbitIterator( W, wt )</A>
(returns an iterator for the <A>W</A>-orbit containing the weight <A>wt</A>).
We write weights as integral linear combinations of fundamental weights, so
in &GAP; weights are represented by lists of integers
(of length equal to the rank of the root system). <P/>
Also we mention the function <A>PositiveRootsAsWeights( R )</A> that returns
the positive roots of <A>R</A> written as weights, i.e., as linear
combinations of the fundamental weights.
<Example><![CDATA[
gap> R:=RootSystem( [ "B", 2, "F", 4, "E", 6 ] );
<root system of type B2 F4 E6>
gap> R:= RootSystem( "A", 2 );
<root system of type A2>
gap> PositiveRoots( R );
[ [ 1, 0 ], [ 0, 1 ], [ 1, 1 ] ]
gap> BilinearFormMat( R );
[ [ 2, -1 ], [ -1, 2 ] ]
gap> W:= WeylGroup( R );
Group([ [ [ -1, 1 ], [ 0, 1 ] ], [ [ 1, 0 ], [ 1, -1 ] ] ])
gap> ConjugateDominantWeight( W, [-3,2] );
[ 2, 1 ]
gap> o:= WeylOrbitIterator( W, [-3,2] );
<iterator>
gap> # Using the iterator we can loop over the orbit:
gap> NextIterator( o );
[ 2, 1 ]
gap> NextIterator( o );
[ -1, -2 ]
gap> PositiveRootsAsWeights( R );
[ [ 2, -1 ], [ -1, 2 ], [ 1, 1 ] ]
]]></Example>
</Description>
</ManSection>
<ManSection>
<Attr Name="BilinearFormMatNF" Arg="R"/>
<Description>
This is the matrix of the <Q>normalized</Q> bilinear form. This means
that all diagonal entries are even, and 2 is the minimum value
occurring on the diagonal. If <A>R</A> is a root system constructed by
<Ref Oper="RootSystem"/>, then this is equal to <A>BilinearFormMat( R )</A>.
</Description>
</ManSection>
<ManSection>
<Attr Name="PositiveRootsNF" Arg="R"/>
<Description>
This is the list of positive roots of the root system <A>R</A>, written
as linear combinations of the simple roots. This means that the simple roots
are unit vectors. If <A>R</A> is a root system constructed by
<Ref Oper="RootSystem"/>, then this is equal to <A>PositiveRoots( R )</A>. <P/>
One of the reasons for writing the positive roots like this is the
following. Let <A>a, b</A> be two elements of <A>PositiveRootsNF( R )</A>,
and let <A>B</A> be the matrix of the bilinear form. Then
<A>a*( B*b )</A> is the result of applying the bilinear form to <A>a, b</A>.
<Example><![CDATA[
gap> R:= RootSystem( SimpleLieAlgebra( "B", 2, Rationals ) );;
gap> PositiveRootsNF( R );
[ [ 1, 0 ], [ 0, 1 ], [ 1, 1 ], [ 1, 2 ] ]
gap> # We note that in this case PositiveRoots( R ) will give the
gap> # positive roots in a different format.
]]></Example>
</Description>
</ManSection>
<ManSection>
<Attr Name="SimpleSystemNF" Arg="R"/>
<Description>
This is the list of simple roots of <A>R</A>, written as unit vectors
(this means that they are elements of <A>PositiveRootsNF( R )</A>).
If <A>R</A> is a root system constructed by <Ref Oper="RootSystem"/>, then
this is equal to <A>SimpleSystem( R )</A>.
</Description>
</ManSection>
<ManSection>
<Attr Name="PositiveRootsInConvexOrder" Arg="R"/>
<Description>
This function returns the positive roots of the root system <A>R</A>,
in the <Q>convex</Q> order. Let <M>w_0=s_1\cdots s_t</M> be a reduced
expression of the longest element in the Weyl group. Then the <M>k</M>-th element of the list returned by this function is <M>s_1\cdots s_{k-1}(\alpha_k)
</M>. (Where the reduced expression used is the one returned by
<A>LongestWeylWord( R )</A>.) If <M>\alpha</M>, <M>\beta</M> and
<M>\alpha+\beta</M> are positive roots, then <M>\alpha+\beta</M> occurs
between <M>\alpha</M> and <M>\beta</M> (whence the name convex order).<P/>
In the output all roots are written in <Q>normal form</Q>, i.e., as
elements of <A>PositiveRootsNF( R )</A>.
<ManSection>
<Attr Name="SimpleRootsAsWeights" Arg="R"/>
<Description>
Returns the simple roots of the root system <A>R</A>, written as
linear combinations of the fundamental weights.
<Example><![CDATA[
gap> R:= RootSystem( "A", 2 );;
gap> SimpleRootsAsWeights( R );
[ [ 2, -1 ], [ -1, 2 ] ]
]]></Example>
</Description>
</ManSection>
</Section>
<Section> <Heading>Weyl groups and their elements</Heading>
Now we describe a few functions that deal with reduced words in the Weyl
group of the root system <A>R</A>. These words are represented as lists of
positive integers <M>i</M>, denoting the <M>i</M>-th simple reflection
(which corresponds to the <M>i</M>-th element of <A>SimpleSystem( R )</A>).
For example <A>[ 3, 2, 1, 3, 1 ]</A> represents
the expression <M>s_3 s_2 s_1 s_3 s_1</M>.
<ManSection>
<Oper Name="ApplyWeylElement" Arg="W, wt, wd"/>
<Description>
Here <A>wd</A> is a (not necessarily reduced) word in the Weyl group <A>W</A>,
and <A>wt</A> is a weight (written as integral linear combination of the
simple weights). This function returns the result of applying <A>wd</A> to
<A>wt</A>. For example, if <A>wt=</A><M>\mu</M>, and
<A>wd = [ 1, 2 ]</A> then this function returns <M>s_1s_2(\mu)</M> (where
<M>s_i</M> is the simple reflection corresponding to the <M>i</M>-th simple
root).
<ManSection>
<Oper Name="LengthOfWeylWord" Arg="W, wd"/>
<Description>
Here <A>wd</A> is a word in the Weyl group <A>W</A>. This function returns
the length of that word.
<ManSection>
<Attr Name="LongestWeylWord" Arg="R"/>
<Description>
Here <A>R</A> is a root system. <A>LongestWeylWord( R )</A> returns the
longest word in the Weyl group of <A>R</A>. <P/>
If this function is called for a root system <A>R</A>, a reduced expression
for the longest element in the Weyl group is calculated (the one which
is the smallest in the lexicographical ordering). However, if you would
like to work with a different reduced expression, then it is possible to
set it by <A>SetLongestWeylWord( R, wd )</A>, where <A>wd</A> is a reduced
expression of the longest element in the Weyl group. Note that you will have
to do this before calling <A>LongestWeylWord</A>, or any function that
may call <A>LongestWeylWord</A> (once the attribute is set, it will not be
possible to change it). Note also that you must be sure that the word you
give is in fact a reduced expression for the
longest element in the Weyl group, as this is not checked (you can check
this with <Ref Oper="LengthOfWeylWord"/>). <P/>
We note that virtually all algorithms for quantized enveloping algebras
depend on the choice of reduced expression for the longest element in the
Weyl group (as the PBW-type basis depends on this).
<ManSection>
<Oper Name="ReducedWordIterator" Arg="W, wd"/>
<Description>
Here <A>W</A> is a Weyl group, and <A>wd</A> a reduced word. This
function returns an iterator for the set of reduced words that represent the
same element as <A>wd</A>. The elements are output in ascending
lexicographical order.
<Example><![CDATA[
gap> R:= RootSystem( "F", 4 );;
gap> it:= ReducedWordIterator( WeylGroup(R), LongestWeylWord(R) );
<iterator>
gap> NextIterator( it );
[ 1, 2, 1, 3, 2, 1, 3, 2, 3, 4, 3, 2, 1, 3, 2, 3, 4, 3, 2, 1, 3, 2, 3, 4 ]
gap> k:= 1;;
gap> while not IsDoneIterator( it ) do
> k:= k+1; w:= NextIterator( it );
> od;
gap> k;
2144892
]]></Example>
So there are 2144892 reduced expressions for the longest element in the Weyl
group of type <M>F_4</M>.
</Description>
</ManSection>
<ManSection>
<Oper Name="ExchangeElement" Arg="W, wd, ind"/>
<Description>
Here <A>W</A> is a Weyl group, and <A>wd</A> is a <E>reduced</E> word in
<A>W</A>, and <A>ind</A> is an index between 1 and the rank of the root
system. Let <A>v</A> denote the word obtained from <A>wd</A> by adding
<A>ind</A> at the end. This function <E>assumes</E> that the length of <A>v</A>
is one less than the length of <A>wd</A>, and returns a reduced expression
for <A>v</A> that is obtained from <A>wd</A> by deleting one entry. Nothing is
guaranteed of the output if the length of <A>v</A> is bigger than the length
of <A>wd</A>.
<ManSection>
<Oper Name="GetBraidRelations" Arg="W, wd1, wd2"/>
<Description>
Here <A>W</A> is a Weyl group, and <A>wd1</A>, <A>wd2</A> are two reduced
words representing the same element in <A>W</A>. This function returns a
list of braid relations that can be applied to <A>wd1</A> to obtain
<A>wd2</A>. Here a braid relation is represented as a list, with at the odd
positions integers that represent positions in a word,
and at the even positions the indices that are on those positions after
applying the relation. For example, let <A>wd</A> be the word
<A>[ 1, 2, 1, 3, 2, 1 ]</A> and let <A>r = [ 3, 3, 4, 1 ]</A> be a relation.
Then the result of applying <A>r</A> to <A>wd</A> is
<A>[ 1, 2, 3, 1, 2, 1]</A> (i.e., on the third position we put a 3, and on
the fourth position a 1).<P/>
We note that the function does not check first whether <A>wd1</A> and
<A>wd2</A> represent the same element in <A>W</A>. If this is not the
case, then an error will occur during the execution of the function, or it
will produce wrong output.
<ManSection>
<Attr Name="LongWords" Arg="R"/>
<Description>
For a root system <A>R</A> this returns a list of triples (of length equal
to the rank of <A>R</A>). Let <A>t</A> be the <A>k</A>-th triple
occurring in this list. The first element of <A>t</A> is an expression
for the longest element of the Weyl group, starting with <A>k</A>. The
second element is a list of braid relations, moving
this expression to the value of <A>LongestWeylWord( R )</A>. The third element is a list of braid relations performing the reverse transformation.
In <Package>QuaGroup</Package> we deal with two types of quantized
enveloping algebra. First there are the quantized enveloping algebras
defined over the field <Ref Var="QuantumField"/>. We say that these
algebras are <Q>generic</Q> quantized enveloping algebras, in
<Package>QuaGroup</Package> they have the category <A>IsGenericQUEA</A>.
Secondly, we deal with the quantized enveloping algebras that are defined
over a different field.
<ManSection>
<Attr Name="QuantizedUEA" Arg="R"/>
<Oper Name="QuantizedUEA" Arg="R, F, v"/>
<Attr Name="QuantizedUEA" Arg="L"/>
<Oper Name="QuantizedUEA" Arg="L, F, v"/>
<Description>
In the first two forms <A>R</A> is a root system. With only
<A>R</A> as input, the corresponding generic quantized enveloping
algebra is constructed. It is stored as an attribute of <A>R</A> (so that
constructing it twice for the same root system yields the same object).
Also the root system is stored in the quantized enveloping algebra as the
attribute <A>RootSystem</A>. <P/>
The attribute <A>GeneratorsOfAlgebra</A> contains the generators of a
PBW-type basis (see Section <Ref Sect="sec2.4"/>), that are constructed
relative to the reduced expression for the longest element in the Weyl
group that is contained in <A>LongestWeylWord( R )</A>.
We refer to <Ref Oper="ObjByExtRep"/> for a
description of the construction of elements of a quantized enveloping
algebra. <P/>
The call <A>QuantizedUEA( R, F, v )</A> returns the quantized universal
enveloping algebra with quantum parameter <A>v</A>, which must lie in the
field <A>F</A>. In this case the elements of <A>GeneratorsOfAlgebra</A> are
the images of the generators of the corresponding generic quantized
enveloping algebra. This means that if <A>v</A> is a root of unity, then the
generators will not generate the whole algebra, but rather a finite
dimensional subalgebra (as for instance <M>E_i^k=0</M> for <M>k</M>
large enough). It is possible to construct elements that do not lie in this
finite dimensional subalgebra using <Ref Oper="ObjByExtRep"/>. <P/>
In the last two cases <A>L</A> must be a semisimple Lie algebra. The two
calls are short for <A>QuantizedUEA( RootSystem( L ) )</A> and
<A>QuantizedUEA( RootSystem( L ), F, v )</A> respectively.
<Example><![CDATA[
gap> # We construct the generic quantized enveloping algebra corresponding
gap> # to the root system of type A2+G2:
gap> R:= RootSystem( [ "A", 2, "G", 2 ] );;
gap> U:= QuantizedUEA( R );
QuantumUEA( <root system of type A2 G2>, Qpar = q )
gap> RootSystem( U );
<root system of type A2 G2>
gap> g:= GeneratorsOfAlgebra( U );
[ F1, F2, F3, F4, F5, F6, F7, F8, F9, K1, (-q+q^-1)*[ K1 ; 1 ]+K1, K2,
(-q+q^-1)*[ K2 ; 1 ]+K2, K3, (-q+q^-1)*[ K3 ; 1 ]+K3, K4,
(-q^3+q^-3)*[ K4 ; 1 ]+K4, E1, E2, E3, E4, E5, E6, E7, E8, E9 ]
gap> # These elements generate a PBW-type basis of U; the nine elements Fi,
gap> # and the nine elements Ei correspond to the roots listed in convex order:
gap> PositiveRootsInConvexOrder( R );
[ [ 1, 0, 0, 0 ], [ 1, 1, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ],
[ 0, 0, 3, 1 ], [ 0, 0, 2, 1 ], [ 0, 0, 3, 2 ], [ 0, 0, 1, 1 ],
[ 0, 0, 0, 1 ] ]
gap> # So, for example, F5 is an element of weight -[ 0, 0, 3, 1 ].
gap> # We can also multiply elements; the result is written on the PBW-basis:
gap> g[17]*g[4];
(-1+q^-6)*F4*[ K4 ; 1 ]+(q^-3)*F4*K4
gap> # Now we construct a non-generic quantized enveloping algebra:
gap> R:= RootSystem( "A", 2 );;
gap> U:= QuantizedUEA( R, CF(3), E(3) );;
gap> g:= GeneratorsOfAlgebra( U );
[ F1, F2, F3, K1, (-E(3)+E(3)^2)*[ K1 ; 1 ]+K1, K2,
(-E(3)+E(3)^2)*[ K2 ; 1 ]+K2, E1, E2, E3 ]
]]></Example>
As can be seen in the example, every element of <M>U</M> is written as a
linear combination of monomials in the PBW-generators; the generators of
<M>U^-</M> come first, then the generators of <M>U^0</M>, and finally the
generators of <M>U^+</M>.
</Description>
</ManSection>
<ManSection>
<Oper Name="ObjByExtRep" Arg="fam, list"/>
<Description>
Here <A>fam</A> is the elements family of a quantized enveloping
algebra <A>U</A>. Secondly, <A>list</A> is a list describing an element of
<A>U</A>. We explain how this description works. First we describe an
indexing system for the generators of <A>U</A>. Let <A>R</A> be the root
system of <A>U</A>.
Let <A>t</A> be the number of positive roots, and <A>rank</A> the rank of
the root system. Then the generators of <A>U</A> are <A>Fk</A>, <A>Ki</A>
(and its inverse), <A>Ek</A>, for <A>k=1...t</A>, <A>i=1..rank</A>.
(See Section <Ref Sect="sec2.4"/>; for the construction of the <A>Fk</A>,
<A>Ek</A>, the value of <A>LongestWeylWord( R )</A> is used.) Now the index
of <A>Fk</A> is <A>k</A>, and the
index of <A>Ek</A> is <A>t+rank+k</A>. Furthermore, elements
of the algebra generated by the <A>Ki</A>, and its inverse, are written as
linear combinations of products of <Q>binomials</Q>, as in Section
<Ref Sect="sec2.5"/>. The element
<Display> K_i^{d}\begin{bmatrix} K_{i} \\ s \end{bmatrix} </Display>
(where <M>d=0,1</M>), is indexed as <A>[ t+i, d ]</A> (what happens to the
<A>s</A> is described later). So an index is either an integer, or a list of
two integers. <P/>
A monomial is a list of indices, each followed by an exponent. First come
the indices of the <A>Fk</A>, (<A>1..t</A>), then come the lists of the form
<A>[ t+i, d ]</A>, and finally the indices of the <A>Ek</A>. Each index is
followed by an exponent. An index of the form <A>[ t+i, d ]</A> is followed
by the <A>s</A> in the above formula. <P/>
The second argument of <A>ObjByExtRep</A> is a list of monomials
followed by coefficients. This function returns the element of <A>U</A>
described by this list.<P/>
Finally we remark that the element
<Display> K_i^{d}\begin{bmatrix} K_{i} \\ s \end{bmatrix} </Display>
is printed as <A>Ki[ Ki ; s ]</A> if <A>d=1</A>, and as <A>[ Ki ; s ]</A>
if <A>d=0</A>.
<Example><![CDATA[
gap> U:= QuantizedUEA( RootSystem("A",2) );;
gap> fam:= ElementsFamily( FamilyObj( U ) );;
gap> list:= [ [ 2, 3, [ 4, 0 ], 8, 6, 11 ], _q^2, # monomial and coefficient
> [ 1, 7, 3, 5, [ 5, 1 ], 3, 8, 9 ], _q^-1 + _q^2 ]; # monomial and coefficient
[ [ 2, 3, [ 4, 0 ], 8, 6, 11 ], q^2, [ 1, 7, 3, 5, [ 5, 1 ], 3, 8, 9 ],
q^2+q^-1 ]
gap> ObjByExtRep( fam, list );
(q^2)*F2^(3)*[ K1 ; 8 ]*E1^(11)+(q^2+q^-1)*F1^(7)*F3^(5)*K2[ K2 ; 3 ]*E3^(9)
]]></Example>
</Description>
</ManSection>
<ManSection>
<Oper Name="ExtRepOfObj" Arg="elm"/>
<Description>
For the element <A>elm</A> of a quantized enveloping algebra, this function
returns the list that defines <A>elm</A> (see <Ref Oper="ObjByExtRep"/>).
<Example><![CDATA[
gap> U:= QuantizedUEA( RootSystem("A",2) );;
gap> g:= GeneratorsOfAlgebra(U);
[ F1, F2, F3, K1, (-q+q^-1)*[ K1 ; 1 ]+K1, K2, (-q+q^-1)*[ K2 ; 1 ]+K2, E1,
E2, E3 ]
gap> ExtRepOfObj( g[5] );
[ [ [ 4, 0 ], 1 ], -q+q^-1, [ [ 4, 1 ], 0 ], 1 ]
]]></Example>
</Description>
</ManSection>
<ManSection>
<Attr Name="QuantumParameter" Arg="U"/>
<Description>
Returns the quantum parameter used in the definition of <A>U</A>.
<Example><![CDATA[
gap> R:= RootSystem("A",2);;
gap> U0:= QuantizedUEA( R, CF(3), E(3) );;
gap> QuantumParameter( U0 );
E(3)
]]></Example>
</Description>
</ManSection>
<ManSection>
<Attr Name="CanonicalMapping" Arg="U"/>
<Description>
Here <A>U</A> is a quantized enveloping algebra. Let <A>U0</A> denote
the corresponding <Q>generic</Q> quantized enveloping algebra. This function
returns the mapping <A>U0 --> U</A> obtained by mapping <A>q</A> (which is
the quantum parameter of <A>U0</A>) to the quantum parameter of <A>U</A>.
<Example><![CDATA[
gap> R:= RootSystem("A", 3 );;
gap> U:= QuantizedUEA( R, CF(5), E(5) );;
gap> f:= CanonicalMapping( U );
MappingByFunction( QuantumUEA( <root system of type A
3>, Qpar = q ), QuantumUEA( <root system of type A3>, Qpar =
E(5) ), function( u ) ... end )
gap> U0:= Source( f );
QuantumUEA( <root system of type A3>, Qpar = q )
gap> g:= GeneratorsOfAlgebra( U0 );;
gap> u:= g[18]*g[9]*g[6];
(q^2)*F6*K2*E6+(q)*K2*[ K3 ; 1 ]
gap> Image( f, u );
(E(5)^2)*F6*K2*E6+(E(5))*K2*[ K3 ; 1 ]
]]></Example>
</Description>
</ManSection>
<ManSection>
<Oper Name="WriteQEAToFile" Arg="U, file"/>
<Description>
Here <A>U</A> is a quantized enveloping algebra, and file is a string
containing the name of a file. This function writes some data to
<A>file</A>, that allows <Ref Oper="ReadQEAFromFile"/> to recover it.
<Example><![CDATA[
gap> U:= QuantizedUEA( RootSystem("A",3) );;
gap> WriteQEAToFile( U, "A3" );
]]></Example>
</Description>
</ManSection>
<ManSection>
<Oper Name="ReadQEAFromFile" Arg="file"/>
<Description>
Here <A>file</A> is a string containing the name of a file, to which a
quantized enveloping algebra has been written by <Ref Oper="WriteQEAToFile"/>.
This function recovers the quantized enveloping algebra.
<Example><![CDATA[
gap> U:= QuantizedUEA( RootSystem("A",3) );;
gap> WriteQEAToFile( U, "A3" );
gap> U0:= ReadQEAFromFile( "A3" );
QuantumUEA( <root system of type A3>, Qpar = q )
]]></Example>
</Description>
</ManSection>
</Section>
<Section> <Heading> Homomorphisms and automorphisms </Heading>
Here we describe functions for creating homomorphisms and
(anti)-automorphisms of a quantized enveloping algebra.
<ManSection>
<Oper Name="QEAHomomorphism" Arg="U, A, list"/>
<Description>
Here <A>U</A> is a generic quantized enveloping algebra (i.e.,
with quantum parameter <A>&uscore;q</A>), <A>A</A> is an algebra with one
over <A>QuantumField</A>, and <A>list</A> is a list of <A>4*rank</A>
elements of <A>A</A> (where <A>rank</A> is the rank of the root system of
<A>U</A>). On the first rank positions there are the images of the
<M>F_{\alpha}</M> (where the <M>\alpha</M> are simple roots, listed
in the order in which they occur in <A>SimpleSystem( R )</A>). On the
positions <A>rank+1...2*rank</A> are the images of the <M>K_{\alpha}</M>.
On the positions <A>2*rank+1...3*rank</A> are the images of the
<M>K_{\alpha}^{-1}</M>, and finally on the positions <A>3*rank+1...4*rank</A>
occur the images of the <M>E_{\alpha}</M>. <P/>
This function returns the homomorphism <A>U -> A</A>, defined by this data.
In the example below we construct a homomorphism from one quantized
enveloping algebra into another. Both are constructed relative to the same
root system, but with different reduced expressions for the longest element
of the Weyl group.
<Example><![CDATA[
gap> R:= RootSystem( "G", 2 );;
gap> SetLongestWeylWord( R, [1,2,1,2,1,2] );
gap> UR:= QuantizedUEA( R );;
gap> S:= RootSystem( "G", 2 );;
gap> SetLongestWeylWord( S, [2,1,2,1,2,1] );
gap> US:= QuantizedUEA( S );;
gap> gS:= GeneratorsOfAlgebra( US );
[ F1, F2, F3, F4, F5, F6, K1, (-q+q^-1)*[ K1 ; 1 ]+K1, K2,
(-q^3+q^-3)*[ K2 ; 1 ]+K2, E1, E2, E3, E4, E5, E6 ]
gap> SimpleSystem( R );
[ [ 1, 0 ], [ 0, 1 ] ]
gap> PositiveRootsInConvexOrder( S );
[ [ 0, 1 ], [ 1, 1 ], [ 3, 2 ], [ 2, 1 ], [ 3, 1 ], [ 1, 0 ] ]
gap> # We see that the simple roots of R occur on positions 6 and 1
gap> # in the list PositiveRootsInConvexOrder( S ); This means that we
gap> # get the following list of images of the homomorphism:
gap> imgs:= [ gS[6], gS[1], # the images of the F_{\alpha}
> gS[7], gS[9], # the images of the K_{\alpha}
> gS[8], gS[10], # the images of the K_{\alpha}^{-1}
> gS[16], gS[11] ]; # the images of the E_{\alpha}
[ F6, F1, K1, K2, (-q+q^-1)*[ K1 ; 1 ]+K1, (-q^3+q^-3)*[ K2 ; 1 ]+K2, E6, E1 ]
gap> h:= QEAHomomorphism( UR, US, imgs );
<homomorphism: QuantumUEA( <root system of type G
2>, Qpar = q ) -> QuantumUEA( <root system of type G2>, Qpar = q )>
gap> Image( h, GeneratorsOfAlgebra( UR )[3] );
(q^10-q^6-q^4+1)*F1*F6^(2)+(q^6-q^2)*F2*F6+(q^4)*F4
]]></Example>
</Description>
</ManSection>
<ManSection>
<Oper Name="QEAAutomorphism" Arg="U, list"/>
<Oper Name="QEAAutomorphism" Arg="U, f"/>
<Description>
In the first form <A>U</A> is a generic quantized enveloping algebra
(i.e., with quantum parameter <A>&uscore;q</A>), and <A>list</A> is a
list of <A>4*rank</A> elements of <A>U</A> (where <A>rank</A> is the
rank of the corresponding root system). On the first <A>rank</A>
positions there are the images of the <M>F_{\alpha}</M> (where the
<M>\alpha</M> are simple roots, listed in the order in which they occur in
<A>SimpleSystem( R )</A>). On the positions <A>rank+1...2*rank</A> are
the images of the <M>K_{\alpha}</M>. On the positions
<A>2*rank+1...3*rank</A> are the images of the <M>K_{\alpha}^{-1}</M>,
and finally on the positions <A>3*rank+1...4*rank</A> occur the images of
the <M>E_{\alpha}</M>.<P/>
In the second form <A>U</A> is a non-generic quantized enveloping algebra,
and <A>f</A> is an automorphism of the corresponding generic quantized
enveloping algebra. The corresponding automorphism of <A>U</A> is
constructed. In this case <A>f</A> must not be the bar-automorphism of the
corresponding generic quantized enveloping algebra (cf.
<Ref Attr="BarAutomorphism"/>), as this automorphism doesn't work in the
non-generic case.<P/>
The image of an element <A>x</A> under an automorphism <A>f</A> is computed
by <A>Image( f, x )</A>. Note that there is no function for calculating
pre-images (in general this seems to be a very hard task). If you
want the inverse of an automorphism, you have to construct it explicitly
(e.g., by <A>QEAAutomorphism( U, list )</A>, where <A>list</A> is a list of
pre-images). <P/>
Below we construct the automorphism <M>\omega</M> (cf. Section
<Ref Sect="sec2.2"/>) of the quantized
enveloping of type <M>A_3</M>, when the quantum parameter is
<A>&uscore;q</A>, and when the quantum parameter is a fifth root of unity.
<Example><![CDATA[
gap> # First we construct the quantized enveloping algebra:
gap> R:= RootSystem( "A", 3 );;
gap> U0:= QuantizedUEA( R );
QuantumUEA( <root system of type A3>, Qpar = q )
gap> g:= GeneratorsOfAlgebra( U0 );
[ F1, F2, F3, F4, F5, F6, K1, (-q+q^-1)*[ K1 ; 1 ]+K1, K2,
(-q+q^-1)*[ K2 ; 1 ]+K2, K3, (-q+q^-1)*[ K3 ; 1 ]+K3, E1, E2, E3, E4, E5,
E6 ]
gap> # Now, for instance, we map F_{\alpha} to E_{\alpha}, where \alpha
gap> # is a simple root. In order to find where those F_{\alpha}, E_{\alpha}
gap> # are in the list of generators, we look at the list of positive roots
gap> # in convex order:
gap> PositiveRootsInConvexOrder( R );
[ [ 1, 0, 0 ], [ 1, 1, 0 ], [ 0, 1, 0 ], [ 1, 1, 1 ], [ 0, 1, 1 ],
[ 0, 0, 1 ] ]
gap> # So the simple roots occur on positions 1, 3, 6. This means that we
gap> # have the following list of images:
gap> imgs:= [ g[13], g[15], g[18], g[8], g[10], g[12], g[7], g[9], g[11],
> g[1], g[3], g[6] ];
[ E1, E3, E6, (-q+q^-1)*[ K1 ; 1 ]+K1, (-q+q^-1)*[ K2 ; 1 ]+K2,
(-q+q^-1)*[ K3 ; 1 ]+K3, K1, K2, K3, F1, F3, F6 ]
gap> f:= QEAAutomorphism( U0, imgs );
<automorphism of QuantumUEA( <root system of type A3>, Qpar = q )>
gap> Image( f, g[2] );
(-q)*E2
gap> # f induces an automorphism of any non-generic quantized enveloping
gap> # algebra with the same root system R:
gap> U1:= QuantizedUEA( R, CF(5), E(5) );
QuantumUEA( <root system of type A3>, Qpar = E(5) )
gap> h:= QEAAutomorphism( U1, f );
<automorphism of QuantumUEA( <root system of type A3>, Qpar = E(5) )>
gap> Image( h, GeneratorsOfAlgebra(U1)[7] );
(-E(5)+E(5)^4)*[ K1 ; 1 ]+K1
]]></Example>
</Description>
</ManSection>
<ManSection>
<Oper Name="QEAAntiAutomorphism" Arg="U, list"/>
<Oper Name="QEAAntiAutomorphism" Arg="U, f"/>
<Description>
These are functions for constructing anti-automorphisms of quantized
enveloping algebras. The same comments apply as for
<Ref Oper="QEAAutomorphism"/>.
</Description>
</ManSection>
<ManSection>
<Attr Name="AutomorphismOmega" Arg="U"/>
<Description>
This is the automorphism <M>\omega</M> (cf. Section <Ref Sect="sec2.2"/>).
<Example><![CDATA[
gap> R:= RootSystem( "A", 3 );;
gap> U:= QuantizedUEA( R, CF(5), E(5) );
QuantumUEA( <root system of type A3>, Qpar = E(5) )
gap> f:= AutomorphismOmega( U );
<automorphism of QuantumUEA( <root system of type A3>, Qpar = E(5) )>
]]></Example>
</Description>
</ManSection>
<ManSection>
<Attr Name="AntiAutomorphismTau" Arg=""/>
<Description>
This is the anti-automorphism <M>\tau</M> (cf. Section <Ref Sect="sec2.2"/>).
<Example><![CDATA[
gap> R:= RootSystem( "A", 3 );;
gap> U:= QuantizedUEA( R, CF(5), E(5) );
QuantumUEA( <root system of type A3>, Qpar = E(5) )
gap> t:= AntiAutomorphismTau( U );
<anti-automorphism of QuantumUEA( <root system of type A3>, Qpar = E(5) )>
]]></Example>
</Description>
</ManSection>
<ManSection>
<Attr Name="BarAutomorphism" Arg="U"/>
<Description>
This is the automorphism <M>\bar{~}</M> defined in Section
<Ref Sect="sec2.2"/> Here <A>U</A> must be a generic quantized enveloping
algebra.
<Example><![CDATA[
gap> U:= QuantizedUEA( RootSystem(["A",2,"B",2]) );;
gap> bar:= BarAutomorphism( U );
<automorphism of QuantumUEA( <root system of type A2 B2>, Qpar = q )>
gap> Image( bar, GeneratorsOfAlgebra( U )[5] );
(q^2-q^-2)*F4*F7+F5
]]></Example>
</Description>
</ManSection>
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