| Quellcodebibliothek Statistik Leitseite products/Sources/formale Sprachen/GAP/pkg/quagroup/doc/ (Algebra von RWTH Aachen Version 4.15.1©) Datei vom 11.0.2024 mit Größe 109 kB |
|
Quelle quagroup.xmlSprache: XML |
|||||||||||||||
|
<?xml version="1.0" encoding="utf-8"?> <?LaTeX ExtraPreamble="\usepackage{amsmath}"?> <!DOCTYPE Book SYSTEM "gapdoc.dtd"> <Book Name="quagroup"> <#Include SYSTEM "title.xml"> <TableOfContents/> <Body> <Chapter> <Heading>Introduction</Heading> 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: <Log><![CDATA[ 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 ]]></Log> 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> </Chapter> <Chapter Label="chap2"> <Heading>Background</Heading> In this chapter we summarize some of the theoretical concepts with which <Package>QuaGroup</Package> operates. <Section Label="sec2.1"> <Heading>Gaussian Binomials</Heading> Let <Math>v</Math> be an indeterminate over <Math>\mathbb{Q}</Math>. For a positive integer <Math>n</M 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> </Section> <Section Label="sec2.2"> <Heading>Quantized enveloping algebras</Heading> 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 respe <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 weig 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 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< 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_{\alp <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</M 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>-mo 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 )</Mat 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). </Section> <Section Label="sec2.4"> <Heading>PBW-type bases </Heading> 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> <Math>U^+</Math> is generated by the <Math>E_{\alpha}</Math>. So a basis of <Math>U_q(\mathfrak{g})</ formed by all elements <Math>FKE</Math>, where <Math>F</Math>, <Math>K</Math>, <Math>E</Math> run through b <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>, <Mat 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> 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 e 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> 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})</M 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_{\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}</Ma <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</Mat over <Math>U_{\epsilon}</Math>. We note that it is not necessarily irreducible. </Section> <Section Label="sec2.6"> <Heading>The canonical basis </Heading> 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\Delt 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>-monomia <Math>x\neq x_i</Math> for all <Math>i</Math>, and <Math>\zeta_i\in q\mathbb{Z}[q]</Math>. The basis <M 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 b Then <Math>\mathcal{L}</Math> is also spanned by all <Math>w_0</Math>-monomials (where <Math>w_0</Ma 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 el <Math>{\bf B}</Math> with principal monomial <Math>x</Math>. Write <Math>X</Math> as a linear combinati <Math>\widetilde{w}_0</Math>-monomials, and let <Math>\widetilde{x}</Math> be the principal mon 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{ 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 th choice of <Math>w_0</Math>. Let <Math>\alpha\in\Delta</Math>, and let <Math>\widetilde{w}_0</Math> b 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}'</Math> is the <Math>\widetilde{w}_0</Math>-monomial constructed from <Math>\widetilde{x}</Math> by increasi 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 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}< 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, s 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}</Math> be the corresponding element of <Math>\mathcal{B}</Math>. Let <Math>y</Math> be the <Math>w_0</Math>-monomi that <Math>\widetilde{F}_{\alpha}(b')=y\bmod q\mathcal{L}</Math>. 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}</Math>, if <Math>\widetilde{F}_{\alpha}(b)=b'</Math>. The graph <Math>\Gamma</Math> 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. <P/> 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\D 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 <Mat 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</Mat 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</Mat <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)</Ma 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 There is an edge with label <Math>\alpha\in\Delta</Math> from <Math>\pi_1</Math> to <Math>\pi_2</Math> <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>. </Section> </Chapter> <Chapter Label="chap3"> <Heading>QuaGroup</Heading> In this chapter we describe the functionality provided by <Package>QuaGroup</Package>. <Section Label="sec:globcst"> <Heading>Global constants</Heading> <ManSection> <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>. <Example><![CDATA[ gap> R:= RootSystem( "G", 2 );; gap> PositiveRootsInConvexOrder( R ); [ [ 1, 0 ], [ 3, 1 ], [ 2, 1 ], [ 3, 2 ], [ 1, 1 ], [ 0, 1 ] ] ]]></Example> </Description> </ManSection> <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). <Example><![CDATA[ gap> W:= WeylGroup( RootSystem( "G", 2 ) ) ;; gap> ApplyWeylElement( W, [ -3, 7 ], [ 1, 1, 2, 1, 2 ] ); [ 15, -11 ] ]]></Example> </Description> </ManSection> <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. <Example><![CDATA[ gap> W:= WeylGroup( RootSystem( "F", 4 ) ) ; <matrix group with 4 generators> gap> LengthOfWeylWord( W, [ 1, 3, 2, 4, 2 ] ); 3 ]]></Example> </Description> </ManSection> <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). <Example><![CDATA[ gap> R:= RootSystem( "G", 2 );; gap> LongestWeylWord( R ); [ 1, 2, 1, 2, 1, 2 ] ]]></Example> </Description> </ManSection> <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>. <Example><![CDATA[ gap> R:= RootSystem( "G", 2 );; gap> wd:= LongestWeylWord( R );; gap> ExchangeElement( WeylGroup(R), wd, 1 ); [ 2, 1, 2, 1, 2 ] ]]></Example> </Description> </ManSection> <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. <Example><![CDATA[ gap> R:= RootSystem( "A", 3 );; gap> wd1:= LongestWeylWord( R ); [ 1, 2, 1, 3, 2, 1 ] gap> wd2:= [ 1, 3, 2, 1, 3, 2 ];; gap> GetBraidRelations( WeylGroup(R), wd1, wd2 ); [ [ 3, 3, 4, 1 ], [ 4, 2, 5, 1, 6, 2 ], [ 2, 3, 3, 2, 4, 3 ], [ 4, 1, 5, 3 ] ] ]]></Example> </Description> </ManSection> <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. <Example><![CDATA[ gap> R:= RootSystem( "A", 3 );; gap> LongWords( R )[3]; [ [ 3, 1, 2, 1, 3, 2 ], [ [ 3, 3, 4, 1 ], [ 4, 2, 5, 1, 6, 2 ], [ 2, 3, 3, 2, 4, 3 ], [ 4, 1, 5, 3 ], [ 1, 3, 2, 1 ] ], [ [ 4, 3, 5, 1 ], [ 1, 1, 2, 3 ], [ 2, 2, 3, 3, 4, 2 ], [ 4, 1, 5, 2, 6, 1 ], [ 3, 1, 4, 3 ] ] ] ]]></Example> </Description> </ManSection> </Section> <Section> <Heading>Quantized enveloping algebras </Heading> 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> <ManSection> <Oper Name="AutomorphismTalpha" Arg="U, ind"/> <Description> This is the automorphism <M>T_{\alpha}</M> (cf. Section <Ref Sect="sec2.4"/>), where <M>\alpha</M> is the <A>ind</A>-th simple root. <Example><![CDATA[ gap> U:= QuantizedUEA( RootSystem( "B", 3 ) );; gap> f:=AutomorphismTalpha( U, 1 ); <automorphism of QuantumUEA( <root system of type B3>, Qpar = q )> gap> a:= GeneratorsOfAlgebra( U )[3]; F3 gap> Image( f, a ); F2 ]]></Example> </Description> </ManSection> <ManSection> <Oper Name="DiagramAutomorphism" Arg="U, perm"/> <Description> This is the automorphism of <A>U</A> induced by a diagram automorphism of the underlying root system. The diagram automorphism is represented by the permutation <A>perm</A>, which is the permutation of the simple roots performed by the diagram automorphism.<P/> In the example below we construct the diagram automorphism of the root system of type <M>A_3</M>, which is represented by the permutation <A>(1,3)</A>. <Example><![CDATA[ gap> R:= RootSystem( "A", 3 );; gap> U:= QuantizedUEA( R );; gap> f:= DiagramAutomorphism( U, (1,3) ); <automorphism of QuantumUEA( <root system of type A3>, Qpar = q )> gap> g:= GeneratorsOfAlgebra( U ); [ 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> Image( f, g[1] ); F6 ]]></Example> </Description> </ManSection> <ManSection> <Oper Name="\*" Arg="f, h"/> <Description> We can compose automorphisms and anti-automorphisms using the infix <A>*</A> operator. The result of composing two automorphisms is an automorphism. The result of composing an automorphism and an anti-automorphism is an anti-automorphism. The result of composing two anti-automorphisms is an automorphism. <Example><![CDATA[ gap> U:= QuantizedUEA( RootSystem( "B", 3 ) );; gap> f:=AutomorphismTalpha( U, 1 ); <automorphism of QuantumUEA( <root system of type B3>, Qpar = q )> gap> h:= AutomorphismOmega( U ); <automorphism of QuantumUEA( <root system of type B3>, Qpar = q )> gap> f*h; <automorphism of QuantumUEA( <root system of type B3>, Qpar = q )> gap> t:= AntiAutomorphismTau( U );; gap> T:= AutomorphismTalpha( U, 2 );; gap> Tinv:= t*T*t; <automorphism of QuantumUEA( <root system of type B3>, Qpar = q )> gap> # (The last call may take a little while.) gap> x:= Image( T, GeneratorsOfAlgebra( U )[1] ); (-q^4+1)*F1*F3+(-q^2)*F2 gap> Image( Tinv, x ); F1 ]]></Example> According to <Cite Key="J96"/>, 8.14(10), <M>\tau \circ T_{\alpha} \circ \tau</M> is the inverse of <M>T_{\alpha}</M>. </Description> </ManSection> </Section> <Section> <Heading>Hopf algebra structure</Heading> Here we describe functions for dealing with the Hopf algebra structure of a quantized enveloping algebra. This structure enables us to construct tensor products, and dual modules of modules over a quantized enveloping algebra. We refer to the next section (Section <Ref Sect="sec3.8"/>) for some functions for creating modules. <ManSection> <Oper Name="TensorPower" Arg="U, d"/> <Description> Here <A>U</A> is a quantized universal enveloping algebra, and <A>d</A> a non-negative integer. This function returns the associative algebra with underlying vector space the <A>d</A>-fold tensor product of <A>U</A> with itself. The product is defined component wise. <Example><![CDATA[ gap> U:= QuantizedUEA( RootSystem( [ "B", 2 ] ) );; gap> T:= TensorPower( U, 3 ); <algebra over QuantumField, with 36 generators> gap> g:= GeneratorsOfAlgebra( T );; gap> x:= g[1]; 1*(1<x>1<x>F1) gap> y:= g[30]; 1*(E2<x>1<x>1) gap> x*y; 1*(E2<x>1<x>F1) ]]></Example> </Description> </ManSection> <ManSection> <Oper Name="UseTwistedHopfStructure" Arg="U, f, finv"/> <Description> Here <A>U</A> is a quantized enveloping algebra, and <A>f</A>, <A>finv</A> two (anti-) automorphisms of <A>U</A>, where <A>finv</A> is the inverse of <A>f</A>. After calling this function the Hopf structure on <A>U</A> is used that is obtained from the <Q>normal</Q> Hopf structure (see Section <Ref Sect="sec2.3"/>) by twisting it with <A>f</A>. <P/> A call to this function sets the attribute <A>HopfStructureTwist</A>, which is the list <A>[ f, finv ]</A>. <Example><![CDATA[ gap> U:= QuantizedUEA( RootSystem("A",2), CF(5), E(5) );; gap> t:= AntiAutomorphismTau( U );; gap> UseTwistedHopfStructure( U, t, t ); ]]></Example> </Description> </ManSection> <ManSection> <Oper Name="ComultiplicationMap" Arg="U, d"/> <Description> This is a homomorphism from the quantized enveloping algebra <A>U</A> to the <A>d</A>-fold tensor power of <A>U</A> with itself. It is obtained by a repeated application of the comultiplication of <A>U</A>. So for <A>d=2</A> we get the comultiplication of <A>U</A>. <Example><![CDATA[ gap> U:= QuantizedUEA( RootSystem("A",2), CF(5), E(5) );; gap> D:= ComultiplicationMap( U, 3 ); <Comultiplication of QuantumUEA( <root system of type A2>, Qpar = E(5) ), degree 3> gap> Image( D, GeneratorsOfAlgebra(U)[4] ); 1*(K1<x>K1<x>K1) ]]></Example> </Description> </ManSection> <ManSection> <Attr Name="AntipodeMap" Arg="U"/> <Description> This is the antipode map of the quantized enveloping algebra <A>U</A>, which is constructed as an anti-automorphism of <A>U</A>. <Example><![CDATA[ gap> U:= QuantizedUEA( RootSystem("A",2), CF(5), E(5) );; gap> a:= AntipodeMap( U ); <anti-automorphism of QuantumUEA( <root system of type A2>, Qpar = E(5) )> ]]></Example> </Description> </ManSection> <ManSection> <Attr Name="CounitMap" Arg="U"/> <Description> This is the counit map of the quantized enveloping algebra <A>U</A>, which is constructed as a function from <A>U</A> to the ground field. <Example><![CDATA[ gap> U:= QuantizedUEA( RootSystem("A",2), CF(5), E(5) );; gap> co:= CounitMap( U ); function( u ) ... end gap> x:= GeneratorsOfAlgebra( U )[4]; K1 gap> co( x ); 1 ]]></Example> </Description> </ManSection> </Section> <Section Label="sec3.8"> <Heading>Modules</Heading> Here we describe some functions for constructing left modules over quantized enveloping algebras. We refer to the &GAP; reference manual for an overview of basic functions for algebra modules, which are also applicable to the modules constructed by the functions described in this section. We mention <A>MatrixOfAction</A>, <A>DirectSumOfAlgebraModules</A>. The action of an element of the algebra on an element of the module is calculated by the infix operator <A>&circum;</A>. <ManSection> <Oper Name="HighestWeightModule" Label="for a quantized env. alg." Arg="U, wt"/> <Description> Here <A>U</A> is a quantized universal enveloping algebra, and <A>wt</A> a dominant weight (i.e., a list of length equal to the rank of the root system, consisting of non-negative integers). This function returns a finite-dimensional highest-weight module of highest weight <A>wt</A> over <A>U</A>. If <A>U</A> is generic then this is the unique irreducible highest-weight module over <A>U</A>. Otherwise it is the Weyl module, cf. Section <Ref Sect="sec2.5"/>. In this last case the module is not necessarily irreducible.<P/> Let <A>V</A> denote the module returned by this function. The first basis element of the attribute <A>Basis( V )</A> is a highest-weight vector; it is written as <A>1*v0</A>. Other basis elements are written as, for example, <A>F2*F9*v0</A>, which means that this vector is the result of letting the PBW-monomial <A>F2*F9</A> act on the highest-weight vector.<P/> <Example><![CDATA[ gap> U:= QuantizedUEA( RootSystem( [ "A", 2, "G", 2 ] ) );; gap> V:= HighestWeightModule( U, [ 0, 1, 0, 2 ] ); <231-dimensional left-module over QuantumUEA( <root system of type A2 G 2>, Qpar = q )> gap> Basis( V )[1]; 1*v0 gap> Basis(V)[23]+(_q^2+_q^-2)*Basis(V)[137]; F3*F5*v0+(q^2+q^-2)*F8^(6)*v0 gap> # We compute the action of an element on a vector: gap> gg:= GeneratorsOfAlgebra( U );; gap> x:= gg[21]*gg[5]; F5*E4+(-q^-1)*F6*K3 gap> x^Basis(V)[1]; (-q^-1)*F6*v0 ]]></Example> </Description> </ManSection> <ManSection> <Attr Name="IrreducibleQuotient" Arg="V"/> <Description> Here <A>V</A> is a highest-weight module over a non-generic quantized enveloping algebra. This function returns the quotient of <A>V</A> by the maximal submodule not containing the highest weight vector. This is not necessarily equal to <A>V</A> if the quantum parameter is a root of 1. <Example><![CDATA[ gap> R:= RootSystem( "A", 2 );; gap> U:= QuantizedUEA( R, CF(3), E(3) );; gap> V:= HighestWeightModule( U, [1,1] ); <8-dimensional left-module over QuantumUEA( <root system of type A2>, Qpar = E(3) )> gap> IrreducibleQuotient( V ); <7-dimensional left-module over QuantumUEA( <root system of type A2>, Qpar = E(3) )> ]]></Example> </Description> </ManSection> <ManSection> <Oper Name="HWModuleByTensorProduct" Arg="U, wt"/> <Description> Here <A>U</A> must be a <E>generic</E> quantized enveloping algebra, and <A>wt</A> a dominant weight. This function returns the irreducible highest-weight module with highest weight <A>wt</A>. The algorithm uses tensor products (whence the name). On some inputs this algorithm is faster than the one use for <Ref Oper="HighestWeightModule" Label="for a quantized env. alg."/>, on some inputs it is slower. I do not know any good heuristics. <P/> The basis supplied with the module returned is the canonical basis. <Example><![CDATA[ gap> U:= QuantizedUEA( RootSystem("G",2) );; gap> V:= HWModuleByTensorProduct( U, [2,1] ); <189-dimensional left-module over QuantumUEA( <root system of type G 2>, Qpar = q )> gap> # (This is a case where this algorithm is a lot faster.) ]]></Example> </Description> </ManSection> <ManSection> <Oper Name="DIYModule" Arg="U, V, acts"/> <Description> Here <A>U</A> is a generic quantized enveloping algebra, and <A>V</A> is a vector space over the field <A>QuantumField</A>. <A>U</A> acts on <A>V</A> and the action is described by the data in the list <A>acts</A>. <A>acts</A> is a list of lists, of length <A>4*l</A>, where <A>l</A> is the rank of the root system. <A>acts</A> describes the actions of the generators <M>[F_1,...,F_l,K_1,...,K_l,K_1^-1,...,K_l^-1, E_1,...,E_l ]</M>. (Here <M>F_k</M> is the generator <M>F_{\alpha_k}</M>, where <M>\alpha_k</M> is the <M>k</M>-th simple root, and likewise for <M>E_k</M>.) The action of each generator is described by a list of length <A>dim V</A>, giving the images of the basis elements of <A>V</A>. If an image is zero then it may be omitted: in that case there is a <Q>hole</Q> in the list. This function returns the <A>U</A>-module defined by the input.<P/> Let <A>R</A> be a root system of type <M>A_1</M>, and <A>U</A> the corresponding quantized enveloping algebra (generated by <M>F, K, K^-1, E</M>). In the example below we construct the 2-dimensional <M>U</M>-module with basis vectors <M>v_1,v_2</M>, and <A>U</A>-action given by <M>Fv_1 = v_2</M>, <M>Fv_2=0</M>, <M>Kv_1 = qv_1</M>, <M>Kv_2=q^{-1}v_2</M>, <M>Ev_1=0</M>, <M>Ev_2=v_1</M>. <Example><![CDATA[ gap> U:= QuantizedUEA( RootSystem("A",1) ); QuantumUEA( <root system of type A1>, Qpar = q ) gap> V:= QuantumField^2; ( QuantumField^2 ) gap> v:= BasisVectors( Basis(V) ); [ [ 1, 0 ], [ 0, 1 ] ] gap> acts:= [ [ v[2], 0*v[1] ], [ _q*v[1], _q^-1*v[2] ], > [ _q^-1*v[1], _q*v[2] ], [ 0*v[1], v[1] ] ];; gap> M:= DIYModule( U, V, acts ); <2-dimensional left-module over QuantumUEA( <root system of type A 1>, Qpar = q )> ]]></Example> </Description> </ManSection> <ManSection> <Oper Name="TensorProductOfAlgebraModules" Arg="V, W"/> <Oper Name="TensorProductOfAlgebraModules" Arg="V, W"/> <Description> Here <A>V</A> and <A>W</A> are two modules over the same quantized enveloping algebra <A>U</A>. This function constructs the tensor product of <A>V</A> and <A>W</A> (as a <A>U</A>-module). For this the comultiplication map of <A>U</A> is used (see <Ref Attr="ComultiplicationMap"/>). <P/> In the second form list is a list of <A>U</A>-modules. In that case the iterated tensor product is constructed. <Example><![CDATA[ gap> U:= QuantizedUEA( RootSystem( [ "A", 2 ] ) );; gap> V1:= HighestWeightModule( U, [ 1, 0 ] );; gap> V2:= HighestWeightModule( U, [ 0, 1 ] );; gap> TensorProductOfAlgebraModules( V1, V2 ); <9-dimensional left-module over QuantumUEA( <root system of type A 2>, Qpar = q )> ]]></Example> </Description> </ManSection> <ManSection> <Oper Name="HWModuleByGenerator" Arg="V, v, hw"/> <Description> Here <A>V</A> is a module over a generic quantized enveloping algebra <A>U</A>, <A>v</A> is a highest-weight vector (i.e., all <M>E_{\alpha}</M><A>v=0</A>), of weight <A>hw</A>, which must be dominant. This function returns a highest-weight module over <A>U</A> isomorphic to the submodule of <A>V</A> generated by <A>v</A>. <Example><![CDATA[ gap> U:= QuantizedUEA( RootSystem("B",2) );; gap> W1:= HighestWeightModule( U, [1,0] );; gap> W2:= HighestWeightModule( U, [0,1] );; gap> T:= TensorProductOfAlgebraModules( W1, W2 ); <20-dimensional left-module over QuantumUEA( <root system of type B 2>, Qpar = q )> gap> HWModuleByGenerator( T, Basis(T)[1], [1,1] ); <16-dimensional left-module over QuantumUEA( <root system of type B 2>, Qpar = q )> ]]></Example> </Description> </ManSection> <ManSection> <Oper Name="InducedQEAModule" Arg="U, V"/> <Description> Here <A>U</A> is a non-generic quantized enveloping algebra, and <A>V</A> a module over the corresponding generic quantized enveloping algebra. This function returns the <A>U</A>-module obtained from <A>V</A> by setting <A>&uscore;q</A> equal to the quantum parameter of <A>U</A>. <Example><![CDATA[ gap> R:= RootSystem("B",2);; gap> U:= QuantizedUEA( R );; gap> U0:= QuantizedUEA( R, CF(3), E(3) );; gap> V:= HighestWeightModule( U, [1,1] );; gap> W:= InducedQEAModule( U0, V ); <16-dimensional left-module over QuantumUEA( <root system of type B2>, Qpar = E(3) )> gap> # This module is isomorphic to the one obtained by gap> # HighestWeightModule( U0, [1,1] ); ]]></Example> </Description> </ManSection> <ManSection> <Attr Name="GenericModule" Arg="W"/> <Description> For an induced module (see <Ref Oper="InducedQEAModule"/>) this function returns the corresponding module over the generic quantized enveloping algebra. </Description> </ManSection> <ManSection> <Attr Name="CanonicalMapping" Arg="W"/> <Description> Here <A>W</A> is an induced module. Let <A>V</A> be the corresponding generic module (<Ref Attr="GenericModule"/>). This function returns the map <A>V --> W</A>, that sets <A>&uscore;q</A> equal to the quantum parameter of the acting algebra of <A>W</A>. <Example><![CDATA[ gap> R:= RootSystem("B",2);; gap> U:= QuantizedUEA( R );; gap> U0:= QuantizedUEA( R, CF(3), E(3) );; gap> V:= HighestWeightModule( U, [1,1] );; gap> W:= InducedQEAModule( U0, V );; gap> f:= CanonicalMapping( W ); MappingByFunction( < 16-dimensional left-module over QuantumUEA( <root system of type B 2>, Qpar = q )>, < 16-dimensional left-module over QuantumUEA( <root system of type B2>, Qpar = E(3) )>, function( v ) ... end ) gap> Image( f, _q^2*Basis(V)[3] ); (E(3)^2)*e.3 ]]></Example> </Description> </ManSection> <ManSection> <Oper Name="U2Module" Arg="U, hw"/> <Description> Here <A>U</A> must be a quantized enveloping algebra of type <M>A_2</M>. This function returns the highest-weight module over <A>U</A> of highest-weight <A>hw</A> (which must be dominant). This function is generally a lot faster than <Ref Oper="HighestWeightModule" Label="for a quantized env. alg."/>. <Example><![CDATA[ gap> U:= QuantizedUEA( RootSystem("A",2) );; gap> A2Module( U, [4,7] ); <260-dimensional left-module over QuantumUEA( <root system of type A 2>, Qpar = q )> ]]></Example> </Description> </ManSection> <ManSection> <Oper Name="MinusculeModule" Arg="U, hw"/> <Description> Here <A>U</A> must be a generic quantized enveloping algebra, and <A>hw</A> a minuscule dominant weight. This function returns the highest-weight module over <A>U</A> of highest-weight <A>hw</A>. This function is generally somewhat faster than <Ref Oper="HighestWeightModule" Label="for a quantized env. alg."/>. <Example><![CDATA[ gap> U:= QuantizedUEA( RootSystem("A",5) );; gap> MinusculeModule( U, [0,0,1,0,0] ); <20-dimensional left-module over QuantumUEA( <root system of type A 5>, Qpar = q )> ]]></Example> </Description> </ManSection> <ManSection> <Attr Name="DualAlgebraModule" Arg="V"/> <Description> Here <A>V</A> is a finite-dimensional left module over a quantized enveloping algebra <A>U</A>. This function returns the dual space of <A>V</A> as an algebra module. For this the antipode map of <A>U</A> is used (see <Ref Attr="AntipodeMap"/>). <P/> Let <A>M</A> denote the module returned by this function. Then <A>M</A> has as basis the dual basis with respect to <A>Basis( V )</A>. An element of this basis is printed as <A>F@v</A>, where <A>v</A> is an element of <A>Basis( V )</A>. This is the function which takes the value <A>1</A> on <A>v</A> and <A>0</A> on all other basis elements. A general element of <A>M</A> is a linear combination of these basis elements. <P/> The elements of <A>M</A> can be viewed as functions which take arguments. However, internally the elements of <A>M</A> are represented as wrapped up functions. The function corresponding to an element <A>m</A> of <A>M</A> is obtained by <A>ExtRepOfObj( m )</A> (the result of which is printed in the same way as <A>m</A>, but is not equal to it). <Example><![CDATA[ gap> U:= QuantizedUEA( RootSystem("A",2) );; gap> V:= HighestWeightModule( U, [1,1] );; gap> M:= DualAlgebraModule( V ); <8-dimensional left-module over QuantumUEA( <root system of type A 2>, Qpar = q )> gap> u:= GeneratorsOfAlgebra( U )[2]; F2 gap> vv:= BasisVectors( Basis( M ) ); [ (1)*F@1*v0, (1)*F@F1*v0, (1)*F@F3*v0, (1)*F@F1*F3*v0, (1)*F@F2*v0, (1)*F@F1*F2*v0, (1)*F@F2*F3*v0, (1)*F@F2^(2)*v0 ] gap> u^vv[3]; <zero function> gap> # (The zero of the dual space is printed as <zero function>). gap> u^vv[4]; (-q^5+q^3)*F@1*v0 gap> # We get the function corresponding to a vector in M by using ExtRepOfObj: gap> f:= ExtRepOfObj( vv[1] ); (1)*F@1*v0 gap> # We can calculate images of this function: gap> List( Basis(V), v -> Image( f, v ) ); [ 1, 0, 0, 0, 0, 0, 0, 0 ] ]]></Example> </Description> </ManSection> <ManSection> <Attr Name="TrivialAlgebraModule" Arg="U"/> <Description> Returns the trivial module over the quantized enveloping algebra <A>U</A>. For this the counit map of <A>U</A> is used. <Example><![CDATA[ gap> U:= QuantizedUEA( RootSystem("A",2) );; gap> V:= TrivialAlgebraModule( U ); <left-module over QuantumUEA( <root system of type A2>, Qpar = q )> ]]></Example> </Description> </ManSection> <ManSection> <Oper Name="WeightsAndVectors" Arg="V"/> <Description> Here <A>V</A> is a left module over a quantized enveloping algebra. <A>WeightsAndVectors( V )</A> is a list of two lists; the first of these is a list of the weights of <A>V</A>, the second a list of corresponding weight vectors. These are again grouped in lists: if the multiplicity of a weight is <A>m</A>, then there are <A>m</A> weight vectors, forming a basis of the corresponding weight space. <P/> Modules constructed by <Ref Oper="HighestWeightModule" Label="for a quantized env. alg."/> come with this attribute set. There is a method installed for computing <A>WeightsAndVectors( V )</A>, for modules <A>V</A> over a generic quantized enveloping algebra, such that all basis vectors (i.e., all elements of <A>Basis( V )</A>) are weight vectors. <Example><![CDATA[ gap> U:= QuantizedUEA( RootSystem( "A", 2 ) );; gap> V:= HighestWeightModule( U, [ 1, 1 ] );; gap> WeightsAndVectors( V ); [ [ [ 1, 1 ], [ -1, 2 ], [ 2, -1 ], [ 0, 0 ], [ -2, 1 ], [ 1, -2 ], [ -1, -1 ] ], [ [ 1*v0 ], [ F1*v0 ], [ F3*v0 ], [ F1*F3*v0, F2*v0 ], [ F1*F2*v0 ], [ F2*F3*v0 ], [ F2^(2)*v0 ] ] ] ]]></Example> </Description> </ManSection> <ManSection> <Attr Name="HighestWeightsAndVectors" Arg="V"/> <Description> Is analogous to <Ref Attr="WeightsAndVectors"/>; now only the highest weights are listed along with the corresponding highest-weight vectors. <P/> There is a method installed for this using <Ref Attr="WeightsAndVectors"/>; which means that it works if and only if <A>WeightsAndVectors( V )</A> works. <Example><![CDATA[ gap> U:= QuantizedUEA( RootSystem( [ "A", 2 ] ) );; gap> V:= HighestWeightModule( U, [ 1, 1 ] );; gap> HighestWeightsAndVectors( V ); [ [ [ 1, 1 ] ], [ [ 1*v0 ] ] ] ]]></Example> </Description> </ManSection> <ManSection> <Attr Name="RMatrix" Arg="V"/> <Description> Here <A>V</A> is a module over the a quantized enveloping algebra <A>U</A>. This function returns the matrix of a linear map <M>\theta : V\otimes V \to V\otimes V</M> that is a solution to the quantum Yang-Baxter equation. We have that <M>\theta\circ P</M> is an isomorphism of <A>U</A>-modules, where <M>P :V\otimes V\to V\otimes V</M> is the linear map such that <M>P(v\otimes w)=w\otimes v</M>. For more details we refer to <Cite Key="J96"/>, Chapter 7. <P/> This function works for modules for which <Ref Attr="WeightsAndVectors"/> works. <Example><![CDATA[ gap> U:= QuantizedUEA( RootSystem("A",1) );; gap> V:= HighestWeightModule( U, [1] );; gap> RMatrix( V ); [ [ 1, 0, 0, 0 ], [ 0, q, -q^2+1, 0 ], [ 0, 0, q, 0 ], [ 0, 0, 0, 1 ] ] ]]></Example> </Description> </ManSection> <ManSection> <Oper Name="IsomorphismOfTensorModules" Arg="V, W"/> <Description> Here <A>V</A>, <A>W</A> are two modules over the same quantized enveloping algebra <A>U</A>. This function returns a linear map <M>\theta : V\otimes W\to W\otimes V</M> that is an isomorphism of U-modules. <P/> This function is only guaranteed to work correctly if the Hopf algebra structure is non-twisted (see <Ref Oper="UseTwistedHopfStructure"/>). <P/> This function works for modules for which <Ref Attr="WeightsAndVectors"/> works. <Example><![CDATA[ gap> U:= QuantizedUEA( RootSystem("B",2) );; gap> V:= HighestWeightModule( U, [1,0] );; gap> W:= HighestWeightModule( U, [0,1] );; gap> h:= IsomorphismOfTensorModules( V, W );; gap> VW:= Source( h ); <20-dimensional left-module over QuantumUEA( <root system of type B 2>, Qpar = q )> gap> Image( h, Basis(VW)[13] ); q*(1*v0<x>F3*v0)+-q^2+1*(F4*v0<x>F2*v0)+-q^3+q^-1*(F3*v0<x>1*v0) ]]></Example> </Description> </ManSection> <ManSection> <Oper Name="WriteModuleToFile" Arg="V, file"/> <Description> Here <A>V</A> is a module over a quantized enveloping algebra, and <A>file</A> is a string containing the name of a file. This function writes some data to <A>file</A>, that allows <Ref Oper="ReadModuleFromFile"/> to recover it.<P/> We remark that this function currently is only implemented for generic quantized enveloping algebras. </Description> </ManSection> <ManSection> <Oper Name="ReadModuleFromFile" Arg="file"/> <Description> Here <A>file</A> is a string containing the name of a file, to which a module over a quantized enveloping algebra has been written by <Ref Oper="WriteModuleToFile"/>. This function recovers the module. More precisely: a new module is constructed that is isomorphic to the old one. In the process the algebra acting on the module is constructed anew (from data written to the file). This algebra can be accessed by <A>LeftActingAlgebra( V )</A>. <P/> We remark that this function currently is only implemented for generic quantized enveloping algebras. <Example><![CDATA[ gap> U:= QuantizedUEA( RootSystem("A",3) );; gap> V:= HighestWeightModule( U, [1,1,1] );; gap> WriteModuleToFile( V, "A3mod" ); gap> W:= ReadModuleFromFile( "A3mod" ); <64-dimensional left-module over QuantumUEA( <root system of type A 3>, Qpar = q )> ]]></Example> </Description> </ManSection> </Section> <Section> <Heading>The path model</Heading> In this section we describe functions for dealing with the path model. We work only with LS-paths, which are represented by two lists, one of weights, and one of rationals (see Section <Ref Sect="sec2.7"/>). <ManSection> <Oper Name="DominantLSPath" Arg="R, wt"/> <Description> Here <A>R</A> is a root system, and <A>wt</A> a dominant weight in the weight lattice of <A>R</A>. This function returns the LS-path that is the line from the origin to <A>wt</A>. <Example><![CDATA[ gap> R:= RootSystem( "G", 2 );; gap> DominantLSPath( R, [1,3] ); <LS path of shape [ 1, 3 ] ending in [ 1, 3 ] > ]]></Example> </Description> </ManSection> <ManSection> <Oper Name="Falpha" Label="for an LS-path" Arg="path, ind"/> <Description> Is the result of applying the path operator <M>f_{\alpha_{<A>ind</A>}}</M> to the LS-path <A>path</A> (where <M>\alpha_{<A>ind</A>}</M> is the <A>ind</A>-th simple root). <P/> The result is <A>fail</A> if <M>f_{\alpha_{<A>ind</A>}}</M><A>(path)=0</A>. <Example><![CDATA[ gap> R:= RootSystem( "G", 2 );; gap> p:=DominantLSPath( R, [1,3] );; gap> p1:=Falpha( p, 1 ); <LS path of shape [ 1, 3 ] ending in [ -1, 4 ] > gap> Falpha( p1, 1 ); fail ]]></Example> </Description> </ManSection> <ManSection> <Oper Name="Ealpha" Label="for an LS-path" Arg="path, ind"/> <Description> Is the result of applying the path operator <M>e_{\alpha_{<A>ind</A>}}</M> to the LS-path <A>path</A> (where <M>\alpha_{<A>ind</A>}</M> is the <A>ind</A>-th simple root). <P/> The result is <A>fail</A> if <M>e_{\alpha_{<A>ind</A>}}</M><A>(path)=0</A>. <Example><![CDATA[ gap> R:= RootSystem( "G", 2 );; gap> p:=DominantLSPath( R, [1,3] );; gap> Ealpha( p, 2 ); fail gap> p1:=Falpha( p, 1 );; gap> Ealpha( p1, 1 ); <LS path of shape [ 1, 3 ] ending in [ 1, 3 ] > ]]></Example> </Description> </ManSection> <ManSection> <Attr Name="LSSequence" Arg="path"/> <Description> returns the two sequences (of weights and rational numbers) that define the LS-path path. <Example><![CDATA[ gap> R:= RootSystem( "G", 2 );; gap> p:=DominantLSPath( R, [1,3] );; gap> p1:= Falpha( Falpha( p, 1 ), 2 );; gap> LSSequence( p1 ); [ [ [ 11, -4 ], [ -1, 4 ] ], [ 0, 1/4, 1 ] ] ]]></Example> </Description> </ManSection> <ManSection> <Attr Name="WeylWord" Arg="path"/> <Description> Here <A>path</A> is an LS-path in the orbit (under the root operators) of a dominant LS-path ending in the dominant weight <M>\lambda</M>. This means that the first direction of path is of the form <M>w(\lambda)</M> for some <M>w</M> in the Weyl group. This function returns a list <M>[i_1,\ldots, i_m ]</M> such that <M>w=s_{i_1}\cdots s_{i_m}</M>. <Example><![CDATA[ gap> R:= RootSystem( "G", 2 );; gap> p:=DominantLSPath( R, [1,3] );; gap> p1:= Falpha( Falpha( Falpha( p, 1 ), 2 ), 1 );; gap> WeylWord( p1 ); [ 1, 2, 1 ] ]]></Example> </Description> </ManSection> <ManSection> <Attr Name="EndWeight" Arg="path"/> <Description> Here <A>path</A> is an LS-path; this function returns the weight that is the endpoint of path <Example><![CDATA[ gap> R:= RootSystem( "G", 2 );; gap> p:=DominantLSPath( R, [1,3] );; gap> p1:= Falpha( Falpha( Falpha( p, 1 ), 2 ), 1 );; gap> EndWeight( p1 ); [ 0, 3 ] ]]></Example> </Description> </ManSection> <ManSection> <Func Name="CrystalGraph" Label="for root system and weight" Arg="R, wt"/> <Description> This function returns a record describing the crystal graph of the highest-weight module with highest weight <A>wt</A>, over the quantized enveloping algebra corresponding to <A>R</A>. It is computed using the path model. Therefore the points in the graph are LS-paths. <P/> Denote the output by <A>r</A>; then <A>r.points</A> is the list of points of the graph. Furthermore, <A>r.edges</A> is a list of edges of the graph; this is a list of elements of the form <A>[ [ i, j ], u ]</A>. This means that there is an arrow from point <A>i</A> (i.e., the point on position <A>i</A> in <A>r.points</A>) to point <A>j</A>, with label <A>u</A>. <Example><![CDATA[ gap> R:= RootSystem( "A", 2 );; gap> CrystalGraph( R, [1,1] ); rec( edges := [ [ [ 1, 2 ], 1 ], [ [ 1, 3 ], 2 ], [ [ 2, 4 ], 2 ], [ [ 3, 5 ], 1 ], [ [ 4, 6 ], 2 ], [ [ 5, 7 ], 1 ], [ [ 6, 8 ], 1 ], [ [ 7, 8 ], 2 ] ], points := [ <LS path of shape [ 1, 1 ] ending in [ 1, 1 ] >, <LS path of shape [ 1, 1 ] ending in [ -1, 2 ] >, <LS path of shape [ 1, 1 ] ending in [ 2, -1 ] >, <LS path of shape [ 1, 1 ] ending in [ 0, 0 ] >, <LS path of shape [ 1, 1 ] ending in [ 0, 0 ] >, <LS path of shape [ 1, 1 ] ending in [ 1, -2 ] >, <LS path of shape [ 1, 1 ] ending in [ -2, 1 ] >, <LS path of shape [ 1, 1 ] ending in [ -1, -1 ] > ] ) ]]></Example> </Description> </ManSection> </Section> <Section> <Heading> Canonical bases </Heading> Here we describe functions for computing the canonical basis of the negative part of a quantized enveloping algebra, and of a module. <ManSection> <Oper Name="Falpha" Label="for a PBW-monomial" Arg="x, ind"/> <Description> Here <A>x</A> is a PBW-monomial in <M>U^-</M> (i.e., a monomial in the <M>F_{\alpha}</M>, where <M>\alpha</M> runs over the positive roots). This function returns the result of applying the <A>ind</A>-th Kashiwara operator <M>\widetilde{F}_{\alpha_{<A>ind</A>}}</M> to <M>x</M> (cf. Section <Ref Sect="sec2.6"/>). <Example><![CDATA[ gap> U:= QuantizedUEA( RootSystem( "F", 4 ) );; gap> x:= One( U ); 1 gap> Falpha( Falpha( x, 3 ), 2 ); F3*F9 ]]></Example> </Description> </ManSection> <ManSection> <Oper Name="Ealpha" Label="for a PBW-monomial" Arg="x, ind"/> <Description> Here <A>x</A> is a PBW-monomial in <M>U^-</M> (i.e., a monomial in the <M>F_{\alpha}</M>, where <M>\alpha</M> runs over the positive roots). This function returns the result of applying the <A>ind</A>-th Kashiwara operator <M>\widetilde{E}_{\alpha_{<A>ind</A>}}</M> to <M>x</M> (cf. Section <Ref Sect="sec2.6"/>). The result is <A>fail</A> if <M>\widetilde{E}_{\alpha_{<A>ind</A>}}</M><A>(x)=0</A>. <Example><![CDATA[ gap> U:= QuantizedUEA( RootSystem( "F", 4 ) );; gap> Ealpha( One( U ), 2 ); fail gap> g:= GeneratorsOfAlgebra( U );; gap> x:= g[1]*g[4]*g[7]*g[17]; F1*F4*F7*F17 gap> Ealpha( x, 3 ); F1*F2*F7*F17 ]]></Example> </Description> </ManSection> <ManSection> <Attr Name="CanonicalBasis" Arg="U"/> <Description> Is the canonical basis of the quantized universal enveloping algebra <A>U</A>. When this is constructed nothing is computed. By using <Ref Oper="PBWElements"/>, <Ref Oper="MonomialElements"/>, <Ref Oper="Strings"/> information about elements of the canonical basis can be obtained. <Example><![CDATA[ gap> U:= QuantizedUEA( RootSystem( "F", 4 ) );; gap> B:= CanonicalBasis( U ); <canonical basis of QuantumUEA( <root system of type F4>, Qpar = q ) > ]]></Example> </Description> </ManSection> <ManSection> <Oper Name="PBWElements" Arg="B, rt"/> <Description> Here <A>B</A> is the canonical basis of a quantized uea, and <A>rt</A> a list of non-negative integers representing an element of the root lattice (e.g., if the simple roots are <M>\alpha</M>, <M>\beta</M> and <A>rt = [ 3, 2 ]</A>, then <A>rt</A> represents <M>3\alpha+2\beta</M>).<P/> It is possible to add the option <A>lowrank</A>, as follows <A>PBWElements( B, rt :lowrank )</A>. In that case a somewhat different method will be used, that is significantly faster if the underlying root system has rank 2,3. It is about equally fast for ranks 4,5; and slower for ranks greater than 5. <Example><![CDATA[ gap> U:= QuantizedUEA( RootSystem( "F", 4 ) );; gap> B:= CanonicalBasis( U );; gap> PBWElements( B, [1,2,1,0] ); [ F1*F3^(2)*F9, F1*F3*F7+(q^4)*F1*F3^(2)*F9, (q^4)*F1*F3^(2)*F9+F2*F3*F9, (q^2)*F1*F3*F7+(q^6+q^2)*F1*F3^(2)*F9+(q^2)*F2*F3*F9+F2*F7, (q^4)*F1*F3*F7+(q^8)*F1*F3^(2)*F9+(q^4)*F2*F3*F9+(q^2)*F2*F7+F3*F4 ] gap> U:= QuantizedUEA( RootSystem("G",2) );; gap> B:= CanonicalBasis( U );; gap> PBWElements( B, [2,3] : lowrank ); [ F1^(2)*F6^(3), F1*F5*F6^(2)+(q^10+q^8)*F1^(2)*F6^(3), (q^2)*F1*F5*F6^(2)+(q^12+q^6)*F1^(2)*F6^(3)+F3*F6^(2), (q^8)*F1*F5*F6^(2)+(q^18)*F1^(2)*F6^(3)+(q^6)*F3*F6^(2)+F5^(2)*F6 ] ]]></Example> </Description> </ManSection> <ManSection> <Oper Name="MonomialElements" Arg="B, rt"/> <Description> This does the same as <Ref Oper="PBWElements"/>, except that the elements are written as linear combinations of monomials in the generators <M>F_{\alpha}</M>, where <M>\alpha</M> runs through the simple roots. <P/> We remark that this information is also computed <Q>behind the scenes</Q> when calling <A>PBWElements( B, rt )</A>. However, it is not computed if the option <A>lowrank</A> is present in the call to <A>PBWElements</A>. <Example><![CDATA[ gap> U:= QuantizedUEA( RootSystem( "F", 4 ) );; gap> B:= CanonicalBasis( U );; gap> MonomialElements( B, [1,2,1,0] ); [ F1*F3^(2)*F9, F1*F3*F9*F3+(-1)*F1*F3^(2)*F9, F3^(2)*F1*F9, F3*F1*F9*F3, F3*F9*F3*F1+(-1)*F3^(2)*F1*F9 ] ]]></Example> </Description> </ManSection> <ManSection> <Oper Name="Strings" Arg="B, rt"/> <Description> Here <A>B</A>, <A>rt</A> are the same as in <Ref Oper="PBWElements"/>. This returns the list of strings corresponding to the elements of <A>B</A> of weight <A>rt</A> (cf. Section <Ref Sect="sec2.6"/>). For example, if on the <M>k</M>-th position of the list returned by this function we have <A>[ 1, 2, 2, 3 ]</A>, then the principal monomial of the <M>k</M>-th element of <A>PBWElements( B, rt )</A> is <M>\widetilde{F}_1^2\widetilde{F}_2^3(1)</M> (where <M>\widetilde{F}_i</M> is the <M>i</M>-th Kashiwara operator). <P/> We remark that this information is also computed <Q>behind the scenes</Q> when calling <A>PBWElements( B, rt )</A>. However, it is not computed if the option <A>lowrank</A> is present in the call to <A>PBWElements</A>. <Example><![CDATA[ gap> U:= QuantizedUEA( RootSystem( "F", 4 ) );; gap> B:= CanonicalBasis( U );; gap> Strings( B, [1,2,1,0] ); [ [ 1, 1, 2, 2, 3, 1 ], [ 1, 1, 2, 1, 3, 1, 2, 1 ], [ 2, 2, 1, 1, 3, 1 ], [ 2, 1, 1, 1, 3, 1, 2, 1 ], [ 2, 1, 3, 1, 2, 1, 1, 1 ] ] gap> Falpha( Falpha( Falpha( Falpha( One(U), 3 ), 1 ), 2 ), 2 ); F2*F3*F9 gap> PBWElements( B, [1,2,1,0] )[3]; (q^4)*F1*F3^(2)*F9+F2*F3*F9 ]]></Example> </Description> </ManSection> <ManSection> <Oper Name="PrincipalMonomial" Arg="u"/> <Description> Here <A>u</A> is an element of the output of <Ref Oper="PBWElements"/>. This function returns the unique monomial of <A>u</A> that has coefficient 1. <Example><![CDATA[ gap> U:= QuantizedUEA( RootSystem("G",2) );; gap> B:= CanonicalBasis( U );; gap> p:= PBWElements( B, [4,4] : lowrank )[4]; (q^9)*F1^(2)*F3*F6^(3)+F1^(2)*F5^(2)*F6^(2)+(q^13+q^11+q^9)*F1^(3)*F5*F6^( 3)+(q^28+q^26+2*q^24+q^22+q^20)*F1^(4)*F6^(4) gap> PrincipalMonomial( p ); F1^(2)*F5^(2)*F6^(2) ]]></Example> </Description> </ManSection> <ManSection> <Oper Name="StringMonomial" Arg="u"/> <Description> Here <A>u</A> is a monomial in the negative part of a quantized enveloping algebra, e.g., as output by <Ref Oper="PrincipalMonomial"/>. This function computes the corresponding <Q>string</Q> (see Section <Ref Sect="sec2.6"/>). The strings are output in the same way as in <Ref Sect="Strings"/>. <Example><![CDATA[ gap> U:= QuantizedUEA( RootSystem("G",2) );; gap> B:= CanonicalBasis( U );; gap> p:= PBWElements( B, [1,2] : lowrank )[2];; gap> m:=PrincipalMonomial( p ); F5*F6 gap> StringMonomial( m ); [ 2, 2, 1, 1 ] gap> Falpha( Falpha( Falpha( One(U), 1 ), 2 ), 2 ); F5*F6 ]]></Example> </Description> </ManSection> <ManSection> <Oper Name="Falpha" Label="for a module element" Arg="V, v, ind"/> <Description> Here <A>V</A> is a module over a quantized enveloping algebra, <A>v</A> an element of it, and <A>ind</A> an index between 1 and the rank of the root system. The function returns the result of applying the <A>ind</A>-th Kashiwara operator <M>\widetilde{F}_{<A>ind</A>}</M> to <A>v</A>. Here the Kashiwara operators are different from the ones described in Section <Ref Sect="sec2.6"/>. We refer to <Cite Key="J96"/>, 9.2 for the definition of the operators used here. <Example><![CDATA[ gap> U:= QuantizedUEA( RootSystem("B",2) );; gap> V:= HighestWeightModule( U, [1,1] );; gap> Falpha( V, Basis(V)[1], 1 ); F1*v0 ]]></Example> </Description> </ManSection> <ManSection> <Oper Name="Ealpha" Label="for a module element" Arg="V, v, ind"/> <Description> Here <A>V</A> is a module over a quantized enveloping algebra, <A>v</A> an element of it, and <A>ind</A> an index between 1 and the rank of the root system. The function returns the result of applying the <A>ind</A>-th Kashiwara operator <M>\widetilde{E}_{<A>ind</A>}</M> to <A>v</A>. Here the Kashiwara operators are different from the ones described in Section <Ref Sect="sec2.6"/>. We refer to <Cite Key="J96"/>, 9.2 for the definition of the operators used here. <Example><![CDATA[ gap> U:= QuantizedUEA( RootSystem("B",2) );; gap> V:= HighestWeightModule( U, [1,1] );; gap> v:= Falpha( V, Basis(V)[2], 2 ); (q^2)*F1*F4*v0+F2*v0 gap> Ealpha( V, v, 2 ); F1*v0 ]]></Example> </Description> </ManSection> <ManSection> <Attr Name="CrystalBasis" Arg="V"/> <Description> Here <A>V</A> is a finite-dimensional left module over a quantized enveloping algebra. This function returns the canonical, or crystal basis of V (see Section <Ref Sect="sec2.6"/>).<P/> This function only works for modules for which <Ref Oper="WeightsAndVectors"/> works. <Example><![CDATA[ gap> U:= QuantizedUEA( RootSystem( "B", 2 ) );; gap> V:= HighestWeightModule( U, [1,1] ); <16-dimensional left-module over QuantumUEA( <root system of type B 2>, Qpar = q )> gap> CrystalBasis( V ); Basis( <16-dimensional left-module over QuantumUEA( <root system of type B 2>, Qpar = q )>, [ 1*v0, F1*v0, F4*v0, F1*F4*v0, (q^2)*F1*F4*v0+F2*v0, F2*F4*v0, (q)*F2*F4*v0+F3*v0, (-q^-4)*F1*F2*v0, (-q^-1)*F1*F3*v0+(-q^-3)*F2^(2)*v0, (-q^-2)*F2^(2)*v0, F3*F4*v0, (-q^-4)*F2*F3*v0+(-q^-2)*F2^(2)*F4*v0, (-q^-2)*F2*F3*v0, (q^-4)*F2^(3)*v0, (-q^-1)*F3^(2)*v0, (q^-5)*F2^(2)*F3*v0 ] ) ]]></Example> </Description> </ManSection> <ManSection> <Attr Name="CrystalVectors" Arg="V"/> <Description> Here <A>V</A> is a finite-dimensional left module over a quantized enveloping algebra. Let <A>C</A> be the crystal basis of <A>V</A> (i.e., output by <Ref Attr="CrystalBasis"/>). This function returns a list of cosets of the basis elements of <A>C</A> modulo <M>qL</M>, where <M>L</M> is the <M>\mathbb{Z}[q]</M>-lattice spanned by <A>C</A>. <P/> The coset of a vector <A>v</A> is printed as <A><v></A>. <P/> The crystal vectors are used to construct the point set of the crystal graph of <A>V</A> (<Ref Func="CrystalGraph" Label="for a module"/>). <P/> This function only works for modules for which <Ref Oper="WeightsAndVectors"/> works. <Example><![CDATA[ gap> U:= QuantizedUEA( RootSystem( "B", 2 ) );; gap> V:= HighestWeightModule( U, [1,1] ); <16-dimensional left-module over QuantumUEA( <root system of type B 2>, Qpar = q )> gap> CrystalVectors( V ); [ <1*v0>, <F1*v0>, <F4*v0>, <F2*v0>, <F1*F4*v0>, <F3*v0>, <(-q^-4)*F1*F2*v0>, <F2*F4*v0>, <F1*F3*v0>, <F3*F4*v0>, <(-q^-1)*F1*F3*v0+(-q^-3)*F2^(2)*v0>, <(-q^-4)*F2*F3*v0+(-q^-2)*F2^(2)*F4*v0>, <F2^(2)*F4*v0>, <(q^-4)*F2^(3)*v0>, <(-q^-1)*F3^(2)*v0>, <(q^-5)*F2^(2)*F3*v0> ] ]]></Example> </Description> </ManSection> <ManSection> <Oper Name="Falpha" Label="for a crystal vector" Arg="v, ind"/> <Description> Here <A>v</A> is a crystal vector, i.e., an element of <A>CrystalVectors( V )</A>, where <A>V</A> is a left module over a quantized enveloping algebra. This function returns the result of applying the <A>ind</A>-th Kashiwara operator <M>\widetilde{F}_{\alpha_{<A>ind</A>}}</M> to v. The result is <A>fail</A> if <M>\widetilde{F}_{\alpha_{<A>ind</A>}}</M><A>(v)=0</A>. <Example><![CDATA[ gap> U:= QuantizedUEA( RootSystem( "B", 2 ) );; gap> V:= HighestWeightModule( U, [1,1] );; gap> c:=CrystalVectors( V );; gap> Falpha( c[2], 2 ); <F2*v0> gap> Falpha( c[3], 2 ); fail gap> Falpha( Falpha( Falpha( c[1], 1 ), 2 ), 1 ); fail gap> p:= DominantLSPath( RootSystem( "B", 2 ), [1,1] ); <LS path of shape [ 1, 1 ] ending in [ 1, 1 ] > gap> Falpha( Falpha( Falpha( p, 1 ), 2 ), 1 ); fail ]]></Example> The last part of this example is an illustration of the fact that the crystal graph of a highest-weight module can be obtained by the path method (see Section <Ref Sect="sec2.7"/>). </Description> </ManSection> <ManSection> <Oper Name="Ealpha" Label="for a crystal vector" Arg="v, ind"/> <Description> Here <A>v</A> is a crystal vector, i.e., an element of <A>CrystalVectors( V )</A>, where <A>V</A> is a left module over a quantized enveloping algebra. This function returns the result of applying the <A>ind</A>-th Kashiwara operator <M>\widetilde{E}_{\alpha_{<A>ind</A>}}</M> to v. The result is <A>fail</A> if <M>\widetilde{E}_{\alpha_{<A>ind</A>}}</M><A>(v)=0</A>. <Example><![CDATA[ gap> U:= QuantizedUEA( RootSystem( "B", 2 ) );; gap> V:= HighestWeightModule( U, [1,1] );; gap> c:=CrystalVectors( V );; gap> Ealpha( c[3], 1 ); fail gap> Ealpha( c[3], 2 ); <1*v0> ]]></Example> </Description> </ManSection> <ManSection> <Func Name="CrystalGraph" Label="for a module" Arg="V"/> <Description> Returns the crystal graph of the module <A>V</A>. The points of this graph are the cosets output by <Ref Attr="CrystalVectors"/>. The edges work in the same way as in <Ref Func="CrystalGraph" Label="for root system and weight"/>. <Example><![CDATA[ gap> U:= QuantizedUEA( RootSystem("A",2) );; gap> V1:= HighestWeightModule( U, [1,0] );; gap> V2:= HighestWeightModule( U, [0,1] );; gap> W:= TensorProductOfAlgebraModules( V1, V2 );; gap> CrystalGraph( W ); rec( edges := [ [ [ 1, 2 ], 1 ], [ [ 1, 3 ], 2 ], [ [ 2, 4 ], 2 ], [ [ 3, 5 ], 1 ], [ [ 4, 6 ], 2 ], [ [ 5, 7 ], 1 ], [ [ 6, 8 ], 1 ], [ [ 7, 8 ], 2 ] ], points := [ <1*(1*v0<x>1*v0)>, <1*(F1*v0<x>1*v0)>, <1*(1*v0<x>F3*v0)>, <1*(1*v0<x>F2*v0)+q^-1*(F2*v0<x>1*v0)>, <-q^-1*(1*v0<x>F2*v0)+q^-1*(F1*v0<x>F3*v0)>, <1*(F2*v0<x>F3*v0)>, <-q^-1*(F1*v0<x>F2*v0)>, <-q^-1*(F2*v0<x>F2*v0)>, <-q^-3*(1*v0<x>F2*v0)+-q^-1*(F1*v0<x>F3*v0)+1*(F2*v0<x>1*v0)> ] ) ]]></Example> </Description> </ManSection> </Section> <Section> <Heading> Universal enveloping algebras </Heading> Here we describe functions for connecting a quantized enveloping algebra to the corresponding universal enveloping algebra. <ManSection> <Attr Name="UEA" Arg="L"/> <Description> This function returns the universal enveloping algebra <A>u</A> of the semisimple Lie algebra <A>L</A>. The generators of <A>u</A> are the generators of a Kostant lattice in the universal enveloping algebra (these generators are obtained from <A>L</A> by <A>LatticeGeneratorsInUEA( L )</A>, see the &GAP; reference manual). <Example><![CDATA[ gap> L:= SimpleLieAlgebra( "B", 2, Rationals ); <Lie algebra of dimension 10 over Rationals> gap> u:= UEA( L ); <algebra over Rationals, with 10 generators> gap> g:= GeneratorsOfAlgebra( u ); [ y1, y2, y3, y4, x1, x2, x3, x4, ( h9/1 ), ( h10/1 ) ] ]]></Example> </Description> </ManSection> <ManSection> <Attr Name="UnderlyingLieAlgebra" Arg="u"/> <Description> For a universal enveloping algebra <A>u</A> constructed by <Ref Attr="UEA"/>, this returns the corresponding semisimple Lie algebra <Example><![CDATA[ gap> L:= SimpleLieAlgebra( "B", 2, Rationals );; gap> u:= UEA( L );; gap> UnderlyingLieAlgebra( u ); <Lie algebra of dimension 10 over Rationals> ]]></Example> </Description> </ManSection> <ManSection> <Oper Name="HighestWeightModule" Label="for a universal env. alg" Arg="u, hw"/> <Description> For a universal enveloping algebra <A>u</A> constructed by <Ref Attr="UEA"/>, this returns the irreducible highest weight module over <A>u</A> with highest weight <A>hw</A>, which must be dominant. This module is the same as the corresponding highest weight module over the semisimple Lie algebra, but in this case the enveloping algebra <A>u</A> acts. <Example><![CDATA[ gap> L:= SimpleLieAlgebra( "B", 2, Rationals );; gap> u:= UEA( L );; gap> HighestWeightModule( u, [2,3] ); <140-dimensional left-module over <algebra over Rationals, with 10 generators>> ]]></Example> </Description> </ManSection> <ManSection> <Attr Name="QUEAToUEAMap" Arg="L"/> <Description> Here <A>L</A> is a semisimple Lie algebra. Set <A>u := UEA( L )</A>, and <A>U := QuantizedUEA( L )</A> (so <A>u, U</A> are the universal enveloping algebra, and <Q>generic</Q> quantized enveloping algebra of <A>L</A> respectively). Then <A>QUEAToUEAMap( L )</A> returns the algebra homomorphism from <A>U</A> to <A>u</A> obtained by mapping <A>q</A> to <A>1</A>, a generator <A>Fi</A>, corresponding to a simple root to the generator <A>yi</A> (corresponding to the same simple root), and likewise for <A>Ei</A> and <A>xi</A>. This means that <A>Ki</A> is mapped to one, and <A>[ Ki : s ]</A> to <A>hi</A> choose <A>s</A>. <P/> The canonical basis of <A>U</A> is mapped to the canonical basis of <A>u</A>. <Example><![CDATA[ gap> L:= SimpleLieAlgebra( "B", 2, Rationals );; gap> f:= QUEAToUEAMap( L ); <mapping: QuantumUEA( <root system of rank 2>, Qpar = q ) -> Algebra( Rationals, [ y1, y2, y3, y4, x1, x2, x3, x4, ( h9/1 ), ( h10/1 ) ] ) > gap> U:= Source( f ); QuantumUEA( <root system of rank 2>, Qpar = q ) gap> u:= Range( f ); <algebra over Rationals, with 10 generators> gap> B:= CanonicalBasis( U );; gap> p:= PBWElements( B, [1,2] ); [ F1*F4^(2), (q^3+q)*F1*F4^(2)+F2*F4, (q^4)*F1*F4^(2)+(q)*F2*F4+F3 ] gap> pu:= List( p, x -> Image( f, x ) ); [ y1*y2^(2), 2*y1*y2^(2)+y2*y3-2*y4, y1*y2^(2)+y2*y3-1*y4 ] gap> V:= HighestWeightModule( u, [2,1] ); <40-dimensional left-module over <algebra over Rationals, with 10 generators>> gap> List( pu, x -> x^Basis(V)[1] ); [ 0*v0, y2*y3*v0-2*y4*v0, y2*y3*v0-1*y4*v0 ] gap> # Which gives us a piece of the canonical basis of V. ]]></Example> </Description> </ManSection> </Section> </Chapter> </Body> <Bibliography Databases="quagroup" /> <TheIndex/> </Book>
¤ Dauer der Verarbeitung: 0.71 Sekunden (vorverarbeitet am 2026-04-25) ¤*© Formatika GbR, Deutschland |
|
||||||||||||||
2026-05-26