In this chapter we describe the functionality provided by QuaGroup.
‣ QuantumField | ( global variable ) |
This is the field Q(q) of rational functions in q, over Q.
gap> QuantumField; QuantumField
‣ _q | ( global variable ) |
This is an indeterminate; QuantumField is the field of rational functions in this indeterminate. The identifier _q is fixed once the package QuaGroup is loaded. The symbol _q is chosen (instead of q) in order to avoid potential name clashes. We note that _q is printed as q.
gap> _q; q gap> _q in QuantumField; true
‣ GaussNumber( n, par ) | ( operation ) |
This function computes for the integer n the Gaussian integer [n]_v=par (cf. Section 2.1).
gap> GaussNumber( 4, _q ); q^3+q+q^-1+q^-3
‣ GaussianFactorial( n, par ) | ( operation ) |
This function computes for the integer n the Gaussian factorial [n]!_v=par.
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
‣ GaussianBinomial( n, k, par ) | ( operation ) |
This function computes for two integers n and k the Gaussian binomial n choose k, where the parameter v is replaced by par.
gap> GaussianBinomial( 5, 2, _q^2 ); q^12+q^8+2*q^4+2+2*q^-4+q^-8+q^-12
In this section we describe some functions for dealing with root systems. These functions supplement the ones already present in the GAP library.
‣ RootSystem( type, rank ) | ( operation ) |
‣ RootSystem( list ) | ( operation ) |
Here type is a capital letter between "A" and "G", and rank is a positive integer (≥ 1 if type="A", ≥ 2 if type="B", "C", ≥ 4 if type="D", 6,7,8 if type="E", 4 if type="F", and 2 if type="G"). This function returns the root system of type type and rank rank. In the second form list is a list of types and ranks, e.g., [ "B", 2, "F", 4, "D", 7 ].
The root system constructed by this function comes with he attributes PositiveRoots, NegativeRoots, SimpleSystem, CartanMatrix, BilinearFormMat. Here the attribute SimpleSystem contains a set of simple roots, written as unit vectors. PositiveRoots is a list of the positive roots, written as linear combinations of the simple roots, and likewise for NegativeRoots. CartanMatrix( R ) is the Cartan matrix of the root system R, where the entry on position ( i, j ) is given by ⟨ α_i, α_j^∨⟩ where α_i is the i-th simple root. BilinearFormMat( R ) is the matrix of the bilinear form, where the entry on position ( i, j ) is given by ( α_i, α_j ) (see Section 2.2).
WeylGroup( R ) returns the Weyl group of the root system R. We refer to the GAP reference manual for an overview of the functions for Weyl groups in the GAP library. We mention the functions ConjugateDominantWeight( W, wt ) (returns the dominant weight in the W-orbit of the weight wt), and WeylOrbitIterator( W, wt ) (returns an iterator for the W-orbit containing the weight wt). 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).
Also we mention the function PositiveRootsAsWeights( R ) that returns the positive roots of R written as weights, i.e., as linear combinations of the fundamental weights.
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 ] ]
‣ BilinearFormMatNF( R ) | ( attribute ) |
This is the matrix of the "normalized" bilinear form. This means that all diagonal entries are even, and 2 is the minimum value occurring on the diagonal. If R is a root system constructed by RootSystem (3.3-1), then this is equal to BilinearFormMat( R ).
‣ PositiveRootsNF( R ) | ( attribute ) |
This is the list of positive roots of the root system R, written as linear combinations of the simple roots. This means that the simple roots are unit vectors. If R is a root system constructed by RootSystem (3.3-1), then this is equal to PositiveRoots( R ).
One of the reasons for writing the positive roots like this is the following. Let a, b be two elements of PositiveRootsNF( R ), and let B be the matrix of the bilinear form. Then a*( B*b ) is the result of applying the bilinear form to a, b.
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.
‣ SimpleSystemNF( R ) | ( attribute ) |
This is the list of simple roots of R, written as unit vectors (this means that they are elements of PositiveRootsNF( R )). If R is a root system constructed by RootSystem (3.3-1), then this is equal to SimpleSystem( R ).
‣ PositiveRootsInConvexOrder( R ) | ( attribute ) |
This function returns the positive roots of the root system R, in the "convex" order. Let w_0=s_1⋯ s_t be a reduced expression of the longest element in the Weyl group. Then the k-th element of the list returned by this function is s_1⋯ s_k-1(α_k). (Where the reduced expression used is the one returned by LongestWeylWord( R ).) If α, β and α+β are positive roots, then α+β occurs between α and β (whence the name convex order).
In the output all roots are written in "normal form", i.e., as elements of PositiveRootsNF( R ).
gap> R:= RootSystem( "G", 2 );; gap> PositiveRootsInConvexOrder( R ); [ [ 1, 0 ], [ 3, 1 ], [ 2, 1 ], [ 3, 2 ], [ 1, 1 ], [ 0, 1 ] ]
‣ SimpleRootsAsWeights( R ) | ( attribute ) |
Returns the simple roots of the root system R, written as linear combinations of the fundamental weights.
gap> R:= RootSystem( "A", 2 );; gap> SimpleRootsAsWeights( R ); [ [ 2, -1 ], [ -1, 2 ] ]
Now we describe a few functions that deal with reduced words in the Weyl group of the root system R. These words are represented as lists of positive integers i, denoting the i-th simple reflection (which corresponds to the i-th element of SimpleSystem( R )). For example [ 3, 2, 1, 3, 1 ] represents the expression s_3 s_2 s_1 s_3 s_1.
‣ ApplyWeylElement( W, wt, wd ) | ( operation ) |
Here wd is a (not necessarily reduced) word in the Weyl group W, and wt is a weight (written as integral linear combination of the simple weights). This function returns the result of applying wd to wt. For example, if wt=μ, and wd = [ 1, 2 ] then this function returns s_1s_2(μ) (where s_i is the simple reflection corresponding to the i-th simple root).
gap> W:= WeylGroup( RootSystem( "G", 2 ) ) ;; gap> ApplyWeylElement( W, [ -3, 7 ], [ 1, 1, 2, 1, 2 ] ); [ 15, -11 ]
‣ LengthOfWeylWord( W, wd ) | ( operation ) |
Here wd is a word in the Weyl group W. This function returns the length of that word.
gap> W:= WeylGroup( RootSystem( "F", 4 ) ) ; <matrix group with 4 generators> gap> LengthOfWeylWord( W, [ 1, 3, 2, 4, 2 ] ); 3
‣ LongestWeylWord( R ) | ( attribute ) |
Here R is a root system. LongestWeylWord( R ) returns the longest word in the Weyl group of R.
If this function is called for a root system R, 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 SetLongestWeylWord( R, wd ), where wd is a reduced expression of the longest element in the Weyl group. Note that you will have to do this before calling LongestWeylWord, or any function that may call LongestWeylWord (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 LengthOfWeylWord (3.4-2)).
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).
gap> R:= RootSystem( "G", 2 );; gap> LongestWeylWord( R ); [ 1, 2, 1, 2, 1, 2 ]
‣ ReducedWordIterator( W, wd ) | ( operation ) |
Here W is a Weyl group, and wd a reduced word. This function returns an iterator for the set of reduced words that represent the same element as wd. The elements are output in ascending lexicographical order.
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
So there are 2144892 reduced expressions for the longest element in the Weyl group of type F_4.
‣ ExchangeElement( W, wd, ind ) | ( operation ) |
Here W is a Weyl group, and wd is a reduced word in W, and ind is an index between 1 and the rank of the root system. Let v denote the word obtained from wd by adding ind at the end. This function assumes that the length of v is one less than the length of wd, and returns a reduced expression for v that is obtained from wd by deleting one entry. Nothing is guaranteed of the output if the length of v is bigger than the length of wd.
gap> R:= RootSystem( "G", 2 );; gap> wd:= LongestWeylWord( R );; gap> ExchangeElement( WeylGroup(R), wd, 1 ); [ 2, 1, 2, 1, 2 ]
‣ GetBraidRelations( W, wd1, wd2 ) | ( operation ) |
Here W is a Weyl group, and wd1, wd2 are two reduced words representing the same element in W. This function returns a list of braid relations that can be applied to wd1 to obtain wd2. 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 wd be the word [ 1, 2, 1, 3, 2, 1 ] and let r = [ 3, 3, 4, 1 ] be a relation. Then the result of applying r to wd is [ 1, 2, 3, 1, 2, 1] (i.e., on the third position we put a 3, and on the fourth position a 1).
We note that the function does not check first whether wd1 and wd2 represent the same element in W. If this is not the case, then an error will occur during the execution of the function, or it will produce wrong output.
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 ] ]
‣ LongWords( R ) | ( attribute ) |
For a root system R this returns a list of triples (of length equal to the rank of R). Let t be the k-th triple occurring in this list. The first element of t is an expression for the longest element of the Weyl group, starting with k. The second element is a list of braid relations, moving this expression to the value of LongestWeylWord( R ). The third element is a list of braid relations performing the reverse transformation.
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 ] ] ]
In QuaGroup we deal with two types of quantized enveloping algebra. First there are the quantized enveloping algebras defined over the field QuantumField (3.1-1). We say that these algebras are "generic" quantized enveloping algebras, in QuaGroup they have the category IsGenericQUEA. Secondly, we deal with the quantized enveloping algebras that are defined over a different field.
‣ QuantizedUEA( R ) | ( attribute ) |
‣ QuantizedUEA( R, F, v ) | ( operation ) |
‣ QuantizedUEA( L ) | ( attribute ) |
‣ QuantizedUEA( L, F, v ) | ( operation ) |
In the first two forms R is a root system. With only R as input, the corresponding generic quantized enveloping algebra is constructed. It is stored as an attribute of R (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 RootSystem.
The attribute GeneratorsOfAlgebra contains the generators of a PBW-type basis (see Section 2.4), that are constructed relative to the reduced expression for the longest element in the Weyl group that is contained in LongestWeylWord( R ). We refer to ObjByExtRep (3.5-2) for a description of the construction of elements of a quantized enveloping algebra.
The call QuantizedUEA( R, F, v ) returns the quantized universal enveloping algebra with quantum parameter v, which must lie in the field F. In this case the elements of GeneratorsOfAlgebra are the images of the generators of the corresponding generic quantized enveloping algebra. This means that if v is a root of unity, then the generators will not generate the whole algebra, but rather a finite dimensional subalgebra (as for instance E_i^k=0 for k large enough). It is possible to construct elements that do not lie in this finite dimensional subalgebra using ObjByExtRep (3.5-2).
In the last two cases L must be a semisimple Lie algebra. The two calls are short for QuantizedUEA( RootSystem( L ) ) and QuantizedUEA( RootSystem( L ), F, v ) respectively.
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 ]
As can be seen in the example, every element of U is written as a linear combination of monomials in the PBW-generators; the generators of U^- come first, then the generators of U^0, and finally the generators of U^+.
‣ ObjByExtRep( fam, list ) | ( operation ) |
Here fam is the elements family of a quantized enveloping algebra U. Secondly, list is a list describing an element of U. We explain how this description works. First we describe an indexing system for the generators of U. Let R be the root system of U. Let t be the number of positive roots, and rank the rank of the root system. Then the generators of U are Fk, Ki (and its inverse), Ek, for k=1...t, i=1..rank. (See Section 2.4; for the construction of the Fk, Ek, the value of LongestWeylWord( R ) is used.) Now the index of Fk is k, and the index of Ek is t+rank+k. Furthermore, elements of the algebra generated by the Ki, and its inverse, are written as linear combinations of products of "binomials", as in Section 2.5. The element
K_i^{d}\begin{bmatrix} K_{i} \\ s \end{bmatrix}
(where d=0,1), is indexed as [ t+i, d ] (what happens to the s is described later). So an index is either an integer, or a list of two integers.
A monomial is a list of indices, each followed by an exponent. First come the indices of the Fk, (1..t), then come the lists of the form [ t+i, d ], and finally the indices of the Ek. Each index is followed by an exponent. An index of the form [ t+i, d ] is followed by the s in the above formula.
The second argument of ObjByExtRep is a list of monomials followed by coefficients. This function returns the element of U described by this list.
Finally we remark that the element
K_i^{d}\begin{bmatrix} K_{i} \\ s \end{bmatrix}
is printed as Ki[ Ki ; s ] if d=1, and as [ Ki ; s ] if d=0.
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)
‣ ExtRepOfObj( elm ) | ( operation ) |
For the element elm of a quantized enveloping algebra, this function returns the list that defines elm (see ObjByExtRep (3.5-2)).
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 ]
‣ QuantumParameter( U ) | ( attribute ) |
Returns the quantum parameter used in the definition of U.
gap> R:= RootSystem("A",2);; gap> U0:= QuantizedUEA( R, CF(3), E(3) );; gap> QuantumParameter( U0 ); E(3)
‣ CanonicalMapping( U ) | ( attribute ) |
Here U is a quantized enveloping algebra. Let U0 denote the corresponding "generic" quantized enveloping algebra. This function returns the mapping U0 --> U obtained by mapping q (which is the quantum parameter of U0) to the quantum parameter of U.
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 ]
‣ WriteQEAToFile( U, file ) | ( operation ) |
Here U is a quantized enveloping algebra, and file is a string containing the name of a file. This function writes some data to file, that allows ReadQEAFromFile (3.5-7) to recover it.
gap> U:= QuantizedUEA( RootSystem("A",3) );; gap> WriteQEAToFile( U, "A3" );
‣ ReadQEAFromFile( file ) | ( operation ) |
Here file is a string containing the name of a file, to which a quantized enveloping algebra has been written by WriteQEAToFile (3.5-6). This function recovers the quantized enveloping algebra.
gap> U:= QuantizedUEA( RootSystem("A",3) );; gap> WriteQEAToFile( U, "A3" ); gap> U0:= ReadQEAFromFile( "A3" ); QuantumUEA( <root system of type A3>, Qpar = q )
Here we describe functions for creating homomorphisms and (anti)-automorphisms of a quantized enveloping algebra.
‣ QEAHomomorphism( U, A, list ) | ( operation ) |
Here U is a generic quantized enveloping algebra (i.e., with quantum parameter _q), A is an algebra with one over QuantumField, and list is a list of 4*rank elements of A (where rank is the rank of the root system of U). On the first rank positions there are the images of the F_α (where the α are simple roots, listed in the order in which they occur in SimpleSystem( R )). On the positions rank+1...2*rank are the images of the K_α. On the positions 2*rank+1...3*rank are the images of the K_α^-1, and finally on the positions 3*rank+1...4*rank occur the images of the E_α.
This function returns the homomorphism U -> 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.
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
‣ QEAAutomorphism( U, list ) | ( operation ) |
‣ QEAAutomorphism( U, f ) | ( operation ) |
In the first form U is a generic quantized enveloping algebra (i.e., with quantum parameter _q), and list is a list of 4*rank elements of U (where rank is the rank of the corresponding root system). On the first rank positions there are the images of the F_α (where the α are simple roots, listed in the order in which they occur in SimpleSystem( R )). On the positions rank+1...2*rank are the images of the K_α. On the positions 2*rank+1...3*rank are the images of the K_α^-1, and finally on the positions 3*rank+1...4*rank occur the images of the E_α.
In the second form U is a non-generic quantized enveloping algebra, and f is an automorphism of the corresponding generic quantized enveloping algebra. The corresponding automorphism of U is constructed. In this case f must not be the bar-automorphism of the corresponding generic quantized enveloping algebra (cf. BarAutomorphism (3.6-6)), as this automorphism doesn't work in the non-generic case.
The image of an element x under an automorphism f is computed by Image( f, x ). 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 QEAAutomorphism( U, list ), where list is a list of pre-images).
Below we construct the automorphism ω (cf. Section 2.2) of the quantized enveloping of type A_3, when the quantum parameter is _q, and when the quantum parameter is a fifth root of unity.
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
‣ QEAAntiAutomorphism( U, list ) | ( operation ) |
‣ QEAAntiAutomorphism( U, f ) | ( operation ) |
These are functions for constructing anti-automorphisms of quantized enveloping algebras. The same comments apply as for QEAAutomorphism (3.6-2).
‣ AutomorphismOmega( U ) | ( attribute ) |
This is the automorphism ω (cf. Section 2.2).
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) )>
‣ AntiAutomorphismTau( ) | ( attribute ) |
This is the anti-automorphism τ (cf. Section 2.2).
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) )>
‣ BarAutomorphism( U ) | ( attribute ) |
This is the automorphism bar~ defined in Section 2.2 Here U must be a generic quantized enveloping algebra.
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
‣ AutomorphismTalpha( U, ind ) | ( operation ) |
This is the automorphism T_α (cf. Section 2.4), where α is the ind-th simple root.
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
‣ DiagramAutomorphism( U, perm ) | ( operation ) |
This is the automorphism of U induced by a diagram automorphism of the underlying root system. The diagram automorphism is represented by the permutation perm, which is the permutation of the simple roots performed by the diagram automorphism.
In the example below we construct the diagram automorphism of the root system of type A_3, which is represented by the permutation (1,3).
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
3.6-9 \*‣ \*( f, h ) | ( operation ) |
We can compose automorphisms and anti-automorphisms using the infix * 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.
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
According to [Jan96], 8.14(10), τ ∘ T_α ∘ τ is the inverse of T_α.
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 3.8) for some functions for creating modules.
‣ TensorPower( U, d ) | ( operation ) |
Here U is a quantized universal enveloping algebra, and d a non-negative integer. This function returns the associative algebra with underlying vector space the d-fold tensor product of U with itself. The product is defined component wise.
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)
‣ UseTwistedHopfStructure( U, f, finv ) | ( operation ) |
Here U is a quantized enveloping algebra, and f, finv two (anti-) automorphisms of U, where finv is the inverse of f. After calling this function the Hopf structure on U is used that is obtained from the "normal" Hopf structure (see Section 2.3) by twisting it with f.
A call to this function sets the attribute HopfStructureTwist, which is the list [ f, finv ].
gap> U:= QuantizedUEA( RootSystem("A",2), CF(5), E(5) );; gap> t:= AntiAutomorphismTau( U );; gap> UseTwistedHopfStructure( U, t, t );
‣ ComultiplicationMap( U, d ) | ( operation ) |
This is a homomorphism from the quantized enveloping algebra U to the d-fold tensor power of U with itself. It is obtained by a repeated application of the comultiplication of U. So for d=2 we get the comultiplication of U.
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)
‣ AntipodeMap( U ) | ( attribute ) |
This is the antipode map of the quantized enveloping algebra U, which is constructed as an anti-automorphism of U.
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) )>
‣ CounitMap( U ) | ( attribute ) |
This is the counit map of the quantized enveloping algebra U, which is constructed as a function from U to the ground field.
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
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 MatrixOfAction, DirectSumOfAlgebraModules. The action of an element of the algebra on an element of the module is calculated by the infix operator ^.
‣ HighestWeightModule( U, wt ) | ( operation ) |
Here U is a quantized universal enveloping algebra, and wt 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 wt over U. If U is generic then this is the unique irreducible highest-weight module over U. Otherwise it is the Weyl module, cf. Section 2.5. In this last case the module is not necessarily irreducible.
Let V denote the module returned by this function. The first basis element of the attribute Basis( V ) is a highest-weight vector; it is written as 1*v0. Other basis elements are written as, for example, F2*F9*v0, which means that this vector is the result of letting the PBW-monomial F2*F9 act on the highest-weight vector.
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
‣ IrreducibleQuotient( V ) | ( attribute ) |
Here V is a highest-weight module over a non-generic quantized enveloping algebra. This function returns the quotient of V by the maximal submodule not containing the highest weight vector. This is not necessarily equal to V if the quantum parameter is a root of 1.
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) )>
‣ HWModuleByTensorProduct( U, wt ) | ( operation ) |
Here U must be a generic quantized enveloping algebra, and wt a dominant weight. This function returns the irreducible highest-weight module with highest weight wt. The algorithm uses tensor products (whence the name). On some inputs this algorithm is faster than the one use for HighestWeightModule (3.8-1), on some inputs it is slower. I do not know any good heuristics.
The basis supplied with the module returned is the canonical basis.
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.)
‣ DIYModule( U, V, acts ) | ( operation ) |
Here U is a generic quantized enveloping algebra, and V is a vector space over the field QuantumField. U acts on V and the action is described by the data in the list acts. acts is a list of lists, of length 4*l, where l is the rank of the root system. acts describes the actions of the generators [F_1,...,F_l,K_1,...,K_l,K_1^-1,...,K_l^-1, E_1,...,E_l ]. (Here F_k is the generator F_α_k, where α_k is the k-th simple root, and likewise for E_k.) The action of each generator is described by a list of length dim V, giving the images of the basis elements of V. If an image is zero then it may be omitted: in that case there is a "hole" in the list. This function returns the U-module defined by the input.
Let R be a root system of type A_1, and U the corresponding quantized enveloping algebra (generated by F, K, K^-1, E). In the example below we construct the 2-dimensional U-module with basis vectors v_1,v_2, and U-action given by Fv_1 = v_2, Fv_2=0, Kv_1 = qv_1, Kv_2=q^-1v_2, Ev_1=0, Ev_2=v_1.
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 )>
‣ TensorProductOfAlgebraModules( V, W ) | ( operation ) |
‣ TensorProductOfAlgebraModules( V, W ) | ( operation ) |
Here V and W are two modules over the same quantized enveloping algebra U. This function constructs the tensor product of V and W (as a U-module). For this the comultiplication map of U is used (see ComultiplicationMap (3.7-3)).
In the second form list is a list of U-modules. In that case the iterated tensor product is constructed.
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 )>
‣ HWModuleByGenerator( V, v, hw ) | ( operation ) |
Here V is a module over a generic quantized enveloping algebra U, v is a highest-weight vector (i.e., all E_αv=0), of weight hw, which must be dominant. This function returns a highest-weight module over U isomorphic to the submodule of V generated by v.
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 )>
‣ InducedQEAModule( U, V ) | ( operation ) |
Here U is a non-generic quantized enveloping algebra, and V a module over the corresponding generic quantized enveloping algebra. This function returns the U-module obtained from V by setting _q equal to the quantum parameter of U.
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] );
‣ GenericModule( W ) | ( attribute ) |
For an induced module (see InducedQEAModule (3.8-7)) this function returns the corresponding module over the generic quantized enveloping algebra.
‣ CanonicalMapping( W ) | ( attribute ) |
Here W is an induced module. Let V be the corresponding generic module (GenericModule (3.8-8)). This function returns the map V --> W, that sets _q equal to the quantum parameter of the acting algebra of W.
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
‣ U2Module( U, hw ) | ( operation ) |
Here U must be a quantized enveloping algebra of type A_2. This function returns the highest-weight module over U of highest-weight hw (which must be dominant). This function is generally a lot faster than HighestWeightModule (3.8-1).
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 )>
‣ MinusculeModule( U, hw ) | ( operation ) |
Here U must be a generic quantized enveloping algebra, and hw a minuscule dominant weight. This function returns the highest-weight module over U of highest-weight hw. This function is generally somewhat faster than HighestWeightModule (3.8-1).
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 )>
‣ DualAlgebraModule( V ) | ( attribute ) |
Here V is a finite-dimensional left module over a quantized enveloping algebra U. This function returns the dual space of V as an algebra module. For this the antipode map of U is used (see AntipodeMap (3.7-4)).
Let M denote the module returned by this function. Then M has as basis the dual basis with respect to Basis( V ). An element of this basis is printed as F@v, where v is an element of Basis( V ). This is the function which takes the value 1 on v and 0 on all other basis elements. A general element of M is a linear combination of these basis elements.
The elements of M can be viewed as functions which take arguments. However, internally the elements of M are represented as wrapped up functions. The function corresponding to an element m of M is obtained by ExtRepOfObj( m ) (the result of which is printed in the same way as m, but is not equal to it).
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 ]
‣ TrivialAlgebraModule( U ) | ( attribute ) |
Returns the trivial module over the quantized enveloping algebra U. For this the counit map of U is used.
gap> U:= QuantizedUEA( RootSystem("A",2) );; gap> V:= TrivialAlgebraModule( U ); <left-module over QuantumUEA( <root system of type A2>, Qpar = q )>
‣ WeightsAndVectors( V ) | ( operation ) |
Here V is a left module over a quantized enveloping algebra. WeightsAndVectors( V ) is a list of two lists; the first of these is a list of the weights of V, the second a list of corresponding weight vectors. These are again grouped in lists: if the multiplicity of a weight is m, then there are m weight vectors, forming a basis of the corresponding weight space.
Modules constructed by HighestWeightModule (3.8-1) come with this attribute set. There is a method installed for computing WeightsAndVectors( V ), for modules V over a generic quantized enveloping algebra, such that all basis vectors (i.e., all elements of Basis( V )) are weight vectors.
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 ] ] ]
‣ HighestWeightsAndVectors( V ) | ( attribute ) |
Is analogous to WeightsAndVectors (3.8-14); now only the highest weights are listed along with the corresponding highest-weight vectors.
There is a method installed for this using WeightsAndVectors (3.8-14); which means that it works if and only if WeightsAndVectors( V ) works.
gap> U:= QuantizedUEA( RootSystem( [ "A", 2 ] ) );; gap> V:= HighestWeightModule( U, [ 1, 1 ] );; gap> HighestWeightsAndVectors( V ); [ [ [ 1, 1 ] ], [ [ 1*v0 ] ] ]
‣ RMatrix( V ) | ( attribute ) |
Here V is a module over the a quantized enveloping algebra U. This function returns the matrix of a linear map θ : V⊗ V -> V⊗ V that is a solution to the quantum Yang-Baxter equation. We have that θ∘ P is an isomorphism of U-modules, where P :V⊗ V-> V⊗ V is the linear map such that P(v⊗ w)=w⊗ v. For more details we refer to [Jan96], Chapter 7.
This function works for modules for which WeightsAndVectors (3.8-14) works.
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 ] ]
‣ IsomorphismOfTensorModules( V, W ) | ( operation ) |
Here V, W are two modules over the same quantized enveloping algebra U. This function returns a linear map θ : V⊗ W-> W⊗ V that is an isomorphism of U-modules.
This function is only guaranteed to work correctly if the Hopf algebra structure is non-twisted (see UseTwistedHopfStructure (3.7-2)).
This function works for modules for which WeightsAndVectors (3.8-14) works.
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)
‣ WriteModuleToFile( V, file ) | ( operation ) |
Here V is a module over a quantized enveloping algebra, and file is a string containing the name of a file. This function writes some data to file, that allows ReadModuleFromFile (3.8-19) to recover it.
We remark that this function currently is only implemented for generic quantized enveloping algebras.
‣ ReadModuleFromFile( file ) | ( operation ) |
Here file is a string containing the name of a file, to which a module over a quantized enveloping algebra has been written by WriteModuleToFile (3.8-18). 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 LeftActingAlgebra( V ).
We remark that this function currently is only implemented for generic quantized enveloping algebras.
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 )>
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 2.7).
‣ DominantLSPath( R, wt ) | ( operation ) |
Here R is a root system, and wt a dominant weight in the weight lattice of R. This function returns the LS-path that is the line from the origin to wt.
gap> R:= RootSystem( "G", 2 );; gap> DominantLSPath( R, [1,3] ); <LS path of shape [ 1, 3 ] ending in [ 1, 3 ] >
‣ Falpha( path, ind ) | ( operation ) |
Is the result of applying the path operator f_α_ind} to the LS-path path (where α_ind is the ind-th simple root).
The result is fail if f_α_ind}(path)=0.
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
‣ Ealpha( path, ind ) | ( operation ) |
Is the result of applying the path operator e_α_ind} to the LS-path path (where α_ind is the ind-th simple root).
The result is fail if e_α_ind}(path)=0.
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 ] >
‣ LSSequence( path ) | ( attribute ) |
returns the two sequences (of weights and rational numbers) that define the LS-path path.
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 ] ]
‣ WeylWord( path ) | ( attribute ) |
Here path is an LS-path in the orbit (under the root operators) of a dominant LS-path ending in the dominant weight λ. This means that the first direction of path is of the form w(λ) for some w in the Weyl group. This function returns a list [i_1,..., i_m ] such that w=s_i_1⋯ s_i_m.
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 ]
‣ EndWeight( path ) | ( attribute ) |
Here path is an LS-path; this function returns the weight that is the endpoint of path
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 ]
‣ CrystalGraph( R, wt ) | ( function ) |
This function returns a record describing the crystal graph of the highest-weight module with highest weight wt, over the quantized enveloping algebra corresponding to R. It is computed using the path model. Therefore the points in the graph are LS-paths.
Denote the output by r; then r.points is the list of points of the graph. Furthermore, r.edges is a list of edges of the graph; this is a list of elements of the form [ [ i, j ], u ]. This means that there is an arrow from point i (i.e., the point on position i in r.points) to point j, with label u.
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 ] > ] )
Here we describe functions for computing the canonical basis of the negative part of a quantized enveloping algebra, and of a module.
‣ Falpha( x, ind ) | ( operation ) |
Here x is a PBW-monomial in U^- (i.e., a monomial in the F_α, where α runs over the positive roots). This function returns the result of applying the ind-th Kashiwara operator widetildeF_α_ind} to x (cf. Section 2.6).
gap> U:= QuantizedUEA( RootSystem( "F", 4 ) );; gap> x:= One( U ); 1 gap> Falpha( Falpha( x, 3 ), 2 ); F3*F9
‣ Ealpha( x, ind ) | ( operation ) |
Here x is a PBW-monomial in U^- (i.e., a monomial in the F_α, where α runs over the positive roots). This function returns the result of applying the ind-th Kashiwara operator widetildeE_α_ind} to x (cf. Section 2.6). The result is fail if widetildeE_α_ind}(x)=0.
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
‣ CanonicalBasis( U ) | ( attribute ) |
Is the canonical basis of the quantized universal enveloping algebra U. When this is constructed nothing is computed. By using PBWElements (3.10-4), MonomialElements (3.10-5), Strings (3.10-6) information about elements of the canonical basis can be obtained.
gap> U:= QuantizedUEA( RootSystem( "F", 4 ) );; gap> B:= CanonicalBasis( U ); <canonical basis of QuantumUEA( <root system of type F4>, Qpar = q ) >
‣ PBWElements( B, rt ) | ( operation ) |
Here B is the canonical basis of a quantized uea, and rt a list of non-negative integers representing an element of the root lattice (e.g., if the simple roots are α, β and rt = [ 3, 2 ], then rt represents 3α+2β).
It is possible to add the option lowrank, as follows PBWElements( B, rt :lowrank ). 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.
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 ]
‣ MonomialElements( B, rt ) | ( operation ) |
This does the same as PBWElements (3.10-4), except that the elements are written as linear combinations of monomials in the generators F_α, where α runs through the simple roots.
We remark that this information is also computed "behind the scenes" when calling PBWElements( B, rt ). However, it is not computed if the option lowrank is present in the call to PBWElements.
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 ]
‣ Strings( B, rt ) | ( operation ) |
Here B, rt are the same as in PBWElements (3.10-4). This returns the list of strings corresponding to the elements of B of weight rt (cf. Section 2.6). For example, if on the k-th position of the list returned by this function we have [ 1, 2, 2, 3 ], then the principal monomial of the k-th element of PBWElements( B, rt ) is widetildeF_1^2widetildeF_2^3(1) (where widetildeF_i is the i-th Kashiwara operator).
We remark that this information is also computed "behind the scenes" when calling PBWElements( B, rt ). However, it is not computed if the option lowrank is present in the call to PBWElements.
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
‣ PrincipalMonomial( u ) | ( operation ) |
Here u is an element of the output of PBWElements (3.10-4). This function returns the unique monomial of u that has coefficient 1.
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)
‣ StringMonomial( u ) | ( operation ) |
Here u is a monomial in the negative part of a quantized enveloping algebra, e.g., as output by PrincipalMonomial (3.10-7). This function computes the corresponding "string" (see Section 2.6). The strings are output in the same way as in 3.10-6.
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
‣ Falpha( V, v, ind ) | ( operation ) |
Here V is a module over a quantized enveloping algebra, v an element of it, and ind an index between 1 and the rank of the root system. The function returns the result of applying the ind-th Kashiwara operator widetildeF_ind to v. Here the Kashiwara operators are different from the ones described in Section 2.6. We refer to [Jan96], 9.2 for the definition of the operators used here.
gap> U:= QuantizedUEA( RootSystem("B",2) );; gap> V:= HighestWeightModule( U, [1,1] );; gap> Falpha( V, Basis(V)[1], 1 ); F1*v0
‣ Ealpha( V, v, ind ) | ( operation ) |
Here V is a module over a quantized enveloping algebra, v an element of it, and ind an index between 1 and the rank of the root system. The function returns the result of applying the ind-th Kashiwara operator widetildeE_ind to v. Here the Kashiwara operators are different from the ones described in Section 2.6. We refer to [Jan96], 9.2 for the definition of the operators used here.
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
‣ CrystalBasis( V ) | ( attribute ) |
Here V is a finite-dimensional left module over a quantized enveloping algebra. This function returns the canonical, or crystal basis of V (see Section 2.6).
This function only works for modules for which WeightsAndVectors (3.8-14) works.
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 ] )
‣ CrystalVectors( V ) | ( attribute ) |
Here V is a finite-dimensional left module over a quantized enveloping algebra. Let C be the crystal basis of V (i.e., output by CrystalBasis (3.10-11)). This function returns a list of cosets of the basis elements of C modulo qL, where L is the Z[q]-lattice spanned by C.
The coset of a vector v is printed as <v>.
The crystal vectors are used to construct the point set of the crystal graph of V (CrystalGraph (3.10-15)).
This function only works for modules for which WeightsAndVectors (3.8-14) works.
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> ]
‣ Falpha( v, ind ) | ( operation ) |
Here v is a crystal vector, i.e., an element of CrystalVectors( V ), where V is a left module over a quantized enveloping algebra. This function returns the result of applying the ind-th Kashiwara operator widetildeF_α_ind} to v. The result is fail if widetildeF_α_ind}(v)=0.
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
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 2.7).
‣ Ealpha( v, ind ) | ( operation ) |
Here v is a crystal vector, i.e., an element of CrystalVectors( V ), where V is a left module over a quantized enveloping algebra. This function returns the result of applying the ind-th Kashiwara operator widetildeE_α_ind} to v. The result is fail if widetildeE_α_ind}(v)=0.
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>
‣ CrystalGraph( V ) | ( function ) |
Returns the crystal graph of the module V. The points of this graph are the cosets output by CrystalVectors (3.10-12). The edges work in the same way as in CrystalGraph (3.9-7).
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)> ] )
Here we describe functions for connecting a quantized enveloping algebra to the corresponding universal enveloping algebra.
‣ UEA( L ) | ( attribute ) |
This function returns the universal enveloping algebra u of the semisimple Lie algebra L. The generators of u are the generators of a Kostant lattice in the universal enveloping algebra (these generators are obtained from L by LatticeGeneratorsInUEA( L ), see the GAP reference manual).
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 ) ]
‣ UnderlyingLieAlgebra( u ) | ( attribute ) |
For a universal enveloping algebra u constructed by UEA (3.11-1), this returns the corresponding semisimple Lie algebra
gap> L:= SimpleLieAlgebra( "B", 2, Rationals );; gap> u:= UEA( L );; gap> UnderlyingLieAlgebra( u ); <Lie algebra of dimension 10 over Rationals>
‣ HighestWeightModule( u, hw ) | ( operation ) |
For a universal enveloping algebra u constructed by UEA (3.11-1), this returns the irreducible highest weight module over u with highest weight hw, 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 u acts.
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>>
‣ QUEAToUEAMap( L ) | ( attribute ) |
Here L is a semisimple Lie algebra. Set u := UEA( L ), and U := QuantizedUEA( L ) (so u, U are the universal enveloping algebra, and "generic" quantized enveloping algebra of L respectively). Then QUEAToUEAMap( L ) returns the algebra homomorphism from U to u obtained by mapping q to 1, a generator Fi, corresponding to a simple root to the generator yi (corresponding to the same simple root), and likewise for Ei and xi. This means that Ki is mapped to one, and [ Ki : s ] to hi choose s.
The canonical basis of U is mapped to the canonical basis of u.
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.
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