In this section we describe some functions for working with left-modules over quantized enveloping algebras. For a general introduction into algebra modules in Magma we refer to Chapter MODULES OVER AN ALGEBRA AND GROUP REPRESENTATIONS.
Given a quantized enveloping algebra U corresponding to a root datum of rank r and a sequence w of non-negative integers of length r, returns the irreducible representation of U with highest weight w. The object returned is a function which given an element of U computes its matrix.
Given a quantized enveloping algebra U corresponding to a root datum of rank r and a sequence w of non-negative integers of length r, returns the irreducible U-module with highest weight w. The object returned is a left module over U.
> R:= RootDatum("G2");
> U:= QuantizedUEA(R);
> f:= HighestWeightRepresentation(U, [1,1]);
> f(U.6);
[ 0 0 0 0 0 0 0]
[ 0 0 0 0 0 0 0]
[ 0 1 0 0 0 0 0]
[ 0 0 0 0 0 0 0]
[ 0 0 0 0 0 0 0]
[ 0 0 0 0 -q 0 0]
[ 0 0 0 0 0 0 0]
> M:= HighestWeightModule(U, [1,0]);
> U.6^M.5;
(0 0 0 0 0 -q 0)
For a module V over a quantized universal enveloping algebra this returns two sequences. The first sequence consists of the weights that occur in V. The second sequence is a sequence of sequences of elements of V, in bijection with the first sequence. The i-th element of the second sequence consists of a basis of the weight space of weight equal to the i-th weight of the first sequence.
This function is analogous to the previous one. Except in this case the first sequence consists of highest weights, i.e., those weights which occur as highest weights of an irreducible constituent of V. The second sequence consists of sequences that contain the corresponding highest weight vectors. So the submodules generated by the vectors in the second sequence form a direct sum decomposition of V.
Given a (left-) module V over a quantized universal enveloping algebra, returns the canonical basis of V. If V is not irreducible, then the union of the canonical bases of the irreducible components of V is returned.
> U:= QuantizedUEA(RootDatum("B2"));
> V:= HighestWeightModule(U, [1,0]);
> C:= CanonicalBasis(V); C;
[
V: (1 0 0 0 0),
V: (0 1 0 0 0),
V: (0 0 1 0 0),
V: (0 0 0 1 0),
V: ( 0 0 0 0 -1/q^2)
]
We can compute the action of elements of U with respect to the
canonical basis by using ModuleWithBasis:
> M:= ModuleWithBasis(C); > ActionMatrix(M, U.1); [0 0 0 0 0] [1 0 0 0 0] [0 0 0 0 0] [0 0 0 0 0] [0 0 0 1 0]
Given a sequence Q of left-modules over a quantized universal enveloping algebra, returns the module M that is the tensor product of the elements of Q. It also returns a map from the Cartesian product of the elements of Q to M. This maps a tuple t to the element of M that is formed by tensoring the elements of t.
> U:= QuantizedUEA(RootDatum("B2"));
> v1:= HighestWeightModule(U, [1,0]);
> V1:= HighestWeightModule(U, [1,0]);
> V2:= HighestWeightModule(U, [0,1]);
> W, f:= TensorProduct([V1,V2]);
> Dimension(W);
20
> HighestWeightsAndVectors(W);
[
(1 1),
(0 1)
]
[
[
W: (1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0)
],
[
W: (0 0 1 0 0 q^4 0 0 -q^7/(q^2 + 1) 0 0 0 0 0
0 0 0 0 0 0)
]
]
> f(<V1.2,V2.4>);
W: (0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0)
So in particular we see that V1 tensor V2 is the direct sum of two
irreducible modules, one with highest weight (1, 1), the other with highest
weight (0, 1). The corresponding highest-weight vectors are also given.