Quantum Groups Using GAP’s QuaGroup Package

AUTHORS:

• Travis Scrimshaw (03-2017): initial version

The documentation for GAP’s QuaGroup package, originally authored by Willem Adriaan de Graaf, can be found at https://www.gap-system.org/Packages/quagroup.html.

class sage.algebras.quantum_groups.quantum_group_gap.CrystalGraphVertex(V, s)

Bases: SageObject

Helper class used as the vertices of a crystal graph.

class sage.algebras.quantum_groups.quantum_group_gap.HighestWeightModule(Q, weight)

Bases: QuantumGroupModule

A highest weight module of a quantum group.

Element

alias of QuaGroupRepresentationElement

an_element()

Return the highest weight vector of self.

EXAMPLES:

sage: Q = QuantumGroup(['A',2])           # optional - gap_packages
sage: V = Q.highest_weight_module([1,1])  # optional - gap_packages
sage: V.highest_weight_vector()           # optional - gap_packages
1*v0

highest_weight_vector()

Return the highest weight vector of self.

EXAMPLES:

sage: Q = QuantumGroup(['A',2])           # optional - gap_packages
sage: V = Q.highest_weight_module([1,1])  # optional - gap_packages
sage: V.highest_weight_vector()           # optional - gap_packages
1*v0

tensor(*V, **options)

Return the tensor product of self with V.

EXAMPLES:

sage: Q = QuantumGroup(['A',2])            # optional - gap_packages
sage: V = Q.highest_weight_module([1,1])   # optional - gap_packages
sage: Vp = Q.highest_weight_module([1,0])  # optional - gap_packages
sage: Vp.tensor(V)                         # optional - gap_packages
Highest weight module of weight Lambda[1] of Quantum Group of type ['A', 2] with q=q
 # Highest weight module of weight Lambda[1] + Lambda[2] of Quantum Group of type ['A', 2] with q=q

class sage.algebras.quantum_groups.quantum_group_gap.HighestWeightSubmodule(ambient, gen, weight)

Bases: QuantumGroupModule

Initialize self.

EXAMPLES:

sage: Q = QuantumGroup(['A',2])  # optional - gap_packages
sage: V = Q.highest_weight_module([1,0])  # optional - gap_packages
sage: T = tensor([V,V])  # optional - gap_packages
sage: S = T.highest_weight_decomposition()[0]  # optional - gap_packages
sage: TestSuite(S).run()  # optional - gap_packages

Element

alias of QuaGroupRepresentationElement

ambient()

Return the ambient module of self.

EXAMPLES:

sage: Q = QuantumGroup(['A',2])                # optional - gap_packages
sage: V = Q.highest_weight_module([1,0])       # optional - gap_packages
sage: T = tensor([V,V])                        # optional - gap_packages
sage: S = T.highest_weight_decomposition()[0]  # optional - gap_packages
sage: S.ambient() is T                         # optional - gap_packages
True

an_element()

Return the highest weight vector of self.

EXAMPLES:

sage: Q = QuantumGroup(['A',2])                # optional - gap_packages
sage: V = Q.highest_weight_module([1,0])       # optional - gap_packages
sage: T = tensor([V,V])                        # optional - gap_packages
sage: S = T.highest_weight_decomposition()[1]  # optional - gap_packages
sage: u = S.highest_weight_vector(); u         # optional - gap_packages
(1)*e.1
sage: u.lift()                                 # optional - gap_packages
-q^-1*(1*v0<x>F[a1]*v0) + 1*(F[a1]*v0<x>1*v0)

crystal_graph(use_ambient=True)

Return the crystal graph of self.

INPUT:

• use_ambient – boolean (default: True); if True, the vertices are given in terms of the ambient module

EXAMPLES:

sage: Q = QuantumGroup(['A',2])                # optional - gap_packages
sage: V = Q.highest_weight_module([1,0])       # optional - gap_packages
sage: T = tensor([V,V])                        # optional - gap_packages
sage: S = T.highest_weight_decomposition()[1]  # optional - gap_packages
sage: G = S.crystal_graph()                    # optional - gap_packages
sage: sorted(G.vertices(sort=False), key=str)            # optional - gap_packages
[<-q^-1*(1*v0<x>F[a1+a2]*v0) + 1*(F[a1+a2]*v0<x>1*v0)>,
 <-q^-1*(1*v0<x>F[a1]*v0) + 1*(F[a1]*v0<x>1*v0)>,
 <-q^-1*(F[a1]*v0<x>F[a1+a2]*v0) + 1*(F[a1+a2]*v0<x>F[a1]*v0)>]
sage: sorted(S.crystal_graph(False).vertices(sort=False), key=str)  # optional - gap_packages
[<(1)*e.1>, <(1)*e.2>, <(1)*e.3>]

highest_weight_vector()

Return the highest weight vector of self.

EXAMPLES:

sage: Q = QuantumGroup(['A',2])                # optional - gap_packages
sage: V = Q.highest_weight_module([1,0])       # optional - gap_packages
sage: T = tensor([V,V])                        # optional - gap_packages
sage: S = T.highest_weight_decomposition()[1]  # optional - gap_packages
sage: u = S.highest_weight_vector(); u         # optional - gap_packages
(1)*e.1
sage: u.lift()                                 # optional - gap_packages
-q^-1*(1*v0<x>F[a1]*v0) + 1*(F[a1]*v0<x>1*v0)

lift()

The lift morphism from self to the ambient space.

EXAMPLES:

sage: Q = QuantumGroup(['A',2])                # optional - gap_packages
sage: V = Q.highest_weight_module([1,0])       # optional - gap_packages
sage: T = tensor([V,V])                        # optional - gap_packages
sage: S = T.highest_weight_decomposition()[0]  # optional - gap_packages
sage: S.lift                                   # optional - gap_packages
Generic morphism:
  From: Highest weight submodule with weight 2*Lambda[1] generated by 1*(1*v0<x>1*v0)
  To:   Highest weight module ... # Highest weight module ...
sage: x = sum(S.basis())                       # optional - gap_packages
sage: x.lift()                                 # optional - gap_packages
1*(1*v0<x>1*v0) + 1*(1*v0<x>F[a1]*v0) + 1*(1*v0<x>F[a1+a2]*v0)
 + q^-1*(F[a1]*v0<x>1*v0) + 1*(F[a1]*v0<x>F[a1]*v0)
 + 1*(F[a1]*v0<x>F[a1+a2]*v0) + q^-1*(F[a1+a2]*v0<x>1*v0)
 + q^-1*(F[a1+a2]*v0<x>F[a1]*v0) + 1*(F[a1+a2]*v0<x>F[a1+a2]*v0)

retract(elt)

The retract map from the ambient space to self.

EXAMPLES:

sage: Q = QuantumGroup(['A',2])           # optional - gap_packages
sage: V = Q.highest_weight_module([1,0])  # optional - gap_packages
sage: T = tensor([V,V])                   # optional - gap_packages
sage: all(S.retract(S.lift(x)) == x       # optional - gap_packages
....:     for S in T.highest_weight_decomposition()
....:     for x in S.basis())
True

class sage.algebras.quantum_groups.quantum_group_gap.LowerHalfQuantumGroup(Q)

Bases: Parent, UniqueRepresentation

The lower half of the quantum group.

class Element(parent, libgap_elt)

Bases: QuaGroupModuleElement

An element of the lower half of the quantum group.

bar()

Return the bar involution on self.

EXAMPLES:

sage: Q = QuantumGroup(['A',2])     # optional - gap_packages
sage: F1, F2 = Q.F_simple()         # optional - gap_packages
sage: B = Q.lower_half()            # optional - gap_packages
sage: x = B(Q.an_element()); x      # optional - gap_packages
1 + (q)*F[a1]
sage: x.bar()                       # optional - gap_packages
1 + (q^-1)*F[a1]
sage: (F1*x).bar() == F1 * x.bar()  # optional - gap_packages
True
sage: (F2*x).bar() == F2 * x.bar()  # optional - gap_packages
True

sage: Q = QuantumGroup(['G',2])     # optional - gap_packages
sage: F1, F2 = Q.F_simple()         # optional - gap_packages
sage: q = Q.q()                     # optional - gap_packages
sage: B = Q.lower_half()            # optional - gap_packages
sage: x = B(q^-2*F1*F2^2*F1)        # optional - gap_packages
sage: x                             # optional - gap_packages
(q + q^-5)*F[a1]*F[a1+a2]*F[a2]
 + (q^8 + q^6 + q^2 + 1)*F[a1]^(2)*F[a2]^(2)
sage: x.bar()                       # optional - gap_packages
(q^5 + q^-1)*F[a1]*F[a1+a2]*F[a2]
 + (q^12 + q^10 + q^6 + q^4)*F[a1]^(2)*F[a2]^(2)

braid_group_action(braid)

Return the action of the braid group element braid projected into self.

INPUT:

• braid – a reduced word of a braid group element

EXAMPLES:

sage: Q = QuantumGroup(['A',2])  # optional - gap_packages
sage: L = Q.lower_half()         # optional - gap_packages
sage: v = L.highest_weight_vector().f_tilde([1,2,2,1]); v  # optional - gap_packages
F[a1]*F[a1+a2]*F[a2]
sage: v.braid_group_action([1])  # optional - gap_packages
(-q^3-q)*F[a2]^(2)
sage: v.braid_group_action([]) == v  # optional - gap_packages
True

monomial_coefficients(copy=True)

Return the dictionary of self whose keys are the basis indices and the values are coefficients.

EXAMPLES:

sage: Q = QuantumGroup(['A',2])         # optional - gap_packages
sage: B = Q.lower_half()                # optional - gap_packages
sage: x = B.retract(Q.an_element()); x  # optional - gap_packages
1 + (q)*F[a1]
sage: sorted(x.monomial_coefficients().items(), key=str)  # optional - gap_packages
[((0, 0, 0), 1), ((1, 0, 0), q)]

tau()

Return the action of the \(\tau\) anti-automorphism on self.

EXAMPLES:

sage: Q = QuantumGroup(['A',2])           # optional - gap_packages
sage: F1, F2 = Q.F_simple()               # optional - gap_packages
sage: B = Q.lower_half()                  # optional - gap_packages
sage: x = B(Q.an_element()); x            # optional - gap_packages
1 + (q)*F[a1]
sage: x.tau()                             # optional - gap_packages
1 + (q)*F[a1]
sage: (F1*x).tau() == x.tau() * F1.tau()  # optional - gap_packages
True
sage: (F2*x).tau() == x.tau() * F2.tau()  # optional - gap_packages
True

sage: Q = QuantumGroup(['G',2])           # optional - gap_packages
sage: F1, F2 = Q.F_simple()               # optional - gap_packages
sage: q = Q.q()                           # optional - gap_packages
sage: B = Q.lower_half()                  # optional - gap_packages
sage: x = B(q^-2*F1*F2^2*F1)              # optional - gap_packages
sage: x                                   # optional - gap_packages
(q + q^-5)*F[a1]*F[a1+a2]*F[a2]
 + (q^8 + q^6 + q^2 + 1)*F[a1]^(2)*F[a2]^(2)
sage: x.tau()                             # optional - gap_packages
(q + q^-5)*F[a1]*F[a1+a2]*F[a2]
 + (q^8 + q^6 + q^2 + 1)*F[a1]^(2)*F[a2]^(2)

algebra_generators()

Return the algebra generators of self.

EXAMPLES:

sage: Q = QuantumGroup(['A',2])  # optional - gap_packages
sage: B = Q.lower_half()         # optional - gap_packages
sage: B.algebra_generators()     # optional - gap_packages
Finite family {1: F[a1], 2: F[a2]}

ambient()

Return the ambient quantum group of self.

EXAMPLES:

sage: Q = QuantumGroup(['A',2])  # optional - gap_packages
sage: B = Q.lower_half()         # optional - gap_packages
sage: B.ambient() is Q           # optional - gap_packages
True

an_element()

Return the highest weight vector of self.

EXAMPLES:

sage: Q = QuantumGroup(['A',2])  # optional - gap_packages
sage: B = Q.lower_half()         # optional - gap_packages
sage: B.highest_weight_vector()  # optional - gap_packages
1

basis()

Return the basis of self.

This returns the PBW basis of self, which is given by monomials in \(\{F_{\alpha}\}\), where \(\alpha\) runs over all positive roots.

EXAMPLES:

sage: Q = QuantumGroup(['A',2])  # optional - gap_packages
sage: B = Q.lower_half()         # optional - gap_packages
sage: basis = B.basis(); basis   # optional - gap_packages
Lazy family (monomial(i))_{i in The Cartesian product of
 (Non negative integers, Non negative integers, Non negative integers)}
sage: basis[1,2,1]               # optional - gap_packages
F[a1]*F[a1+a2]^(2)*F[a2]
sage: basis[1,2,4]               # optional - gap_packages
F[a1]*F[a1+a2]^(2)*F[a2]^(4)
sage: basis[1,0,4]               # optional - gap_packages
F[a1]*F[a2]^(4)

canonical_basis_elements()

Construct the monomial elements of self indexed by k.

EXAMPLES:

sage: Q = QuantumGroup(['A',2])            # optional - gap_packages
sage: B = Q.lower_half()                   # optional - gap_packages
sage: C = B.canonical_basis_elements(); C  # optional - gap_packages
Lazy family (Canonical basis(i))_{i in The Cartesian product of
 (Non negative integers, Non negative integers)}
sage: C[2,1]                               # optional - gap_packages
[F[a1]^(2)*F[a2], F[a1]*F[a1+a2] + (q^2)*F[a1]^(2)*F[a2]]
sage: C[1,2]                               # optional - gap_packages
[F[a1]*F[a2]^(2), (q^2)*F[a1]*F[a2]^(2) + F[a1+a2]*F[a2]]

gens()

Return the algebra generators of self.

EXAMPLES:

sage: Q = QuantumGroup(['A',2])  # optional - gap_packages
sage: B = Q.lower_half()         # optional - gap_packages
sage: B.algebra_generators()     # optional - gap_packages
Finite family {1: F[a1], 2: F[a2]}

highest_weight_vector()

Return the highest weight vector of self.

EXAMPLES:

sage: Q = QuantumGroup(['A',2])  # optional - gap_packages
sage: B = Q.lower_half()         # optional - gap_packages
sage: B.highest_weight_vector()  # optional - gap_packages
1

lift(elt)

Lift elt to the ambient quantum group of self.

EXAMPLES:

sage: Q = QuantumGroup(['A',2])      # optional - gap_packages
sage: B = Q.lower_half()             # optional - gap_packages
sage: x = B.lift(B.an_element()); x  # optional - gap_packages
1
sage: x.parent() is Q                # optional - gap_packages
True

one()

Return the highest weight vector of self.

EXAMPLES:

sage: Q = QuantumGroup(['A',2])  # optional - gap_packages
sage: B = Q.lower_half()         # optional - gap_packages
sage: B.highest_weight_vector()  # optional - gap_packages
1

retract(elt)

Retract elt from the ambient quantum group to self.

EXAMPLES:

sage: Q = QuantumGroup(['A',2])  # optional - gap_packages
sage: B = Q.lower_half()         # optional - gap_packages
sage: x = Q.an_element(); x      # optional - gap_packages
1 + (q)*F[a1] + E[a1] + (q^2-1-q^-2 + q^-4)*[ K1 ; 2 ]
 + K1 + (-q^-1 + q^-3)*K1[ K1 ; 1 ]
sage: B.retract(x)               # optional - gap_packages
1 + (q)*F[a1]

zero()

Return the zero element of self.

EXAMPLES:

sage: Q = QuantumGroup(['A',2])  # optional - gap_packages
sage: B = Q.lower_half()         # optional - gap_packages
sage: B.zero()                   # optional - gap_packages
0

class sage.algebras.quantum_groups.quantum_group_gap.QuaGroupModuleElement(parent, libgap_elt)

Bases: Element

Base class for elements created using QuaGroup.

e_tilde(i)

Return the action of the Kashiwara operator \(\widetilde{e}_i\) on self.

INPUT:

• i – an element of the index set or a list to perform a string of operators

EXAMPLES:

sage: Q = QuantumGroup(['B',2])           # optional - gap_packages
sage: x = Q.one().f_tilde([1,2,1,1,2,2])  # optional - gap_packages
sage: x.e_tilde([2,2,1,2])                # optional - gap_packages
F[a1]^(2)

f_tilde(i)

Return the action of the Kashiwara operator \(\widetilde{f}_i\) on self.

INPUT:

• i – an element of the index set or a list to perform a string of operators

EXAMPLES:

sage: Q = QuantumGroup(['B',2])     # optional - gap_packages
sage: Q.one().f_tilde(1)            # optional - gap_packages
F[a1]
sage: Q.one().f_tilde(2)            # optional - gap_packages
F[a2]
sage: Q.one().f_tilde([1,2,1,1,2])  # optional - gap_packages
F[a1]*F[a1+a2]^(2)

gap()

Return the gap representation of self.

EXAMPLES:

sage: Q = QuantumGroup(['B',3])  # optional - gap_packages
sage: x = Q.an_element()         # optional - gap_packages
sage: x.gap()                    # optional - gap_packages
1+(q)*F1+E1+(q^4-1-q^-4+q^-8)*[ K1 ; 2 ]+K1+(-q^-2+q^-6)*K1[ K1 ; 1 ]

class sage.algebras.quantum_groups.quantum_group_gap.QuaGroupRepresentationElement(parent, libgap_elt)

Bases: QuaGroupModuleElement

Element of a quantum group representation.

monomial_coefficients(copy=True)

Return the dictionary of self whose keys are the basis indices and the values are coefficients.

EXAMPLES:

sage: Q = QuantumGroup(['A',2])           # optional - gap_packages
sage: V = Q.highest_weight_module([1,1])  # optional - gap_packages
sage: v = V.highest_weight_vector()       # optional - gap_packages
sage: F1, F2 = Q.F_simple()               # optional - gap_packages
sage: q = Q.q()                           # optional - gap_packages
sage: x = v + F1*v + q*F2*F1*v; x         # optional - gap_packages
1*v0 + F[a1]*v0 + (q^2)*F[a1]*F[a2]*v0 + (q)*F[a1+a2]*v0
sage: sorted(x.monomial_coefficients().items(), key=str)  # optional - gap_packages
[(0, 1), (1, 1), (3, q^2), (4, q)]

class sage.algebras.quantum_groups.quantum_group_gap.QuantumGroup(cartan_type, q)

Bases: UniqueRepresentation, Parent

A Drinfel’d-Jimbo quantum group (implemented using the optional GAP package QuaGroup).

EXAMPLES:

We check the quantum Serre relations. We first we import the \(q\)-binomial using the \(q\)-int for quantum groups:

sage: from sage.algebras.quantum_groups.q_numbers import q_binomial

We verify the Serre relations for type \(A_2\):

sage: Q = algebras.QuantumGroup(['A',2])      # optional - gap_packages
sage: F1,F12,F2 = Q.F()                       # optional - gap_packages
sage: q = Q.q()                               # optional - gap_packages
sage: F1^2*F2 - q_binomial(2,1,q) * F1*F2*F1 + F2*F1^2  # optional - gap_packages
0

We verify the Serre relations for type \(B_2\):

sage: Q = algebras.QuantumGroup(['B',2])      # optional - gap_packages
sage: F1, F12, F122, F2 = Q.F()               # optional - gap_packages
sage: F1^2*F2 - q_binomial(2,1,q^2) * F1*F2*F1 + F2*F1^2  # optional - gap_packages
0
sage: (F2^3*F1 - q_binomial(3,1,q) * F2^2*F1*F2  # optional - gap_packages
....:  + q_binomial(3,2,q) * F2*F1*F2^2 - F1*F2^3)
0

REFERENCES:

• Wikipedia article Quantum_group

E()

Return the family of generators \(\{E_{\alpha}\}_{\alpha \in \Phi}\), where \(\Phi\) is the root system of self.

EXAMPLES:

sage: Q = QuantumGroup(['B',2])         # optional - gap_packages
sage: list(Q.E())                       # optional - gap_packages
[E[a1], E[a1+a2], E[a1+2*a2], E[a2]]

E_simple()

Return the family of generators \(\{E_i := E_{\alpha_i}\}_{i \in I}\).

EXAMPLES:

sage: Q = QuantumGroup(['B',2])         # optional - gap_packages
sage: Q.E_simple()                      # optional - gap_packages
Finite family {1: E[a1], 2: E[a2]}

class Element(parent, libgap_elt)

Bases: QuaGroupModuleElement

bar()

Return the bar involution on self.

The bar involution is defined by

\[\overline{E_i} = E_i, \qquad\qquad \overline{F_i} = F_i, \qquad\qquad \overline{K_i} = K_i^{-1}.\]

EXAMPLES:

sage: Q = QuantumGroup(['A',2])        # optional - gap_packages
sage: [gen.bar() for gen in Q.gens()]  # optional - gap_packages
[F[a1],
 (q-q^-1)*F[a1]*F[a2] + F[a1+a2],
 F[a2],
 (-q + q^-1)*[ K1 ; 1 ] + K1, K1,
 (-q + q^-1)*[ K2 ; 1 ] + K2, K2,
 E[a1],
 (-q^2 + 1)*E[a1]*E[a2] + (q^2)*E[a1+a2],
 E[a2]]

braid_group_action(braid)

Return the action of the braid group element braid.

The braid group operator \(T_i \colon U_q(\mathfrak{g}) \to U_q(\mathfrak{g})\) is defined by

\[\begin{split}\begin{aligned} T_i(E_i) &= -F_iK_i, \\ T_i(E_j) &= \sum_{k=0}^{-a_{ij}} (-1)^k q_i^{-k} E_i^{(-a_{ij}-k)} E_j E_i^{(k)} \text{ if } i \neq j,\\ T_i(K_j) &= K_jK_i^{a_{ij}}, \\ T_i(F_i) &= -K_i^{-1}E_i, \\ T_i(F_j) &= \sum_{k=0}^{-a_{ij}} (-1)^k q_i^{-k} F_i^{(k)} F_j F_i^{(-a_{ij}-k)} \text{ if } i \neq j, \end{aligned}\end{split}\]

where \(a_{ij} = \langle \alpha_j, \alpha_i^\vee \rangle\) is the \((i,j)\)-entry of the Cartan matrix associated to \(\mathfrak{g}\).

INPUT:

• braid – a reduced word of a braid group element

EXAMPLES:

sage: Q = QuantumGroup(['A',2])     # optional - gap_packages
sage: F1 = Q.F_simple()[1]          # optional - gap_packages
sage: F1.braid_group_action([1])    # optional - gap_packages
(q-q^-1)*[ K1 ; 1 ]*E[a1] + (-1)*K1*E[a1]
sage: F1.braid_group_action([1,2])  # optional - gap_packages
F[a2]
sage: F1.braid_group_action([2,1])  # optional - gap_packages
(-q^3 + 3*q-3*q^-1 + q^-3)*[ K1 ; 1 ]*[ K2 ; 1 ]*E[a1]*E[a2]
 + (q^3-2*q + q^-1)*[ K1 ; 1 ]*[ K2 ; 1 ]*E[a1+a2]
 + (q^2-2 + q^-2)*[ K1 ; 1 ]*K2*E[a1]*E[a2]
 + (-q^2 + 1)*[ K1 ; 1 ]*K2*E[a1+a2]
 + (q^2-2 + q^-2)*K1*[ K2 ; 1 ]*E[a1]*E[a2]
 + (-q^2 + 1)*K1*[ K2 ; 1 ]*E[a1+a2]
 + (-q + q^-1)*K1*K2*E[a1]*E[a2] + (q)*K1*K2*E[a1+a2]
sage: F1.braid_group_action([1,2,1]) == F1.braid_group_action([2,1,2])  # optional - gap_packages
True
sage: F1.braid_group_action([]) == F1  # optional - gap_packages
True

omega()

Return the action of the \(\omega\) automorphism on self.

The \(\omega\) automorphism is defined by

\[\omega(E_i) = F_i, \qquad\qquad \omega(F_i) = E_i, \qquad\qquad \omega(K_i) = K_i^{-1}.\]

EXAMPLES:

sage: Q = QuantumGroup(['A',2])          # optional - gap_packages
sage: [gen.omega() for gen in Q.gens()]  # optional - gap_packages
[E[a1],
 (-q)*E[a1+a2],
 E[a2],
 (-q + q^-1)*[ K1 ; 1 ] + K1,
 K1,
 (-q + q^-1)*[ K2 ; 1 ] + K2,
 K2,
 F[a1],
 (-q^-1)*F[a1+a2],
 F[a2]]

tau()

Return the action of the \(\tau\) anti-automorphism on self.

The \(\tau\) anti-automorphism is defined by

\[\tau(E_i) = E_i, \qquad\qquad \tau(F_i) = F_i, \qquad\qquad \tau(K_i) = K_i^{-1}.\]

EXAMPLES:

sage: Q = QuantumGroup(['A',2])        # optional - gap_packages
sage: [gen.tau() for gen in Q.gens()]  # optional - gap_packages
[F[a1],
 (-q^2 + 1)*F[a1]*F[a2] + (-q)*F[a1+a2],
 F[a2],
 (-q + q^-1)*[ K1 ; 1 ] + K1,
 K1,
 (-q + q^-1)*[ K2 ; 1 ] + K2,
 K2,
 E[a1],
 (q-q^-1)*E[a1]*E[a2] + (-q)*E[a1+a2],
 E[a2]]

F()

Return the family of generators \(\{F_{\alpha}\}_{\alpha \in \Phi}\), where \(\Phi\) is the root system of self.

EXAMPLES:

sage: Q = QuantumGroup(['G',2])         # optional - gap_packages
sage: list(Q.F())                       # optional - gap_packages
[F[a1], F[3*a1+a2], F[2*a1+a2], F[3*a1+2*a2], F[a1+a2], F[a2]]

F_simple()

Return the family of generators \(\{F_i := F_{\alpha_i}\}_{i \in I}\).

EXAMPLES:

sage: Q = QuantumGroup(['G',2])         # optional - gap_packages
sage: Q.F_simple()                      # optional - gap_packages
Finite family {1: F[a1], 2: F[a2]}

K()

Return the family of generators \(\{K_i\}_{i \in I}\).

EXAMPLES:

sage: Q = QuantumGroup(['A',3])         # optional - gap_packages
sage: Q.K()                             # optional - gap_packages
Finite family {1: K1, 2: K2, 3: K3}
sage: Q.K_inverse()                     # optional - gap_packages
Finite family {1: (-q + q^-1)*[ K1 ; 1 ] + K1,
               2: (-q + q^-1)*[ K2 ; 1 ] + K2,
               3: (-q + q^-1)*[ K3 ; 1 ] + K3}

K_inverse()

Return the family of generators \(\{K_i^{-1}\}_{i \in I}\).

EXAMPLES:

sage: Q = QuantumGroup(['A',3])         # optional - gap_packages
sage: Q.K_inverse()                     # optional - gap_packages
Finite family {1: (-q + q^-1)*[ K1 ; 1 ] + K1,
               2: (-q + q^-1)*[ K2 ; 1 ] + K2,
               3: (-q + q^-1)*[ K3 ; 1 ] + K3}

algebra_generators()

Return the algebra generators of self.

EXAMPLES:

sage: Q = QuantumGroup(['A',2])         # optional - gap_packages
sage: list(Q.algebra_generators())      # optional - gap_packages
[F[a1], F[a2],
 K1, K2,
 (-q + q^-1)*[ K1 ; 1 ] + K1, (-q + q^-1)*[ K2 ; 1 ] + K2,
 E[a1], E[a2]]

antipode(elt)

Return the antipode of elt.

The antipode \(S \colon U_q(\mathfrak{g}) \to U_q(\mathfrak{g})\) is the anti-automorphism defined by

\[S(E_i) = -K_i^{-1}E_i, \qquad S(F_i) = -F_iK_i, \qquad S(K_i) = K_i^{-1}.\]

EXAMPLES:

sage: Q = QuantumGroup(['B',2])         # optional - gap_packages
sage: [Q.antipode(f) for f in Q.F()]    # optional - gap_packages
[(-1)*F[a1]*K1,
 (-q^6 + q^2)*F[a1]*F[a2]*K1*K2 + (-q^4)*F[a1+a2]*K1*K2,
 (-q^8 + q^6 + q^4-q^2)*F[a1]*F[a2]^(2)*K1
  + (-q^9 + 2*q^7-2*q^3 + q)*F[a1]*F[a2]^(2)*K1*K2[ K2 ; 1 ]
  + (-q^5 + q^3)*F[a1+a2]*F[a2]*K1
  + (-q^6 + 2*q^4-q^2)*F[a1+a2]*F[a2]*K1*K2[ K2 ; 1 ]
  + (-q^4)*F[a1+2*a2]*K1 + (-q^5 + q^3)*F[a1+2*a2]*K1*K2[ K2 ; 1 ],
 (-1)*F[a2]*K2]

cartan_type()

Return the Cartan type of self.

EXAMPLES:

sage: Q = QuantumGroup(['A',2])     # optional - gap_packages
sage: Q.cartan_type()               # optional - gap_packages
['A', 2]

coproduct(elt, n=1)

Return the coproduct of elt (iterated n times).

The comultiplication \(\Delta \colon U_q(\mathfrak{g}) \to U_q(\mathfrak{g}) \otimes U_q(\mathfrak{g})\) is defined by

\[\begin{split}\begin{aligned} \Delta(E_i) &= E_i \otimes 1 + K_i \otimes E_i, \\ \Delta(F_i) &= F_i \otimes K_i^{-1} + 1 \otimes F_i, \\ \Delta(K_i) &= K_i \otimes K_i. \end{aligned}\end{split}\]

EXAMPLES:

sage: Q = QuantumGroup(['B',2])         # optional - gap_packages
sage: [Q.coproduct(e) for e in Q.E()]   # optional - gap_packages
[1*(E[a1]<x>1) + 1*(K1<x>E[a1]),
 1*(E[a1+a2]<x>1) + 1*(K1*K2<x>E[a1+a2]) + q^2-q^-2*(K2*E[a1]<x>E[a2]),
 q^4-q^2-1 + q^-2*(E[a1]<x>E[a2]^(2)) + 1*(E[a1+2*a2]<x>1)
  + 1*(K1<x>E[a1+2*a2]) + q-q^-1*(K1*K2[ K2 ; 1 ]<x>E[a1+2*a2])
  + q-q^-1*(K2*E[a1+a2]<x>E[a2]) + q^5-2*q^3
  + 2*q^-1-q^-3*(K2[ K2 ; 1 ]*E[a1]<x>E[a2]^(2)),
 1*(E[a2]<x>1) + 1*(K2<x>E[a2])]
sage: [Q.coproduct(f, 2) for f in Q.F_simple()]  # optional - gap_packages
[1*(1<x>1<x>F[a1]) + -q^2 + q^-2*(1<x>F[a1]<x>[ K1 ; 1 ])
  + 1*(1<x>F[a1]<x>K1) + q^4-2 + q^-4*(F[a1]<x>[ K1 ; 1 ]<x>[ K1 ; 1 ])
  + -q^2 + q^-2*(F[a1]<x>[ K1 ; 1 ]<x>K1) + -q^2
  + q^-2*(F[a1]<x>K1<x>[ K1 ; 1 ]) + 1*(F[a1]<x>K1<x>K1),
 1*(1<x>1<x>F[a2]) + -q + q^-1*(1<x>F[a2]<x>[ K2 ; 1 ])
  + 1*(1<x>F[a2]<x>K2) + q^2-2 + q^-2*(F[a2]<x>[ K2 ; 1 ]<x>[ K2 ; 1 ])
  + -q + q^-1*(F[a2]<x>[ K2 ; 1 ]<x>K2) + -q
  + q^-1*(F[a2]<x>K2<x>[ K2 ; 1 ]) + 1*(F[a2]<x>K2<x>K2)]

counit(elt)

Return the counit of elt.

The counit \(\varepsilon \colon U_q(\mathfrak{g}) \to \QQ(q)\) is defined by

\[\varepsilon(E_i) = \varepsilon(F_i) = 0, \qquad \varepsilon(K_i) = 1.\]

EXAMPLES:

sage: Q = QuantumGroup(['B',2])         # optional - gap_packages
sage: x = Q.an_element()^2              # optional - gap_packages
sage: Q.counit(x)                       # optional - gap_packages
4
sage: Q.counit(Q.one())                 # optional - gap_packages
1
sage: Q.counit(Q.zero())                # optional - gap_packages
0

gap()

Return the gap representation of self.

EXAMPLES:

sage: Q = QuantumGroup(['A',2])         # optional - gap_packages
sage: Q.gap()                           # optional - gap_packages
QuantumUEA( <root system of type A2>, Qpar = q )

gens()

Return the generators of self.

EXAMPLES:

sage: Q = QuantumGroup(['A',2])         # optional - gap_packages
sage: Q.gens()                          # optional - gap_packages
(F[a1], F[a1+a2], F[a2],
 K1, (-q + q^-1)*[ K1 ; 1 ] + K1,
 K2, (-q + q^-1)*[ K2 ; 1 ] + K2,
 E[a1], E[a1+a2], E[a2])

highest_weight_module(weight)

Return the highest weight module of weight weight of self.

EXAMPLES:

sage: Q = QuantumGroup(['A',2])         # optional - gap_packages
sage: Q.highest_weight_module([1,3])    # optional - gap_packages
Highest weight module of weight Lambda[1] + 3*Lambda[2] of
 Quantum Group of type ['A', 2] with q=q

lower_half()

Return the lower half of the quantum group self.

EXAMPLES:

sage: Q = QuantumGroup(['A',2])         # optional - gap_packages
sage: Q.lower_half()                    # optional - gap_packages
Lower Half of Quantum Group of type ['A', 2] with q=q

one()

Return the multiplicative identity of self.

EXAMPLES:

sage: Q = QuantumGroup(['A',2])         # optional - gap_packages
sage: Q.one()                           # optional - gap_packages
1

q()

Return the parameter \(q\).

EXAMPLES:

sage: Q = QuantumGroup(['A',3])           # optional - gap_packages
sage: Q.q()                               # optional - gap_packages
q
sage: zeta3 = CyclotomicField(3).gen()    # optional - gap_packages
sage: Q = QuantumGroup(['B',2], q=zeta3)  # optional - gap_packages
sage: Q.q()                               # optional - gap_packages
zeta3

some_elements()

Return some elements of self.

EXAMPLES:

sage: Q = QuantumGroup(['A',1])  # optional - gap_packages
sage: Q.some_elements()          # optional - gap_packages
[1 + (q)*F[a1] + E[a1] + (q^2-1-q^-2 + q^-4)*[ K1 ; 2 ]
  + K1 + (-q^-1 + q^-3)*K1[ K1 ; 1 ],
 K1, F[a1], E[a1]]

zero()

Return the multiplicative identity of self.

EXAMPLES:

sage: Q = QuantumGroup(['A',2])         # optional - gap_packages
sage: Q.zero()                          # optional - gap_packages
0

class sage.algebras.quantum_groups.quantum_group_gap.QuantumGroupHomset(X, Y, category=None, check=True, base=None)

Bases: HomsetWithBase

The homset whose domain is a quantum group.

class sage.algebras.quantum_groups.quantum_group_gap.QuantumGroupModule(Q, category)

Bases: Parent, UniqueRepresentation

Abstract base class for quantum group representations.

R_matrix()

Return the \(R\)-matrix of self.

EXAMPLES:

sage: Q = QuantumGroup(['A',1])         # optional - gap_packages
sage: V = Q.highest_weight_module([1])  # optional - gap_packages
sage: V.R_matrix()                      # optional - gap_packages
[       1        0        0        0]
[       0        q -q^2 + 1        0]
[       0        0        q        0]
[       0        0        0        1]

basis()

Return a basis of self.

EXAMPLES:

sage: Q = QuantumGroup(['A',2])           # optional - gap_packages
sage: V = Q.highest_weight_module([1,1])  # optional - gap_packages
sage: V.basis()                           # optional - gap_packages
Family (1*v0, F[a1]*v0, F[a2]*v0, F[a1]*F[a2]*v0, F[a1+a2]*v0,
        F[a1]*F[a1+a2]*v0, F[a1+a2]*F[a2]*v0, F[a1+a2]^(2)*v0)

crystal_basis()

Return the crystal basis of self.

EXAMPLES:

sage: Q = QuantumGroup(['A',2])           # optional - gap_packages
sage: V = Q.highest_weight_module([1,1])  # optional - gap_packages
sage: V.crystal_basis()                   # optional - gap_packages
Family (1*v0, F[a1]*v0, F[a2]*v0, F[a1]*F[a2]*v0,
        (q)*F[a1]*F[a2]*v0 + F[a1+a2]*v0, F[a1+a2]*F[a2]*v0,
        (-q^-2)*F[a1]*F[a1+a2]*v0, (-q^-1)*F[a1+a2]^(2)*v0)

crystal_graph()

Return the crystal graph of self.

EXAMPLES:

sage: Q = QuantumGroup(['A',2])           # optional - gap_packages
sage: V = Q.highest_weight_module([1,1])  # optional - gap_packages
sage: G = V.crystal_graph(); G            # optional - gap_packages
Digraph on 8 vertices

sage: B = crystals.Tableaux(['A',2], shape=[2,1])     # optional - gap_packages
sage: G.is_isomorphic(B.digraph(), edge_labels=True)  # optional - gap_packages
True

gap()

Return the gap representation of self.

EXAMPLES:

sage: Q = QuantumGroup(['A',2])           # optional - gap_packages
sage: V = Q.highest_weight_module([1,1])  # optional - gap_packages
sage: V.gap()                             # optional - gap_packages
<8-dimensional left-module over QuantumUEA( <root system of type A2>,
 Qpar = q )>

zero()

Return the zero element of self.

EXAMPLES:

sage: Q = QuantumGroup(['A',2])           # optional - gap_packages
sage: V = Q.highest_weight_module([1,1])  # optional - gap_packages
sage: V.zero()                            # optional - gap_packages
0*v0

class sage.algebras.quantum_groups.quantum_group_gap.QuantumGroupMorphism(parent, im_gens, check=True)

Bases: Morphism

A morphism whose domain is a quantum group.

im_gens()

Return the image of the generators under self.

EXAMPLES:

sage: Q = QuantumGroup(['A',1])   # optional - gap_packages
sage: F, K, Ki, E = Q.gens()      # optional - gap_packages
sage: phi = Q.hom([E, Ki, K, F])  # optional - gap_packages
sage: phi.im_gens()               # optional - gap_packages
(E[a1], (-q + q^-1)*[ K1 ; 1 ] + K1, K1, F[a1])

class sage.algebras.quantum_groups.quantum_group_gap.TensorProductOfHighestWeightModules(*modules, **options)

Bases: QuantumGroupModule

Initialize self.

EXAMPLES:

sage: Q = QuantumGroup(['A',2])  # optional - gap_packages
sage: V = Q.highest_weight_module([1,1])  # optional - gap_packages
sage: T = tensor([V,V])  # optional - gap_packages
sage: TestSuite(T).run()  # optional - gap_packages

Element

alias of QuaGroupRepresentationElement

highest_weight_decomposition()

Return the highest weight decomposition of self.

EXAMPLES:

sage: Q = QuantumGroup(['A',2])           # optional - gap_packages
sage: V = Q.highest_weight_module([1,0])  # optional - gap_packages
sage: T = tensor([V,V])                   # optional - gap_packages
sage: T.highest_weight_decomposition()    # optional - gap_packages
[Highest weight submodule with weight 2*Lambda[1] generated by 1*(1*v0<x>1*v0),
 Highest weight submodule with weight Lambda[2] generated by -q^-1*(1*v0<x>F[a1]*v0) + 1*(F[a1]*v0<x>1*v0)]

highest_weight_vectors()

Return the highest weight vectors of self.

EXAMPLES:

sage: Q = QuantumGroup(['A',2])           # optional - gap_packages
sage: V = Q.highest_weight_module([1,0])  # optional - gap_packages
sage: T = tensor([V,V])                   # optional - gap_packages
sage: T.highest_weight_vectors()          # optional - gap_packages
[1*(1*v0<x>1*v0), -q^-1*(1*v0<x>F[a1]*v0) + 1*(F[a1]*v0<x>1*v0)]

some_elements()

Return the highest weight vectors of self.

EXAMPLES:

sage: Q = QuantumGroup(['A',2])           # optional - gap_packages
sage: V = Q.highest_weight_module([1,0])  # optional - gap_packages
sage: T = tensor([V,V])                   # optional - gap_packages
sage: T.highest_weight_vectors()          # optional - gap_packages
[1*(1*v0<x>1*v0), -q^-1*(1*v0<x>F[a1]*v0) + 1*(F[a1]*v0<x>1*v0)]

tensor_factors()

Return the factors of self.

EXAMPLES:

sage: Q = QuantumGroup(['A',2])           # optional - gap_packages
sage: V = Q.highest_weight_module([1,0])  # optional - gap_packages
sage: T = tensor([V,V])                   # optional - gap_packages
sage: T.tensor_factors()                  # optional - gap_packages
(Highest weight module of weight Lambda[1] of Quantum Group of type ['A', 2] with q=q,
 Highest weight module of weight Lambda[1] of Quantum Group of type ['A', 2] with q=q)

sage.algebras.quantum_groups.quantum_group_gap.projection_lower_half(Q)

Return the projection onto the lower half of the quantum group.

EXAMPLES:

sage: from sage.algebras.quantum_groups.quantum_group_gap import projection_lower_half
sage: Q = QuantumGroup(['G',2])               # optional - gap_packages
sage: phi = projection_lower_half(Q); phi     # optional - gap_packages
Quantum group homomorphism endomorphism of Quantum Group of type ['G', 2] with q=q
  Defn: F[a1] |--> F[a1]
        F[a2] |--> F[a2]
        K1 |--> 0
        K2 |--> 0
        (-q + q^-1)*[ K1 ; 1 ] + K1 |--> 0
        (-q^3 + q^-3)*[ K2 ; 1 ] + K2 |--> 0
        E[a1] |--> 0
        E[a2] |--> 0
sage: all(phi(f) == f for f in Q.F())         # optional - gap_packages
True
sage: all(phi(e) == Q.zero() for e in Q.E())  # optional - gap_packages
True
sage: all(phi(K) == Q.zero() for K in Q.K())  # optional - gap_packages
True