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
withV
.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
); ifTrue
, 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 intoself
.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 byk
.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 ofself
.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 toself
.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:
- 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
(iteratedn
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
ofself
.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