Quantum Group Representations¶
AUTHORS:
- Travis Scrimshaw (2018): initial version 
- class sage.categories.quantum_group_representations.QuantumGroupRepresentations(base, name=None)[source]¶
- Bases: - Category_module- The category of quantum group representations. - class ParentMethods[source]¶
- Bases: - object- cartan_type()[source]¶
- Return the Cartan type of - self.- EXAMPLES: - sage: # needs sage.combinat sage.graphs sage.modules sage: from sage.algebras.quantum_groups.representations import MinusculeRepresentation sage: C = crystals.Tableaux(['C',4], shape=[1]) sage: R = ZZ['q'].fraction_field() sage: V = MinusculeRepresentation(R, C) sage: V.cartan_type() ['C', 4] - >>> from sage.all import * >>> # needs sage.combinat sage.graphs sage.modules >>> from sage.algebras.quantum_groups.representations import MinusculeRepresentation >>> C = crystals.Tableaux(['C',Integer(4)], shape=[Integer(1)]) >>> R = ZZ['q'].fraction_field() >>> V = MinusculeRepresentation(R, C) >>> V.cartan_type() ['C', 4] 
 - index_set()[source]¶
- Return the index set of - self.- EXAMPLES: - sage: # needs sage.combinat sage.graphs sage.modules sage: from sage.algebras.quantum_groups.representations import MinusculeRepresentation sage: C = crystals.Tableaux(['C',4], shape=[1]) sage: R = ZZ['q'].fraction_field() sage: V = MinusculeRepresentation(R, C) sage: V.index_set() (1, 2, 3, 4) - >>> from sage.all import * >>> # needs sage.combinat sage.graphs sage.modules >>> from sage.algebras.quantum_groups.representations import MinusculeRepresentation >>> C = crystals.Tableaux(['C',Integer(4)], shape=[Integer(1)]) >>> R = ZZ['q'].fraction_field() >>> V = MinusculeRepresentation(R, C) >>> V.index_set() (1, 2, 3, 4) 
 - q()[source]¶
- Return the quantum parameter \(q\) of - self.- EXAMPLES: - sage: # needs sage.combinat sage.graphs sage.modules sage: from sage.algebras.quantum_groups.representations import MinusculeRepresentation sage: C = crystals.Tableaux(['C',4], shape=[1]) sage: R = ZZ['q'].fraction_field() sage: V = MinusculeRepresentation(R, C) sage: V.q() q - >>> from sage.all import * >>> # needs sage.combinat sage.graphs sage.modules >>> from sage.algebras.quantum_groups.representations import MinusculeRepresentation >>> C = crystals.Tableaux(['C',Integer(4)], shape=[Integer(1)]) >>> R = ZZ['q'].fraction_field() >>> V = MinusculeRepresentation(R, C) >>> V.q() q 
 
 - class TensorProducts(category, *args)[source]¶
- Bases: - TensorProductsCategory- The category of quantum group representations constructed by tensor product of quantum group representations. - Warning - We use the reversed coproduct in order to match the tensor product rule on crystals. - class ParentMethods[source]¶
- Bases: - object- cartan_type()[source]¶
- Return the Cartan type of - self.- EXAMPLES: - sage: # needs sage.combinat sage.graphs sage.modules sage: from sage.algebras.quantum_groups.representations import MinusculeRepresentation sage: C = crystals.Tableaux(['C',2], shape=[1]) sage: R = ZZ['q'].fraction_field() sage: V = MinusculeRepresentation(R, C) sage: T = tensor([V,V]) sage: T.cartan_type() ['C', 2] - >>> from sage.all import * >>> # needs sage.combinat sage.graphs sage.modules >>> from sage.algebras.quantum_groups.representations import MinusculeRepresentation >>> C = crystals.Tableaux(['C',Integer(2)], shape=[Integer(1)]) >>> R = ZZ['q'].fraction_field() >>> V = MinusculeRepresentation(R, C) >>> T = tensor([V,V]) >>> T.cartan_type() ['C', 2] 
 
 - extra_super_categories()[source]¶
- EXAMPLES: - sage: from sage.categories.quantum_group_representations import QuantumGroupRepresentations sage: Cat = QuantumGroupRepresentations(ZZ['q'].fraction_field()) sage: Cat.TensorProducts().extra_super_categories() [Category of quantum group representations over Fraction Field of Univariate Polynomial Ring in q over Integer Ring] - >>> from sage.all import * >>> from sage.categories.quantum_group_representations import QuantumGroupRepresentations >>> Cat = QuantumGroupRepresentations(ZZ['q'].fraction_field()) >>> Cat.TensorProducts().extra_super_categories() [Category of quantum group representations over Fraction Field of Univariate Polynomial Ring in q over Integer Ring] 
 
 - class WithBasis(base_category)[source]¶
- Bases: - CategoryWithAxiom_over_base_ring- The category of quantum group representations with a distinguished basis. - class ElementMethods[source]¶
- Bases: - object- K(i, power=1)[source]¶
- Return the action of \(K_i\) on - selfto the power- power.- INPUT: - i– an element of the index set
- power– (default: 1) the power of \(K_i\)
 - EXAMPLES: - sage: # needs sage.combinat sage.graphs sage.modules sage: from sage.algebras.quantum_groups.representations import AdjointRepresentation sage: K = crystals.KirillovReshetikhin(['D',4,2], 1,1) sage: R = ZZ['q'].fraction_field() sage: V = AdjointRepresentation(R, K) sage: v = V.an_element(); v 2*B[[]] + 2*B[[[1]]] + 3*B[[[2]]] sage: v.K(0) 2*B[[]] + 2/q^2*B[[[1]]] + 3*B[[[2]]] sage: v.K(1) 2*B[[]] + 2*q^2*B[[[1]]] + 3/q^2*B[[[2]]] sage: v.K(1, 2) 2*B[[]] + 2*q^4*B[[[1]]] + 3/q^4*B[[[2]]] sage: v.K(1, -1) 2*B[[]] + 2/q^2*B[[[1]]] + 3*q^2*B[[[2]]] - >>> from sage.all import * >>> # needs sage.combinat sage.graphs sage.modules >>> from sage.algebras.quantum_groups.representations import AdjointRepresentation >>> K = crystals.KirillovReshetikhin(['D',Integer(4),Integer(2)], Integer(1),Integer(1)) >>> R = ZZ['q'].fraction_field() >>> V = AdjointRepresentation(R, K) >>> v = V.an_element(); v 2*B[[]] + 2*B[[[1]]] + 3*B[[[2]]] >>> v.K(Integer(0)) 2*B[[]] + 2/q^2*B[[[1]]] + 3*B[[[2]]] >>> v.K(Integer(1)) 2*B[[]] + 2*q^2*B[[[1]]] + 3/q^2*B[[[2]]] >>> v.K(Integer(1), Integer(2)) 2*B[[]] + 2*q^4*B[[[1]]] + 3/q^4*B[[[2]]] >>> v.K(Integer(1), -Integer(1)) 2*B[[]] + 2/q^2*B[[[1]]] + 3*q^2*B[[[2]]] 
 - e(i)[source]¶
- Return the action of \(e_i\) on - self.- INPUT: - i– an element of the index set
 - EXAMPLES: - sage: # needs sage.combinat sage.graphs sage.modules sage: from sage.algebras.quantum_groups.representations import AdjointRepresentation sage: C = crystals.Tableaux(['G',2], shape=[1,1]) sage: R = ZZ['q'].fraction_field() sage: V = AdjointRepresentation(R, C) sage: v = V.an_element(); v 2*B[[[1], [2]]] + 2*B[[[1], [3]]] + 3*B[[[2], [3]]] sage: v.e(1) ((3*q^4+3*q^2+3)/q^2)*B[[[1], [3]]] sage: v.e(2) 2*B[[[1], [2]]] - >>> from sage.all import * >>> # needs sage.combinat sage.graphs sage.modules >>> from sage.algebras.quantum_groups.representations import AdjointRepresentation >>> C = crystals.Tableaux(['G',Integer(2)], shape=[Integer(1),Integer(1)]) >>> R = ZZ['q'].fraction_field() >>> V = AdjointRepresentation(R, C) >>> v = V.an_element(); v 2*B[[[1], [2]]] + 2*B[[[1], [3]]] + 3*B[[[2], [3]]] >>> v.e(Integer(1)) ((3*q^4+3*q^2+3)/q^2)*B[[[1], [3]]] >>> v.e(Integer(2)) 2*B[[[1], [2]]] 
 - f(i)[source]¶
- Return the action of \(f_i\) on - self.- INPUT: - i– an element of the index set
 - EXAMPLES: - sage: # needs sage.combinat sage.graphs sage.modules sage: from sage.algebras.quantum_groups.representations import AdjointRepresentation sage: K = crystals.KirillovReshetikhin(['D',4,1], 2,1) sage: R = ZZ['q'].fraction_field() sage: V = AdjointRepresentation(R, K) sage: v = V.an_element(); v 2*B[[]] + 2*B[[[1], [2]]] + 3*B[[[1], [3]]] sage: v.f(0) ((2*q^2+2)/q)*B[[[1], [2]]] sage: v.f(1) 3*B[[[2], [3]]] sage: v.f(2) 2*B[[[1], [3]]] sage: v.f(3) 3*B[[[1], [4]]] sage: v.f(4) 3*B[[[1], [-4]]] - >>> from sage.all import * >>> # needs sage.combinat sage.graphs sage.modules >>> from sage.algebras.quantum_groups.representations import AdjointRepresentation >>> K = crystals.KirillovReshetikhin(['D',Integer(4),Integer(1)], Integer(2),Integer(1)) >>> R = ZZ['q'].fraction_field() >>> V = AdjointRepresentation(R, K) >>> v = V.an_element(); v 2*B[[]] + 2*B[[[1], [2]]] + 3*B[[[1], [3]]] >>> v.f(Integer(0)) ((2*q^2+2)/q)*B[[[1], [2]]] >>> v.f(Integer(1)) 3*B[[[2], [3]]] >>> v.f(Integer(2)) 2*B[[[1], [3]]] >>> v.f(Integer(3)) 3*B[[[1], [4]]] >>> v.f(Integer(4)) 3*B[[[1], [-4]]] 
 
 - class ParentMethods[source]¶
- Bases: - object- tensor(*factors)[source]¶
- Return the tensor product of - selfwith the representations- factors.- EXAMPLES: - sage: # needs sage.combinat sage.graphs sage.modules sage: from sage.algebras.quantum_groups.representations import ( ....: MinusculeRepresentation, AdjointRepresentation) sage: R = ZZ['q'].fraction_field() sage: CM = crystals.Tableaux(['D',4], shape=[1]) sage: CA = crystals.Tableaux(['D',4], shape=[1,1]) sage: V = MinusculeRepresentation(R, CM) sage: V.tensor(V, V) V((1, 0, 0, 0)) # V((1, 0, 0, 0)) # V((1, 0, 0, 0)) sage: A = MinusculeRepresentation(R, CA) sage: V.tensor(A) V((1, 0, 0, 0)) # V((1, 1, 0, 0)) sage: B = crystals.Tableaux(['A',2], shape=[1]) sage: W = MinusculeRepresentation(R, B) sage: tensor([W,V]) Traceback (most recent call last): ... ValueError: all factors must be of the same Cartan type sage: tensor([V,A,W]) Traceback (most recent call last): ... ValueError: all factors must be of the same Cartan type - >>> from sage.all import * >>> # needs sage.combinat sage.graphs sage.modules >>> from sage.algebras.quantum_groups.representations import ( ... MinusculeRepresentation, AdjointRepresentation) >>> R = ZZ['q'].fraction_field() >>> CM = crystals.Tableaux(['D',Integer(4)], shape=[Integer(1)]) >>> CA = crystals.Tableaux(['D',Integer(4)], shape=[Integer(1),Integer(1)]) >>> V = MinusculeRepresentation(R, CM) >>> V.tensor(V, V) V((1, 0, 0, 0)) # V((1, 0, 0, 0)) # V((1, 0, 0, 0)) >>> A = MinusculeRepresentation(R, CA) >>> V.tensor(A) V((1, 0, 0, 0)) # V((1, 1, 0, 0)) >>> B = crystals.Tableaux(['A',Integer(2)], shape=[Integer(1)]) >>> W = MinusculeRepresentation(R, B) >>> tensor([W,V]) Traceback (most recent call last): ... ValueError: all factors must be of the same Cartan type >>> tensor([V,A,W]) Traceback (most recent call last): ... ValueError: all factors must be of the same Cartan type 
 
 - class TensorProducts(category, *args)[source]¶
- Bases: - TensorProductsCategory- The category of quantum group representations with a distinguished basis constructed by tensor product of quantum group representations with a distinguished basis. - class ParentMethods[source]¶
- Bases: - object- K_on_basis(i, b, power=1)[source]¶
- Return the action of \(K_i\) on the basis element indexed by - bto the power- power.- INPUT: - i– an element of the index set
- b– an element of basis keys
- power– (default: 1) the power of \(K_i\)
 - EXAMPLES: - sage: # needs sage.combinat sage.graphs sage.modules sage: from sage.algebras.quantum_groups.representations import ( ....: MinusculeRepresentation, AdjointRepresentation) sage: R = ZZ['q'].fraction_field() sage: CM = crystals.Tableaux(['A',2], shape=[1]) sage: VM = MinusculeRepresentation(R, CM) sage: CA = crystals.Tableaux(['A',2], shape=[2,1]) sage: VA = AdjointRepresentation(R, CA) sage: v = tensor([sum(VM.basis()), VA.module_generator()]); v B[[[1]]] # B[[[1, 1], [2]]] + B[[[2]]] # B[[[1, 1], [2]]] + B[[[3]]] # B[[[1, 1], [2]]] sage: v.K(1) # indirect doctest q^2*B[[[1]]] # B[[[1, 1], [2]]] + B[[[2]]] # B[[[1, 1], [2]]] + q*B[[[3]]] # B[[[1, 1], [2]]] sage: v.K(2, -1) # indirect doctest 1/q*B[[[1]]] # B[[[1, 1], [2]]] + 1/q^2*B[[[2]]] # B[[[1, 1], [2]]] + B[[[3]]] # B[[[1, 1], [2]]] - >>> from sage.all import * >>> # needs sage.combinat sage.graphs sage.modules >>> from sage.algebras.quantum_groups.representations import ( ... MinusculeRepresentation, AdjointRepresentation) >>> R = ZZ['q'].fraction_field() >>> CM = crystals.Tableaux(['A',Integer(2)], shape=[Integer(1)]) >>> VM = MinusculeRepresentation(R, CM) >>> CA = crystals.Tableaux(['A',Integer(2)], shape=[Integer(2),Integer(1)]) >>> VA = AdjointRepresentation(R, CA) >>> v = tensor([sum(VM.basis()), VA.module_generator()]); v B[[[1]]] # B[[[1, 1], [2]]] + B[[[2]]] # B[[[1, 1], [2]]] + B[[[3]]] # B[[[1, 1], [2]]] >>> v.K(Integer(1)) # indirect doctest q^2*B[[[1]]] # B[[[1, 1], [2]]] + B[[[2]]] # B[[[1, 1], [2]]] + q*B[[[3]]] # B[[[1, 1], [2]]] >>> v.K(Integer(2), -Integer(1)) # indirect doctest 1/q*B[[[1]]] # B[[[1, 1], [2]]] + 1/q^2*B[[[2]]] # B[[[1, 1], [2]]] + B[[[3]]] # B[[[1, 1], [2]]] 
 - e_on_basis(i, b)[source]¶
- Return the action of \(e_i\) on the basis element indexed by - b.- INPUT: - i– an element of the index set
- b– an element of basis keys
 - EXAMPLES: - sage: # needs sage.combinat sage.graphs sage.modules sage: from sage.algebras.quantum_groups.representations import ( ....: MinusculeRepresentation, AdjointRepresentation) sage: R = ZZ['q'].fraction_field() sage: CM = crystals.Tableaux(['D',4], shape=[1]) sage: VM = MinusculeRepresentation(R, CM) sage: CA = crystals.Tableaux(['D',4], shape=[1,1]) sage: VA = AdjointRepresentation(R, CA) sage: v = tensor([VM.an_element(), VA.an_element()]); v 4*B[[[1]]] # B[[[1], [2]]] + 4*B[[[1]]] # B[[[1], [3]]] + 6*B[[[1]]] # B[[[2], [3]]] + 4*B[[[2]]] # B[[[1], [2]]] + 4*B[[[2]]] # B[[[1], [3]]] + 6*B[[[2]]] # B[[[2], [3]]] + 6*B[[[3]]] # B[[[1], [2]]] + 6*B[[[3]]] # B[[[1], [3]]] + 9*B[[[3]]] # B[[[2], [3]]] sage: v.e(1) # indirect doctest 4*B[[[1]]] # B[[[1], [2]]] + ((4*q+6)/q)*B[[[1]]] # B[[[1], [3]]] + 6*B[[[1]]] # B[[[2], [3]]] + 6*q*B[[[2]]] # B[[[1], [3]]] + 9*B[[[3]]] # B[[[1], [3]]] sage: v.e(2) # indirect doctest 4*B[[[1]]] # B[[[1], [2]]] + ((6*q+4)/q)*B[[[2]]] # B[[[1], [2]]] + 6*B[[[2]]] # B[[[1], [3]]] + 9*B[[[2]]] # B[[[2], [3]]] + 6*q*B[[[3]]] # B[[[1], [2]]] sage: v.e(3) # indirect doctest 0 sage: v.e(4) # indirect doctest 0 - >>> from sage.all import * >>> # needs sage.combinat sage.graphs sage.modules >>> from sage.algebras.quantum_groups.representations import ( ... MinusculeRepresentation, AdjointRepresentation) >>> R = ZZ['q'].fraction_field() >>> CM = crystals.Tableaux(['D',Integer(4)], shape=[Integer(1)]) >>> VM = MinusculeRepresentation(R, CM) >>> CA = crystals.Tableaux(['D',Integer(4)], shape=[Integer(1),Integer(1)]) >>> VA = AdjointRepresentation(R, CA) >>> v = tensor([VM.an_element(), VA.an_element()]); v 4*B[[[1]]] # B[[[1], [2]]] + 4*B[[[1]]] # B[[[1], [3]]] + 6*B[[[1]]] # B[[[2], [3]]] + 4*B[[[2]]] # B[[[1], [2]]] + 4*B[[[2]]] # B[[[1], [3]]] + 6*B[[[2]]] # B[[[2], [3]]] + 6*B[[[3]]] # B[[[1], [2]]] + 6*B[[[3]]] # B[[[1], [3]]] + 9*B[[[3]]] # B[[[2], [3]]] >>> v.e(Integer(1)) # indirect doctest 4*B[[[1]]] # B[[[1], [2]]] + ((4*q+6)/q)*B[[[1]]] # B[[[1], [3]]] + 6*B[[[1]]] # B[[[2], [3]]] + 6*q*B[[[2]]] # B[[[1], [3]]] + 9*B[[[3]]] # B[[[1], [3]]] >>> v.e(Integer(2)) # indirect doctest 4*B[[[1]]] # B[[[1], [2]]] + ((6*q+4)/q)*B[[[2]]] # B[[[1], [2]]] + 6*B[[[2]]] # B[[[1], [3]]] + 9*B[[[2]]] # B[[[2], [3]]] + 6*q*B[[[3]]] # B[[[1], [2]]] >>> v.e(Integer(3)) # indirect doctest 0 >>> v.e(Integer(4)) # indirect doctest 0 
 - f_on_basis(i, b)[source]¶
- Return the action of \(f_i\) on the basis element indexed by - b.- INPUT: - i– an element of the index set
- b– an element of basis keys
 - EXAMPLES: - sage: # needs sage.combinat sage.graphs sage.modules sage: from sage.algebras.quantum_groups.representations import ( ....: MinusculeRepresentation, AdjointRepresentation) sage: R = ZZ['q'].fraction_field() sage: KM = crystals.KirillovReshetikhin(['B',3,1], 3,1) sage: VM = MinusculeRepresentation(R, KM) sage: KA = crystals.KirillovReshetikhin(['B',3,1], 2,1) sage: VA = AdjointRepresentation(R, KA) sage: v = tensor([VM.an_element(), VA.an_element()]); v 4*B[[+++, []]] # B[[]] + 4*B[[+++, []]] # B[[[1], [2]]] + 6*B[[+++, []]] # B[[[1], [3]]] + 4*B[[++-, []]] # B[[]] + 4*B[[++-, []]] # B[[[1], [2]]] + 6*B[[++-, []]] # B[[[1], [3]]] + 6*B[[+-+, []]] # B[[]] + 6*B[[+-+, []]] # B[[[1], [2]]] + 9*B[[+-+, []]] # B[[[1], [3]]] sage: v.f(0) # indirect doctest ((4*q^4+4)/q^2)*B[[+++, []]] # B[[[1], [2]]] + ((4*q^4+4)/q^2)*B[[++-, []]] # B[[[1], [2]]] + ((6*q^4+6)/q^2)*B[[+-+, []]] # B[[[1], [2]]] sage: v.f(1) # indirect doctest 6*B[[+++, []]] # B[[[2], [3]]] + 6*B[[++-, []]] # B[[[2], [3]]] + 9*B[[+-+, []]] # B[[[2], [3]]] + 6*B[[-++, []]] # B[[]] + 6*B[[-++, []]] # B[[[1], [2]]] + 9*q^2*B[[-++, []]] # B[[[1], [3]]] sage: v.f(2) # indirect doctest 4*B[[+++, []]] # B[[[1], [3]]] + 4*B[[++-, []]] # B[[[1], [3]]] + 4*B[[+-+, []]] # B[[]] + 4*q^2*B[[+-+, []]] # B[[[1], [2]]] + ((6*q^2+6)/q^2)*B[[+-+, []]] # B[[[1], [3]]] sage: v.f(3) # indirect doctest 6*B[[+++, []]] # B[[[1], [0]]] + 4*B[[++-, []]] # B[[]] + 4*B[[++-, []]] # B[[[1], [2]]] + 6*q^2*B[[++-, []]] # B[[[1], [3]]] + 6*B[[++-, []]] # B[[[1], [0]]] + 9*B[[+-+, []]] # B[[[1], [0]]] + 6*B[[+--, []]] # B[[]] + 6*B[[+--, []]] # B[[[1], [2]]] + 9*q^2*B[[+--, []]] # B[[[1], [3]]] - >>> from sage.all import * >>> # needs sage.combinat sage.graphs sage.modules >>> from sage.algebras.quantum_groups.representations import ( ... MinusculeRepresentation, AdjointRepresentation) >>> R = ZZ['q'].fraction_field() >>> KM = crystals.KirillovReshetikhin(['B',Integer(3),Integer(1)], Integer(3),Integer(1)) >>> VM = MinusculeRepresentation(R, KM) >>> KA = crystals.KirillovReshetikhin(['B',Integer(3),Integer(1)], Integer(2),Integer(1)) >>> VA = AdjointRepresentation(R, KA) >>> v = tensor([VM.an_element(), VA.an_element()]); v 4*B[[+++, []]] # B[[]] + 4*B[[+++, []]] # B[[[1], [2]]] + 6*B[[+++, []]] # B[[[1], [3]]] + 4*B[[++-, []]] # B[[]] + 4*B[[++-, []]] # B[[[1], [2]]] + 6*B[[++-, []]] # B[[[1], [3]]] + 6*B[[+-+, []]] # B[[]] + 6*B[[+-+, []]] # B[[[1], [2]]] + 9*B[[+-+, []]] # B[[[1], [3]]] >>> v.f(Integer(0)) # indirect doctest ((4*q^4+4)/q^2)*B[[+++, []]] # B[[[1], [2]]] + ((4*q^4+4)/q^2)*B[[++-, []]] # B[[[1], [2]]] + ((6*q^4+6)/q^2)*B[[+-+, []]] # B[[[1], [2]]] >>> v.f(Integer(1)) # indirect doctest 6*B[[+++, []]] # B[[[2], [3]]] + 6*B[[++-, []]] # B[[[2], [3]]] + 9*B[[+-+, []]] # B[[[2], [3]]] + 6*B[[-++, []]] # B[[]] + 6*B[[-++, []]] # B[[[1], [2]]] + 9*q^2*B[[-++, []]] # B[[[1], [3]]] >>> v.f(Integer(2)) # indirect doctest 4*B[[+++, []]] # B[[[1], [3]]] + 4*B[[++-, []]] # B[[[1], [3]]] + 4*B[[+-+, []]] # B[[]] + 4*q^2*B[[+-+, []]] # B[[[1], [2]]] + ((6*q^2+6)/q^2)*B[[+-+, []]] # B[[[1], [3]]] >>> v.f(Integer(3)) # indirect doctest 6*B[[+++, []]] # B[[[1], [0]]] + 4*B[[++-, []]] # B[[]] + 4*B[[++-, []]] # B[[[1], [2]]] + 6*q^2*B[[++-, []]] # B[[[1], [3]]] + 6*B[[++-, []]] # B[[[1], [0]]] + 9*B[[+-+, []]] # B[[[1], [0]]] + 6*B[[+--, []]] # B[[]] + 6*B[[+--, []]] # B[[[1], [2]]] + 9*q^2*B[[+--, []]] # B[[[1], [3]]] 
 
 - extra_super_categories()[source]¶
- EXAMPLES: - sage: from sage.categories.quantum_group_representations import QuantumGroupRepresentations sage: Cat = QuantumGroupRepresentations(ZZ['q'].fraction_field()) sage: Cat.WithBasis().TensorProducts().extra_super_categories() [Category of quantum group representations with basis over Fraction Field of Univariate Polynomial Ring in q over Integer Ring] - >>> from sage.all import * >>> from sage.categories.quantum_group_representations import QuantumGroupRepresentations >>> Cat = QuantumGroupRepresentations(ZZ['q'].fraction_field()) >>> Cat.WithBasis().TensorProducts().extra_super_categories() [Category of quantum group representations with basis over Fraction Field of Univariate Polynomial Ring in q over Integer Ring] 
 
 
 - example()[source]¶
- Return an example of a quantum group representation as per - Category.example.- EXAMPLES: - sage: from sage.categories.quantum_group_representations import QuantumGroupRepresentations sage: Cat = QuantumGroupRepresentations(ZZ['q'].fraction_field()) sage: Cat.example() # needs sage.combinat sage.graphs sage.modules V((2, 1, 0)) - >>> from sage.all import * >>> from sage.categories.quantum_group_representations import QuantumGroupRepresentations >>> Cat = QuantumGroupRepresentations(ZZ['q'].fraction_field()) >>> Cat.example() # needs sage.combinat sage.graphs sage.modules V((2, 1, 0)) 
 - super_categories()[source]¶
- Return the super categories of - self.- EXAMPLES: - sage: from sage.categories.quantum_group_representations import QuantumGroupRepresentations sage: QuantumGroupRepresentations(ZZ['q'].fraction_field()).super_categories() [Category of vector spaces over Fraction Field of Univariate Polynomial Ring in q over Integer Ring] - >>> from sage.all import * >>> from sage.categories.quantum_group_representations import QuantumGroupRepresentations >>> QuantumGroupRepresentations(ZZ['q'].fraction_field()).super_categories() [Category of vector spaces over Fraction Field of Univariate Polynomial Ring in q over Integer Ring]