Lambda Bracket Algebras¶
AUTHORS:
- Reimundo Heluani (2019-10-05): Initial implementation. 
- class sage.categories.lambda_bracket_algebras.LambdaBracketAlgebras(base, name=None)[source]¶
- Bases: - Category_over_base_ring- The category of Lambda bracket algebras. - This is an abstract base category for Lie conformal algebras and super Lie conformal algebras. - class ElementMethods[source]¶
- Bases: - object- T(n=1)[source]¶
- The - n-th derivative of- self.- INPUT: - n– integer (default: \(1\)); how many times to apply \(T\) to this element
 - OUTPUT: - \(T^n a\) where \(a\) is this element. Notice that we use the divided powers notation \(T^{(j)} = \frac{T^j}{j!}\). - EXAMPLES: - sage: # needs sage.combinat sage.modules sage: Vir = lie_conformal_algebras.Virasoro(QQ) sage: Vir.inject_variables() Defining L, C sage: L.T() TL sage: L.T(3) 6*T^(3)L sage: C.T() 0 - >>> from sage.all import * >>> # needs sage.combinat sage.modules >>> Vir = lie_conformal_algebras.Virasoro(QQ) >>> Vir.inject_variables() Defining L, C >>> L.T() TL >>> L.T(Integer(3)) 6*T^(3)L >>> C.T() 0 
 - bracket(rhs)[source]¶
- The \(\lambda\)-bracket of these two elements. - EXAMPLES: - The brackets of the Virasoro Lie conformal algebra: - sage: Vir = lie_conformal_algebras.Virasoro(QQ); L = Vir.0 # needs sage.combinat sage.modules sage: L.bracket(L) # needs sage.combinat sage.modules {0: TL, 1: 2*L, 3: 1/2*C} sage: L.bracket(L.T()) # needs sage.combinat sage.modules {0: 2*T^(2)L, 1: 3*TL, 2: 4*L, 4: 2*C} - >>> from sage.all import * >>> Vir = lie_conformal_algebras.Virasoro(QQ); L = Vir.gen(0) # needs sage.combinat sage.modules >>> L.bracket(L) # needs sage.combinat sage.modules {0: TL, 1: 2*L, 3: 1/2*C} >>> L.bracket(L.T()) # needs sage.combinat sage.modules {0: 2*T^(2)L, 1: 3*TL, 2: 4*L, 4: 2*C} - Now with a current algebra: - sage: # needs sage.combinat sage.modules sage: V = lie_conformal_algebras.Affine(QQ, 'A1') sage: V.gens() (B[alpha[1]], B[alphacheck[1]], B[-alpha[1]], B['K']) sage: E = V.0; H = V.1; F = V.2 sage: H.bracket(H) {1: 2*B['K']} sage: E.bracket(F) {0: B[alphacheck[1]], 1: B['K']} - >>> from sage.all import * >>> # needs sage.combinat sage.modules >>> V = lie_conformal_algebras.Affine(QQ, 'A1') >>> V.gens() (B[alpha[1]], B[alphacheck[1]], B[-alpha[1]], B['K']) >>> E = V.gen(0); H = V.gen(1); F = V.gen(2) >>> H.bracket(H) {1: 2*B['K']} >>> E.bracket(F) {0: B[alphacheck[1]], 1: B['K']} 
 - nproduct(rhs, n)[source]¶
- The - n-th product of these two elements.- EXAMPLES: - sage: # needs sage.combinat sage.modules sage: Vir = lie_conformal_algebras.Virasoro(QQ); L = Vir.0 sage: L.nproduct(L, 3) 1/2*C sage: L.nproduct(L.T(), 0) 2*T^(2)L sage: V = lie_conformal_algebras.Affine(QQ, 'A1') sage: E = V.0; H = V.1; F = V.2 sage: E.nproduct(H, 0) == - 2*E True sage: E.nproduct(F, 1) B['K'] - >>> from sage.all import * >>> # needs sage.combinat sage.modules >>> Vir = lie_conformal_algebras.Virasoro(QQ); L = Vir.gen(0) >>> L.nproduct(L, Integer(3)) 1/2*C >>> L.nproduct(L.T(), Integer(0)) 2*T^(2)L >>> V = lie_conformal_algebras.Affine(QQ, 'A1') >>> E = V.gen(0); H = V.gen(1); F = V.gen(2) >>> E.nproduct(H, Integer(0)) == - Integer(2)*E True >>> E.nproduct(F, Integer(1)) B['K'] 
 
 - class ParentMethods[source]¶
- Bases: - object- ideal(*gens, **kwds)[source]¶
- The ideal of this Lambda bracket algebra generated by - gens.- Todo - Ideals of Lie Conformal Algebras are not implemented yet. - EXAMPLES: - sage: Vir = lie_conformal_algebras.Virasoro(QQ) # needs sage.combinat sage.modules sage: Vir.ideal() # needs sage.combinat sage.modules Traceback (most recent call last): ... NotImplementedError: ideals of Lie Conformal algebras are not implemented yet - >>> from sage.all import * >>> Vir = lie_conformal_algebras.Virasoro(QQ) # needs sage.combinat sage.modules >>> Vir.ideal() # needs sage.combinat sage.modules Traceback (most recent call last): ... NotImplementedError: ideals of Lie Conformal algebras are not implemented yet 
 
 - class SubcategoryMethods[source]¶
- Bases: - object- FinitelyGenerated()[source]¶
- The category of finitely generated Lambda bracket algebras. - EXAMPLES: - sage: LieConformalAlgebras(QQ).FinitelyGenerated() Category of finitely generated Lie conformal algebras over Rational Field - >>> from sage.all import * >>> LieConformalAlgebras(QQ).FinitelyGenerated() Category of finitely generated Lie conformal algebras over Rational Field 
 - FinitelyGeneratedAsLambdaBracketAlgebra()[source]¶
- The category of finitely generated Lambda bracket algebras. - EXAMPLES: - sage: LieConformalAlgebras(QQ).FinitelyGenerated() Category of finitely generated Lie conformal algebras over Rational Field - >>> from sage.all import * >>> LieConformalAlgebras(QQ).FinitelyGenerated() Category of finitely generated Lie conformal algebras over Rational Field 
 
 - WithBasis[source]¶
- alias of - LambdaBracketAlgebrasWithBasis
 - super_categories()[source]¶
- The list of super categories of this category. - EXAMPLES: - sage: from sage.categories.lambda_bracket_algebras import LambdaBracketAlgebras sage: LambdaBracketAlgebras(QQ).super_categories() [Category of vector spaces over Rational Field] - >>> from sage.all import * >>> from sage.categories.lambda_bracket_algebras import LambdaBracketAlgebras >>> LambdaBracketAlgebras(QQ).super_categories() [Category of vector spaces over Rational Field]