Lambda Bracket Algebras#
AUTHORS:
Reimundo Heluani (2019-10-05): Initial implementation.
- class sage.categories.lambda_bracket_algebras.LambdaBracketAlgebras(base, name=None)#
Bases:
Category_over_base_ringThe category of Lambda bracket algebras.
This is an abstract base category for Lie conformal algebras and super Lie conformal algebras.
- class ElementMethods#
Bases:
object- T(n=1)#
The
n-th derivative ofself.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: Vir = lie_conformal_algebras.Virasoro(QQ) # optional - sage.combinat sage.modules sage: Vir.inject_variables() # optional - sage.combinat sage.modules Defining L, C sage: L.T() # optional - sage.combinat sage.modules TL sage: L.T(3) # optional - sage.combinat sage.modules 6*T^(3)L sage: C.T() # optional - sage.combinat sage.modules 0
- bracket(rhs)#
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 # optional - sage.combinat sage.modules sage: L.bracket(L) # optional - sage.combinat sage.modules {0: TL, 1: 2*L, 3: 1/2*C} sage: L.bracket(L.T()) # optional - sage.combinat sage.modules {0: 2*T^(2)L, 1: 3*TL, 2: 4*L, 4: 2*C}
Now with a current algebra:
sage: V = lie_conformal_algebras.Affine(QQ, 'A1') # optional - sage.combinat sage.modules sage: V.gens() # optional - sage.combinat sage.modules (B[alpha[1]], B[alphacheck[1]], B[-alpha[1]], B['K']) sage: E = V.0; H = V.1; F = V.2 # optional - sage.combinat sage.modules sage: H.bracket(H) # optional - sage.combinat sage.modules {1: 2*B['K']} sage: E.bracket(F) # optional - sage.combinat sage.modules {0: B[alphacheck[1]], 1: B['K']}
- nproduct(rhs, n)#
The
n-th product of these two elements.EXAMPLES:
sage: Vir = lie_conformal_algebras.Virasoro(QQ); L = Vir.0 # optional - sage.combinat sage.modules sage: L.nproduct(L, 3) # optional - sage.combinat sage.modules 1/2*C sage: L.nproduct(L.T(), 0) # optional - sage.combinat sage.modules 2*T^(2)L sage: V = lie_conformal_algebras.Affine(QQ, 'A1') # optional - sage.combinat sage.modules sage: E = V.0; H = V.1; F = V.2 # optional - sage.combinat sage.modules sage: E.nproduct(H, 0) == - 2*E # optional - sage.combinat sage.modules True sage: E.nproduct(F, 1) # optional - sage.combinat sage.modules B['K']
- FinitelyGeneratedAsLambdaBracketAlgebra#
- class ParentMethods#
Bases:
object- ideal(*gens, **kwds)#
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) # optional - sage.combinat sage.modules sage: Vir.ideal() # optional - sage.combinat sage.modules Traceback (most recent call last): ... NotImplementedError: ideals of Lie Conformal algebras are not implemented yet
- class SubcategoryMethods#
Bases:
object- FinitelyGenerated()#
The category of finitely generated Lambda bracket algebras.
EXAMPLES:
sage: LieConformalAlgebras(QQ).FinitelyGenerated() Category of finitely generated lie conformal algebras over Rational Field
- FinitelyGeneratedAsLambdaBracketAlgebra()#
The category of finitely generated Lambda bracket algebras.
EXAMPLES:
sage: LieConformalAlgebras(QQ).FinitelyGenerated() Category of finitely generated lie conformal algebras over Rational Field
- WithBasis#
alias of
LambdaBracketAlgebrasWithBasis
- super_categories()#
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]