Lambda Bracket Algebras¶
AUTHORS:
Reimundo Heluani (2019-10-05): Initial implementation.
-
class
sage.categories.lambda_bracket_algebras.
LambdaBracketAlgebras
(base, name=None)¶ Bases:
sage.categories.category_types.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
¶ 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) sage: Vir.inject_variables() Defining L, C sage: L.T() TL sage: L.T(3) 6*T^(3)L sage: C.T() 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 sage: L.bracket(L) {0: TL, 1: 2*L, 3: 1/2*C} sage: L.bracket(L.T()) {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') 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']}
-
nproduct
(rhs, n)¶ The
n
-th product of these two elements.EXAMPLES:
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']
-
-
FinitelyGeneratedAsLambdaBracketAlgebra
¶ alias of
sage.categories.finitely_generated_lambda_bracket_algebras.FinitelyGeneratedLambdaBracketAlgebras
-
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) sage: Vir.ideal() 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
sage.categories.lambda_bracket_algebras_with_basis.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]
-
class