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_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 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

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              # 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}

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']}

nproduct(rhs, n)#

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']

FinitelyGeneratedAsLambdaBracketAlgebra#: alias of 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)                         # 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

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]