Freely Generated Lie Conformal Algebras - Algebras

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# Freely Generated Lie Conformal Algebras

AUTHORS:

- Reimundo Heluani (2019-08-09): Initial implementation

class sage.algebras.lie_conformal_algebras.freely_generated_lie_conformal_algebra.FreelyGeneratedLieConformalAlgebra(R, index_set=None, central_elements=None, category=None, element_class=None, prefix=None, **kwds)

  Bases: `LieConformalAlgebraWithBasis`

  Base class for a central extension of a freely generated Lie conformal algebra.

  This class provides minimal functionality, it sets up the family of Lie conformal algebra generators.

  Note

  We now only accept direct sums of free modules plus some central generators \(C_i\) such that \(TC_i = 0\).

  central_elements()

    The central generators of this Lie conformal algebra.

    EXAMPLES:

    Sage

        sage: Vir = lie_conformal_algebras.Virasoro(QQ)
        sage: Vir.central_elements()
        (C,)
        sage: V = lie_conformal_algebras.Affine(QQ, 'A1')
        sage: V.central_elements()
        (B['K'],)

    Python

        >>> from sage.all import *
        >>> Vir = lie_conformal_algebras.Virasoro(QQ)
        >>> Vir.central_elements()
        (C,)
        >>> V = lie_conformal_algebras.Affine(QQ, 'A1')
        >>> V.central_elements()
        (B['K'],)

  lie_conformal_algebra_generators()

    The generators of this Lie conformal algebra.

    OUTPUT: a (possibly infinite) family of generators (as an \(R[T]\)-module) of this Lie conformal algebra.

    EXAMPLES:

    Sage

        sage: Vir = lie_conformal_algebras.Virasoro(QQ)
        sage: Vir.lie_conformal_algebra_generators()
        (L, C)
        sage: V = lie_conformal_algebras.Affine(QQ,'A1')
        sage: V.lie_conformal_algebra_generators()
        (B[alpha[1]], B[alphacheck[1]], B[-alpha[1]], B['K'])

    Python

        >>> from sage.all import *
        >>> Vir = lie_conformal_algebras.Virasoro(QQ)
        >>> Vir.lie_conformal_algebra_generators()
        (L, C)
        >>> V = lie_conformal_algebras.Affine(QQ,'A1')
        >>> V.lie_conformal_algebra_generators()
        (B[alpha[1]], B[alphacheck[1]], B[-alpha[1]], B['K'])