Lie Conformal Algebras With Structure Coefficients#

AUTHORS:

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

class sage.algebras.lie_conformal_algebras.lie_conformal_algebra_with_structure_coefs.LieConformalAlgebraWithStructureCoefficients(R, s_coeff, index_set=None, central_elements=None, category=None, element_class=None, prefix=None, names=None, latex_names=None, parity=None, **kwds)[source]#

Bases: FinitelyFreelyGeneratedLCA

A Lie conformal algebra with a set of specified structure coefficients.

INPUT:

  • R – a ring (Default: None); The base ring of this Lie conformal algebra. Behaviour is undefined if it is not a field of characteristic zero.

  • s_coeff – Dictionary (Default: None); a dictionary containing the \(\lambda\) brackets of the generators of this Lie conformal algebra. The family encodes a dictionary whose keys are pairs of either names or indices of the generators and the values are themselves dictionaries. For a pair of generators \(a\) and \(b\), the value of s_coeff[('a','b')] is a dictionary whose keys are positive integer numbers and the corresponding value for the key \(j\) is a dictionary itself representing the j-th product \(a_{(j)}b\). Thus, for a positive integer number \(j\), the value of s_coeff[('a','b')][j] is a dictionary whose entries are pairs ('c',n) where 'c' is the name of a generator and \(n\) is a positive number. The value for this key is the coefficient of \(\frac{T^{n}}{n!} c\) in \(a_{(j)}b\). For example the s_coeff for the Virasoro Lie conformal algebra is:

    {('L','L'):{0:{('L',1):1}, 1:{('L',0):2}, 3:{('C',0):1/2}}}

    Do not include central elements in this dictionary. Also, if the key ('a','b') is present, there is no need to include ('b','a') as it is defined by skew-symmetry. Any missing pair (besides the ones defined by skew-symmetry) is assumed to have vanishing \(\lambda\)-bracket.

  • names – tuple of str (Default: None); The list of names for generators of this Lie conformal algebra. Do not include central elements in this list.

  • central_elements – tuple of str (Default: None); A list of names for central elements of this Lie conformal algebra.

  • index_set – enumerated set (Default: None); an indexing set for the generators of this Lie conformal algebra. Do not include central elements in this list.

  • parity – tuple of \(0\) or \(1\) (Default: tuple of \(0\));

    a tuple specifying the parity of each non-central generator.

EXAMPLES:

  • We construct the \(\beta-\gamma\) system by directly giving the \(\lambda\)-brackets of the generators:

    sage: betagamma_dict = {('b','a'):{0:{('K',0):1}}}
sage: V = LieConformalAlgebra(QQ, betagamma_dict, names=('a','b'), weights=(1,0), central_elements=('K',))
sage: V.category()
Category of H-graded finitely generated Lie conformal algebras with basis over Rational Field
sage: V.inject_variables()
Defining a, b, K
sage: a.bracket(b)
{0: -K}

    >>> from sage.all import *
>>> betagamma_dict = {('b','a'):{Integer(0):{('K',Integer(0)):Integer(1)}}}
>>> V = LieConformalAlgebra(QQ, betagamma_dict, names=('a','b'), weights=(Integer(1),Integer(0)), central_elements=('K',))
>>> V.category()
Category of H-graded finitely generated Lie conformal algebras with basis over Rational Field
>>> V.inject_variables()
Defining a, b, K
>>> a.bracket(b)
{0: -K}

  • We construct the centerless Virasoro Lie conformal algebra:

    sage: virdict =  {('L','L'):{0:{('L',1):1}, 1:{('L',0): 2}}}
sage: R = LieConformalAlgebra(QQbar, virdict, names='L')
sage: R.inject_variables()
Defining L
sage: L.bracket(L)
{0: TL, 1: 2*L}

    >>> from sage.all import *
>>> virdict =  {('L','L'):{Integer(0):{('L',Integer(1)):Integer(1)}, Integer(1):{('L',Integer(0)): Integer(2)}}}
>>> R = LieConformalAlgebra(QQbar, virdict, names='L')
>>> R.inject_variables()
Defining L
>>> L.bracket(L)
{0: TL, 1: 2*L}

  • The construction checks that skew-symmetry is violated:

    sage: wrongdict =  {('L','L'):{0:{('L',1):2}, 1:{('L',0): 2}}}
sage: LieConformalAlgebra(QQbar, wrongdict, names='L')
Traceback (most recent call last):
...
ValueError: two distinct values given for one and the same bracket. Skew-symmetry is not satisfied?

    >>> from sage.all import *
>>> wrongdict =  {('L','L'):{Integer(0):{('L',Integer(1)):Integer(2)}, Integer(1):{('L',Integer(0)): Integer(2)}}}
>>> LieConformalAlgebra(QQbar, wrongdict, names='L')
Traceback (most recent call last):
...
ValueError: two distinct values given for one and the same bracket. Skew-symmetry is not satisfied?
structure_coefficients()[source]#

The structure coefficients of this Lie conformal algebra.

EXAMPLES:

sage: Vir = lie_conformal_algebras.Virasoro(AA)
sage: Vir.structure_coefficients()
Finite family {('L', 'L'): ((0, TL), (1, 2*L), (3, 1/2*C))}

sage: lie_conformal_algebras.NeveuSchwarz(QQ).structure_coefficients()
Finite family {('G', 'G'): ((0, 2*L), (2, 2/3*C)),  ('G', 'L'): ((0, 1/2*TG), (1, 3/2*G)),  ('L', 'G'): ((0, TG), (1, 3/2*G)),  ('L', 'L'): ((0, TL), (1, 2*L), (3, 1/2*C))}

>>> from sage.all import *
>>> Vir = lie_conformal_algebras.Virasoro(AA)
>>> Vir.structure_coefficients()
Finite family {('L', 'L'): ((0, TL), (1, 2*L), (3, 1/2*C))}

>>> lie_conformal_algebras.NeveuSchwarz(QQ).structure_coefficients()
Finite family {('G', 'G'): ((0, 2*L), (2, 2/3*C)),  ('G', 'L'): ((0, 1/2*TG), (1, 3/2*G)),  ('L', 'G'): ((0, TG), (1, 3/2*G)),  ('L', 'L'): ((0, TL), (1, 2*L), (3, 1/2*C))}