# 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)#

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}

• 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}

• 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?

structure_coefficients()#

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))}