Graded Lie Conformal Algebras¶
A (super) Lie conformal algebra \(V\) is called \(H\)-graded if there exists a decomposition \(V = \oplus_n V_n\) such that the \(\lambda\)- bracket is graded of degree \(-1\), that is for homogeneous elements \(a \in V_p\), \(b \in V_q\) with \(\lambda\)-brackets:
we have \(c_n \in V_{p+q-n-1}\). This situation arises typically when \(V\) has a vector \(L \in V\) that generates the Virasoro Lie conformal algebra. Such that for every \(a \in V\) we have
In this situation \(V\) is graded by the eigenvalues \(\Delta_a\) of \(L_{(1)}\), the \((1)\)-th product with \(L\). When the higher order terms \(O(\lambda^2)\) vanish we say that \(a\) is a primary vector of conformal weight or degree \(\Delta_a\).
Note
Although arbitrary gradings are allowed, many of the constructions we implement in these classes work only for positive rational gradings.
AUTHORS:
- Reimundo Heluani (2019-08-09): Initial implementation. 
- class sage.algebras.lie_conformal_algebras.graded_lie_conformal_algebra.GradedLieConformalAlgebra(R, s_coeff, index_set=None, central_elements=None, category=None, prefix=None, names=None, latex_names=None, parity=None, weights=None, **kwds)[source]¶
- Bases: - LieConformalAlgebraWithStructureCoefficients- An H-Graded Lie conformal algebra. - INPUT: - R– a commutative 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); as in the input of- LieConformalAlgebra
- names– tuple of strings (default:- None); as in the input of- LieConformalAlgebra
- central_elements– tuple of strings (default:- None); as in the input of- LieConformalAlgebra
- index_set– enumerated set (default:- None); as in the input of- LieConformalAlgebra
- weights– tuple of nonnegative rational numbers (default: tuple of- 1); a list of degrees for this Lie conformal algebra. This tuple needs to have the same cardinality as- index_setor- names. Central elements are assumed to have weight- 0.
- category– the category that this Lie conformal algebra belongs to
- parity– tuple of- 0or- 1(default: tuple of- 0); a tuple specifying the parity of each non-central generator
 - EXAMPLES: - sage: bosondict = {('a','a'):{1:{('K',0):1}}} sage: R = LieConformalAlgebra(QQ,bosondict,names=('a',),central_elements=('K',), weights=(1,)) sage: R.inject_variables() Defining a, K sage: a.T(3).degree() 4 sage: K.degree() 0 sage: R.category() Category of H-graded finitely generated Lie conformal algebras with basis over Rational Field - >>> from sage.all import * >>> bosondict = {('a','a'):{Integer(1):{('K',Integer(0)):Integer(1)}}} >>> R = LieConformalAlgebra(QQ,bosondict,names=('a',),central_elements=('K',), weights=(Integer(1),)) >>> R.inject_variables() Defining a, K >>> a.T(Integer(3)).degree() 4 >>> K.degree() 0 >>> R.category() Category of H-graded finitely generated Lie conformal algebras with basis over Rational Field