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 \(c_n \in V_{p+qn1}\). 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 (20190809): 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)¶

An HGraded 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 zeros_coeff
– a dictionary (default:None
); as in the input ofLieConformalAlgebra
names
– tuple ofstr
(default:None
); as in the input ofLieConformalAlgebra
central_elements
– tuple ofstr
(default:None
); as in the input ofLieConformalAlgebra
index_set
– enumerated set (default:None
); as in the input ofLieConformalAlgebra
weights
– tuple of nonnegative rational numbers (default: tuple of1
); a list of degrees for this Lie conformal algebra. This tuple needs to have the same cardinality asindex_set
ornames
. Central elements are assumed to have weight0
.category
The category that this Lie conformal algebra belongs to.parity
– tuple of0
or1
(Default: tuple of0
); a tuple specifying the parity of each noncentral 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 Hgraded finitely generated Lie conformal algebras with basis over Rational Field