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:

$[a_\lambda b] = \sum \frac{\lambda^n}{n!} c_n,$

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

$[L_\lambda a] = Ta + \lambda \Delta_a a + O(\lambda^2).$

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)

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 – a dictionary (default: None); as in the input of LieConformalAlgebra

• names – tuple of str (default: None); as in the input of LieConformalAlgebra

• central_elements – tuple of str (default: None); as in the input of LieConformalAlgebra

• index_set – enumerated set (default: None); as in the input of LieConformalAlgebra

• weights – tuple of non-negative 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_set or names. Central elements are assumed to have weight 0.

• category The category that this Lie conformal algebra belongs to.

• parity – tuple of 0 or 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