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:

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

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)#

Bases: sage.algebras.lie_conformal_algebras.lie_conformal_algebra_with_structure_coefs.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 – 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