# Weyl Lie Conformal Algebra¶

Given a commutative ring $$R$$, a free $$R$$-module $$M$$ and a non-degenerate, skew-symmetric, bilinear pairing $$\langle \cdot,\cdot\rangle: M \otimes_R M \rightarrow R$$. The Weyl Lie conformal algebra associated to this datum is the free $$R[T]$$-module generated by $$M$$ plus a central vector $$K$$. The non-vanishing $$\lambda$$-brackets are given by:

$[v_\lambda w] = \langle v, w\rangle K.$

This is not an H-graded Lie conformal algebra. The choice of a Lagrangian decomposition $$M = L \oplus L^*$$ determines an H-graded structure. For this H-graded Lie conformal algebra see the Bosonic Ghosts Lie conformal algebra

AUTHORS:

• Reimundo Heluani (2019-08-09): Initial implementation.

class sage.algebras.lie_conformal_algebras.weyl_lie_conformal_algebra.WeylLieConformalAlgebra(R, ngens=None, gram_matrix=None, names=None, index_set=None)

The Weyl Lie conformal algebra.

INPUT:

• R – a commutative ring; the base ring of this Lie conformal algebra.

• ngens: an even positive Integer (default $$2$$); The number of non-central generators of this Lie conformal algebra.

• gram_matrix: a matrix (default: None); A non-singular skew-symmetric square matrix with coefficients in $$R$$.

• names – a list or tuple of str; alternative names for the generators

• index_set – an enumerated set; alternative indexing set for the generators

OUTPUT:

The Weyl Lie conformal algebra with generators

$$\alpha_i$$, $$i=1,...,ngens$$ and $$\lambda$$-brackets

$[{\alpha_i}_{\lambda} \alpha_j] = M_{ij} K,$

where $$M$$ is the gram_matrix above.

Note

The returned Lie conformal algebra is not $$H$$-graded. For a related $$H$$-graded Lie conformal algebra see BosonicGhostsLieConformalAlgebra.

EXAMPLES:

sage: lie_conformal_algebras.Weyl(QQ)
The Weyl Lie conformal algebra with generators (alpha0, alpha1, K) over Rational Field
sage: R = lie_conformal_algebras.Weyl(QQbar, gram_matrix=Matrix(QQ,[[0,1],[-1,0]]), names = ('a','b'))
sage: R.inject_variables()
Defining a, b, K
sage: a.bracket(b)
{0: K}
sage: b.bracket(a)
{0: -K}

sage: R = lie_conformal_algebras.Weyl(QQbar, ngens=4)
sage: R.gram_matrix()
[ 0  0| 1  0]
[ 0  0| 0  1]
[-----+-----]
[-1  0| 0  0]
[ 0 -1| 0  0]
sage: R.inject_variables()
Defining alpha0, alpha1, alpha2, alpha3, K
sage: alpha0.bracket(alpha2)
{0: K}

sage: R = lie_conformal_algebras.Weyl(QQ); R.category()
Category of finitely generated Lie conformal algebras with basis over Rational Field
False
sage: R.inject_variables()
Defining alpha0, alpha1, K
sage: alpha0.degree()
Traceback (most recent call last):
...
AttributeError: 'WeylLieConformalAlgebra_with_category.element_class' object has no attribute 'degree'

gram_matrix()

The Gram matrix that specifies the $$\lambda$$-brackets of the generators.

EXAMPLES:

sage: R = lie_conformal_algebras.Weyl(QQbar, ngens=4)
sage: R.gram_matrix()
[ 0  0| 1  0]
[ 0  0| 0  1]
[-----+-----]
[-1  0| 0  0]
[ 0 -1| 0  0]