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)#

Bases: LieConformalAlgebraWithStructureCoefficients

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
sage: R in LieConformalAlgebras(QQ).Graded()
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]