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:
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 ofstr
; alternative names for the generatorsindex_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'
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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]