Bosonic Ghosts Lie Conformal Algebra#

The Bosonic-ghosts or \(\beta-\gamma\)-system Lie conformal algebra with \(2n\) generators is the H-graded Lie conformal algebra generated by \(\beta_i, \gamma_i, i = 1,\ldots,n\) and a central element \(K\), with non-vanishing \(\lambda\)-brackets:

\[[{\beta_i}_\lambda \gamma_j] = \delta_{ij} K.\]

The generators \(\beta_i\) have degree \(1\) while the generators \(\gamma_i\) have degree \(0\).


  • Reimundo Heluani (2020-06-15): Initial implementation.

class sage.algebras.lie_conformal_algebras.bosonic_ghosts_lie_conformal_algebra.BosonicGhostsLieConformalAlgebra(R, ngens=2, names=None, index_set=None)#

Bases: GradedLieConformalAlgebra

The Bosonic ghosts or \(\beta-\gamma\)-system Lie conformal algebra.


  • R – a commutative ring.

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

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

  • index_set – an enumerated set; An indexing set for the generators.


The Bosonic Ghosts Lie conformal algebra with generators \(\beta_i,\gamma_i, i=1,\ldots,n\) and \(K\), where \(2n\) is ngens.


sage: R = lie_conformal_algebras.BosonicGhosts(QQ); R
The Bosonic ghosts Lie conformal algebra with generators (beta, gamma, K) over Rational Field
sage: R.inject_variables(); beta.bracket(gamma)
Defining beta, gamma, K
{0: K}

sage: R = lie_conformal_algebras.BosonicGhosts(QQbar, ngens = 4, names = 'abcd'); R
The Bosonic ghosts Lie conformal algebra with generators (a, b, c, d, K) over Algebraic Field
sage: R.structure_coefficients()
Finite family {('a', 'c'): ((0, K),),  ('b', 'd'): ((0, K),),  ('c', 'a'): ((0, -K),),  ('d', 'b'): ((0, -K),)}