# Fermionic Ghosts Super Lie Conformal Algebra¶

The Fermionic-ghosts or b–c system super Lie conformal algebra with $$2n$$ generators is the H-graded super Lie conformal algebra generated by odd vectors $$b_i, c_i, i = 1,\ldots,n$$ and a central element $$K$$, with non-vanishing $$\lambda$$-brackets:

$[{b_i}_\lambda c_j] = \delta_{ij} K.$

The generators $$b_i$$ have degree $$1$$ while the generators $$c_i$$ have degree $$0$$.

AUTHORS:

• Reimundo Heluani (2020-06-03): Initial implementation.

class sage.algebras.lie_conformal_algebras.fermionic_ghosts_lie_conformal_algebra.FermionicGhostsLieConformalAlgebra(R, ngens=2, names=None, index_set=None)

The Fermionic ghosts or $$bc$$-system super 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.

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

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

OUTPUT:

The Fermionic Ghosts super Lie conformal algebra with generators $$b_i,c_i, i=1,\ldots,n$$ and $$K$$ where $$2n$$ is ngens.

EXAMPLES:

sage: R = lie_conformal_algebras.FermionicGhosts(QQ); R
The Fermionic ghosts Lie conformal algebra with generators (b, c, K) over Rational Field
sage: R.inject_variables()
Defining b, c, K
sage: b.bracket(c) == c.bracket(b)
True
sage: b.degree()
1
sage: c.degree()
0
sage: R.category()
Category of H-graded super finitely generated Lie conformal algebras with basis over Rational Field

sage: R = lie_conformal_algebras.FermionicGhosts(QQbar, ngens=4, names = 'abcd');R
The Fermionic 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),)}