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:
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)[source]#
Bases:
GradedLieConformalAlgebra
The Fermionic ghosts or \(bc\)-system super Lie conformal algebra.
INPUT:
R
– a commutative ring; the base ring of this Lie conformal algebrangens
– an even positive Integer (default:2
); The number of non-central generators of this Lie conformal algebra.names
– a tuple ofstr
; alternative names for the generatorsindex_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),)}
>>> from sage.all import * >>> R = lie_conformal_algebras.FermionicGhosts(QQ); R The Fermionic ghosts Lie conformal algebra with generators (b, c, K) over Rational Field >>> R.inject_variables() Defining b, c, K >>> b.bracket(c) == c.bracket(b) True >>> b.degree() 1 >>> c.degree() 0 >>> R.category() Category of H-graded super finitely generated Lie conformal algebras with basis over Rational Field >>> R = lie_conformal_algebras.FermionicGhosts(QQbar, ngens=Integer(4), names = 'abcd');R The Fermionic ghosts Lie conformal algebra with generators (a, b, c, d, K) over Algebraic Field >>> R.structure_coefficients() Finite family {('a', 'c'): ((0, K),), ('b', 'd'): ((0, K),), ('c', 'a'): ((0, K),), ('d', 'b'): ((0, K),)}