# Neveu-Schwarz Super Lie Conformal Algebra#

The $$N=1$$ or Neveu-Schwarz super Lie conformal algebra is a super extension of the Virasoro Lie conformal algebra with generators $$L$$ and $$C$$ by an odd primary generator $$G$$ of conformal weight $$3/2$$. The remaining $$\lambda$$-bracket is given by:

$[G_\lambda G] = 2L + \frac{\lambda^2}{3} C.$

AUTHORS:

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

class sage.algebras.lie_conformal_algebras.neveu_schwarz_lie_conformal_algebra.NeveuSchwarzLieConformalAlgebra(R)[source]#

The Neveu-Schwarz super Lie conformal algebra.

INPUT:

• R – a commutative Ring; the base ring of this Lie conformal algebra.

EXAMPLES:

sage: R = lie_conformal_algebras.NeveuSchwarz(AA); R
The Neveu-Schwarz super Lie conformal algebra over Algebraic Real Field
sage: R.structure_coefficients()
Finite family {('G', 'G'): ((0, 2*L), (2, 2/3*C)),  ('G', 'L'): ((0, 1/2*TG), (1, 3/2*G)),  ('L', 'G'): ((0, TG), (1, 3/2*G)),  ('L', 'L'): ((0, TL), (1, 2*L), (3, 1/2*C))}
sage: R.inject_variables()
Defining L, G, C
sage: G.nproduct(G,0)
2*L
sage: G.degree()
3/2

>>> from sage.all import *
>>> R = lie_conformal_algebras.NeveuSchwarz(AA); R
The Neveu-Schwarz super Lie conformal algebra over Algebraic Real Field
>>> R.structure_coefficients()
Finite family {('G', 'G'): ((0, 2*L), (2, 2/3*C)),  ('G', 'L'): ((0, 1/2*TG), (1, 3/2*G)),  ('L', 'G'): ((0, TG), (1, 3/2*G)),  ('L', 'L'): ((0, TL), (1, 2*L), (3, 1/2*C))}
>>> R.inject_variables()
Defining L, G, C
>>> G.nproduct(G,Integer(0))
2*L
>>> G.degree()
3/2