Free Fermions Super Lie Conformal Algebra.¶

Given an $$R$$-module $$M$$ with a skew-symmetric, bilinear pairing $$\langle\cdot, \cdot\rangle: M\otimes_R M \rightarrow R$$. The Free Fermions super Lie conformal algebra associated to this datum is the free $$R[T]$$-super module generated by $$\Pi M$$ (a purely odd copy of $$M$$) plus a central vector $$K$$ satisfying $$TK=0$$. The remaining $$\lambda$$-brackets are given by:

$[v_\lambda w] = \langle v,w \rangle K,$

where $$v,w \in M$$.

This is an H-graded Lie conformal algebra where every generator $$v \in M$$ has degree $$1/2$$.

AUTHORS:

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

class sage.algebras.lie_conformal_algebras.free_fermions_lie_conformal_algebra.FreeFermionsLieConformalAlgebra(R, ngens=None, gram_matrix=None, names=None, index_set=None)

The Free Fermions Super Lie conformal algebra.

INPUT:

• R: a commutative ring.

• ngens: a positive Integer (default 1); the number of non-central generators of this Lie conformal algebra.

• gram_matrix: a symmetric square matrix with coefficients in R (default: identity_matrix(ngens)); the Gram matrix of the inner product

OUTPUT:

The Free Fermions Lie conformal algebra with generators

$$\psi_i$$, $$i=1,...,n$$ and $$\lambda$$-brackets

$[{\psi_i}_{\lambda} \psi_j] = M_{ij} K,$

where $$n$$ is the number of generators ngens and $$M$$ is the gram_matrix. This super Lie conformal algebra is $$H$$-graded where every generator has degree $$1/2$$.

EXAMPLES:

sage: R = lie_conformal_algebras.FreeFermions(QQbar); R
The free Fermions super Lie conformal algebra with generators (psi, K) over Algebraic Field
sage: R.inject_variables()
Defining psi, K
sage: psi.bracket(psi)
{0: K}

sage: R = lie_conformal_algebras.FreeFermions(QQbar,gram_matrix=Matrix([[0,1],[1,0]])); R
The free Fermions super Lie conformal algebra with generators (psi_0, psi_1, K) over Algebraic Field
sage: R.inject_variables()
Defining psi_0, psi_1, K
sage: psi_0.bracket(psi_1)
{0: K}
sage: psi_0.degree()
1/2
sage: R.category()
Category of H-graded super finitely generated Lie conformal algebras with basis over Algebraic Field

gram_matrix()

The Gram matrix that specifies the $$\lambda$$-brackets of the generators.

EXAMPLES:

sage: R = lie_conformal_algebras.FreeFermions(QQ,ngens=2);
sage: R.gram_matrix()
[1 0]
[0 1]