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)[source]#

Bases: GradedLieConformalAlgebra

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

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

Python

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

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

gram_matrix()[source]#

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

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> R = lie_conformal_algebras.FreeFermions(QQ,ngens=Integer(2));
>>> R.gram_matrix()
[1 0]
[0 1]