Free Fermions Super Lie Conformal Algebra. - Algebras

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# 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)

  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()

    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]