Free Bosons Lie Conformal Algebra - Algebras

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# Free Bosons Lie Conformal Algebra

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

\[[v_\lambda w] = \lambda (v,w)K,\]

where \(v,w \in M\).

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

AUTHORS:

- Reimundo Heluani (2019-08-09): Initial implementation.

class sage.algebras.lie_conformal_algebras.free_bosons_lie_conformal_algebra.FreeBosonsLieConformalAlgebra(R, ngens=None, gram_matrix=None, names=None, index_set=None)

  Bases: `GradedLieConformalAlgebra`

  The Free Bosons 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

  - `names` – tuple of strings; alternative names for the generators

  - `index_set` – an enumerated set; alternative indexing set for the generators

  OUTPUT:

  The Free Bosons Lie conformal algebra with generators

    \(\alpha_i\), \(i=1,...,n\) and \(\lambda\)-brackets

    \[[{\alpha_i}_{\lambda} \alpha_j] = \lambda M_{ij} K,\]

  where \(n\) is the number of generators `ngens` and \(M\) is the `gram_matrix`. This Lie conformal algebra is \(H\)-graded where every generator has conformal weight \(1\).

  EXAMPLES:

  Sage

      sage: R = lie_conformal_algebras.FreeBosons(AA); R
      The free Bosons Lie conformal algebra with generators (alpha, K) over Algebraic Real Field
      sage: R.inject_variables()
      Defining alpha, K
      sage: alpha.bracket(alpha)
      {1: K}
      sage: M = identity_matrix(QQ,2); R = lie_conformal_algebras.FreeBosons(QQ,gram_matrix=M, names='alpha,beta'); R
      The free Bosons Lie conformal algebra with generators (alpha, beta, K) over Rational Field
      sage: R.inject_variables(); alpha.bracket(beta)
      Defining alpha, beta, K
      {}
      sage: alpha.bracket(alpha)
      {1: K}
      sage: R = lie_conformal_algebras.FreeBosons(QQbar, ngens=3); R
      The free Bosons Lie conformal algebra with generators (alpha0, alpha1, alpha2, K) over Algebraic Field

  Python

      >>> from sage.all import *
      >>> R = lie_conformal_algebras.FreeBosons(AA); R
      The free Bosons Lie conformal algebra with generators (alpha, K) over Algebraic Real Field
      >>> R.inject_variables()
      Defining alpha, K
      >>> alpha.bracket(alpha)
      {1: K}
      >>> M = identity_matrix(QQ,Integer(2)); R = lie_conformal_algebras.FreeBosons(QQ,gram_matrix=M, names='alpha,beta'); R
      The free Bosons Lie conformal algebra with generators (alpha, beta, K) over Rational Field
      >>> R.inject_variables(); alpha.bracket(beta)
      Defining alpha, beta, K
      {}
      >>> alpha.bracket(alpha)
      {1: K}
      >>> R = lie_conformal_algebras.FreeBosons(QQbar, ngens=Integer(3)); R
      The free Bosons Lie conformal algebra with generators (alpha0, alpha1, alpha2, K) over Algebraic Field

  gram_matrix()

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

    EXAMPLES:

    Sage

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

    Python

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