Abelian Lie Conformal Algebra - Algebras

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# Abelian Lie Conformal Algebra

For a commutative ring \(R\) and a free \(R\)-module \(M\). The Abelian Lie conformal algebra generated by \(M\) is the free \(R[T]\) module generated by \(M\) with vanishing \(\lambda\)-brackets.

AUTHORS:

- Reimundo Heluani (2020-06-15): Initial implementation.

class sage.algebras.lie_conformal_algebras.abelian_lie_conformal_algebra.AbelianLieConformalAlgebra(R, ngens=1, weights=None, parity=None, names=None, index_set=None)

  Bases: `GradedLieConformalAlgebra`

  The Abelian Lie conformal algebra.

  INPUT:

  - `R` – a commutative ring; the base ring of this Lie conformal algebra

  - `ngens` – positive integer (default: \(1\)); the number of generators of this Lie conformal algebra

  - `weights` – list of positive rational numbers (default: \(1\) for each generator); the weights of the generators. The resulting Lie conformal algebra is \(H\)-graded.

  - `parity` – `None` or a list of `0` or `1` (default: `None`); the parity of the generators. If not `None` the resulting Lie Conformal algebra is a Super Lie conformal algebra

  - `names` – tuple of strings or `None` (default: `None`); the list of names of the generators of this algebra.

  - `index_set` – an enumerated set or `None` (default: `None`); a set indexing the generators of this Lie conformal algebra

  OUTPUT:

  The Abelian Lie conformal algebra with generators \(a_i\), \(i=1,...,n\) and vanishing \(\lambda\)-brackets, where \(n\) is `ngens`.

  EXAMPLES:

  Sage

      sage: R = lie_conformal_algebras.Abelian(QQ,2); R
      The Abelian Lie conformal algebra with generators (a0, a1) over Rational Field
      sage: R.inject_variables()
      Defining a0, a1
      sage: a0.bracket(a1.T(2))
      {}

  Python

      >>> from sage.all import *
      >>> R = lie_conformal_algebras.Abelian(QQ,Integer(2)); R
      The Abelian Lie conformal algebra with generators (a0, a1) over Rational Field
      >>> R.inject_variables()
      Defining a0, a1
      >>> a0.bracket(a1.T(Integer(2)))
      {}

  Todo

  implement its own class to speed up arithmetics in this case.