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)[source]#

Bases: GradedLieConformalAlgebra

The Abelian Lie conformal algebra.

INPUT:

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

  • ngens – a positive integer (default: 1); the number of generators of this Lie conformal algebra

  • weights – a 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 – a tuple of str 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: 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))
{}

>>> 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.