# 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)

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.

• parityNone 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))
{}


Todo

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