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– 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–- Noneor a list of- 0or- 1(default:- None); the parity of the generators. If not- Nonethe 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: 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.