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:
GradedLieConformalAlgebraThe Abelian Lie conformal algebra.
INPUT:
R– a commutative ring; the base ring of this Lie conformal algebrangens– positive integer (default: \(1\)); the number of generators of this Lie conformal algebraweights– 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 of0or1(default:None); the parity of the generators. If notNonethe resulting Lie Conformal algebra is a Super Lie conformal algebranames– tuple of strings orNone(default:None); the list of names of the generators of this algebra.index_set– an enumerated set orNone(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.