Examples of a finite dimensional Lie algebra with basis

sage.categories.examples.finite_dimensional_lie_algebras_with_basis.AbelianLieAlgebra

An example of a finite dimensional Lie algebra with basis: the abelian Lie algebra.

Let \(R\) be a commutative ring, and \(M\) an \(R\)-module. The abelian Lie algebra on \(M\) is the \(R\)-Lie algebra obtained by endowing \(M\) with the trivial Lie bracket (\([a, b] = 0\) for all \(a, b \in M\)).

This class illustrates a minimal implementation of a finite dimensional Lie algebra with basis.

INPUT:

  • R – base ring
  • n – (optional) a nonnegative integer (default: None)
  • M – an \(R\)-module (default: the free \(R\)-module of rank n) to serve as the ground space for the Lie algebra
  • ambient – (optional) a Lie algebra; if this is set, then the resulting Lie algebra is declared a Lie subalgebra of ambient

OUTPUT:

The abelian Lie algebra on \(M\).

sage.categories.examples.finite_dimensional_lie_algebras_with_basis.Example

An example of a finite dimensional Lie algebra with basis: the abelian Lie algebra.

Let \(R\) be a commutative ring, and \(M\) an \(R\)-module. The abelian Lie algebra on \(M\) is the \(R\)-Lie algebra obtained by endowing \(M\) with the trivial Lie bracket (\([a, b] = 0\) for all \(a, b \in M\)).

This class illustrates a minimal implementation of a finite dimensional Lie algebra with basis.

INPUT:

  • R – base ring
  • n – (optional) a nonnegative integer (default: None)
  • M – an \(R\)-module (default: the free \(R\)-module of rank n) to serve as the ground space for the Lie algebra
  • ambient – (optional) a Lie algebra; if this is set, then the resulting Lie algebra is declared a Lie subalgebra of ambient

OUTPUT:

The abelian Lie algebra on \(M\).