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 ringn
– (optional) a nonnegative integer (default:None
)M
– an \(R\)-module (default: the free \(R\)-module of rankn
) to serve as the ground space for the Lie algebraambient
– (optional) a Lie algebra; if this is set, then the resulting Lie algebra is declared a Lie subalgebra ofambient
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 ringn
– (optional) a nonnegative integer (default:None
)M
– an \(R\)-module (default: the free \(R\)-module of rankn
) to serve as the ground space for the Lie algebraambient
– (optional) a Lie algebra; if this is set, then the resulting Lie algebra is declared a Lie subalgebra ofambient
OUTPUT:
The abelian Lie algebra on \(M\).