LieAlgebras[DerivedAlgebra] - find the derived algebra of a Lie algebra
LieAlgName - (optional) name or string, the name of a Lie algebra 𝔤
S - a list of vectors defining a basis for a subalgebra of 𝔤
The derived algebra of a Lie algebra 𝔤 is the span of the set of vectors x, y for all x,y∈ 𝔤. The derived algebra is an ideal in 𝔤.
DerivedAlgebra(LieAlgName) calculates the derived algebra of the Lie algebra 𝔤 defined by LieAlgName. If no argument is given, then the derived algebra of the current Lie algebra is found.
DerivedAlgebra(S) calculates the derived algebra of the Lie subalgebra S (viewed as a Lie algebra in its own right).
A list of vectors defining a basis for the derived algebra of 𝔤 (or S) is returned. If the derived algebra is trivial, then an empty list is returned.
The command DerivedAlgebra is part of the DifferentialGeometry:-LieAlgebras package. It can be used in the form DerivedAlgebra(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:-LieAlgebras:-DerivedAlgebra(...).
First we initialize a Lie algebra.
L1 ≔ e2,e4=e1,e3,e4=e3
We calculate the derived algebra of Alg1.
We calculate the derived algebra of the subalgebra [e1, e2, e4].
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