DerivedAlgebra - Maple Help

LieAlgebras[DerivedAlgebra] - find the derived algebra of a Lie algebra

Calling Sequences

DerivedAlgebra(LieAlgName)

DerivedAlgebra(S)

Parameters

LieAlgName  - (optional) name or string, the name of a Lie algebra $\mathrm{𝔤}$

S           - a list of vectors defining a basis for a subalgebra of $\mathrm{𝔤}$

Description

 • The derived algebra of a Lie algebra is the span of the set of vectors for all . The derived algebra is an ideal in $\mathrm{𝔤}$.
 • DerivedAlgebra(LieAlgName) calculates the derived algebra of the Lie algebra 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 $\mathrm{𝔤}$ (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(...).

Examples

 > $\mathrm{with}\left(\mathrm{DifferentialGeometry}\right):$$\mathrm{with}\left(\mathrm{LieAlgebras}\right):$

Example 1.

First we initialize a Lie algebra.

 > $\mathrm{L1}≔\mathrm{_DG}\left(\left[\left["LieAlgebra",\mathrm{Alg1},\left[4\right]\right],\left[\left[\left[2,4,1\right],1\right],\left[\left[3,4,3\right],1\right]\right]\right]\right)$
 ${\mathrm{L1}}{≔}\left[\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e3}}\right]$ (2.1)
 > $\mathrm{DGsetup}\left(\mathrm{L1}\right):$

We calculate the derived algebra of Alg1.

 Alg1 > $\mathrm{DerivedAlgebra}\left(\right)$
 $\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]$ (2.2)

We calculate the derived algebra of the subalgebra [e1, e2, e4].

 Alg1 > $\mathrm{DerivedAlgebra}\left(\left[\mathrm{e1},\mathrm{e2},\mathrm{e4}\right]\right)$
 $\left[{\mathrm{e1}}\right]$ (2.3)