Lesson 6: Decompositions of Lie Algebras - Maple Help

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LieAlgebra Lessons

Lesson 6:  Decompositions

Overview

In this lesson you will learn to do the following.

 – Determine if a Lie algebra is decomposable into a direct sum of 2 or more Lie algebras and, if so, explicitly find the decomposition.
 – Calculate a Levi decomposition of a Lie algebra into a semi-direct sum of its radical and a semi-simple Lie algebra.

Decompose a Lie into a direct sum of Lie algebras

An (internal) direct sum decomposition of a Lie algebra g is given by a collection of subalgebras g1, g2, ..., gN such that [i] g = g1 + g2 + ... + gN, [ii] the intersection of any two distinct summands gi and gj, i <> j is trivial, and [iii] the Lie bracket of any two distinct gi and gj is trivial, that is, [gi, gj] = 0.

The decomposition of a Lie algebra can obtained using the Decompose command.  Note that the Decompose command  admits a number of keyword optional arguments which are described in detail in the help page.  In section we shall define a define a 7 dimensional Lie algebra Alg and use the Decompose command to

show that Alg is a direct sum of 3 dimensional Lie algebra which is semi-simple, a 2 dimensional solvable algebra, and a 2 dimensional Abelian algebra.

 > with(DifferentialGeometry): with(LieAlgebras):

Define and initialize the 7 dimensional algebra Alg.

 > L := _DG([["LieAlgebra", Alg1, [7]], [[[1, 2, 1], 2], [[1, 2, 2], 1], [[1, 2, 3], 3], [[1, 2, 4], 3], [[1, 2, 5], 1], [[1, 2, 6], -2], [[1, 2, 7], -4], [[1, 4, 5], 1/3], [[1, 4, 6], -2/3], [[1, 5, 4], -4], [[1, 6, 4], 1], [[1, 7, 4], -1], [[1, 7, 5], 1/3], [[1, 7, 6], -2/3], [[2, 3, 1], 2], [[2, 3, 2], 1], [[2, 3, 3], 3], [[2, 3, 4], 3], [[2, 3, 5], 2/3], [[2, 3, 6], -4/3], [[2, 3, 7], -4], [[2, 5, 1], 4], [[2, 5, 3], 4], [[2, 5, 4], 4], [[2, 5, 5], 1], [[2, 5, 6], -2], [[2, 5, 7], -4], [[2, 6, 1], -1], [[2, 6, 3], -1], [[2, 6, 4], -1], [[2, 6, 7], 1], [[2, 7, 1], 1], [[2, 7, 3], 1], [[2, 7, 4], 1], [[2, 7, 7], -1], [[4, 5, 1], 4], [[4, 5, 3], 4], [[4, 5, 4], 4], [[4, 5, 5], 1], [[4, 5, 6], -2], [[4, 5, 7], -4], [[4, 6, 1], -1], [[4, 6, 3], -1], [[4, 6, 4], -1], [[4, 6, 7], 1], [[4, 7, 1], 1], [[4, 7, 3], 1], [[4, 7, 4], 1], [[4, 7, 7], -1], [[5, 6, 4], -3], [[5, 7, 1], -4], [[5, 7, 3], -4], [[5, 7, 4], -1], [[5, 7, 5], -1], [[5, 7, 6], 2], [[5, 7, 7], 4], [[6, 7, 1], 1], [[6, 7, 3], 1], [[6, 7, 4], 1], [[6, 7, 7], -1]]]);
 ${L}{≔}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{2}{}{\mathrm{e1}}{+}{\mathrm{e2}}{+}{3}{}{\mathrm{e3}}{+}{3}{}{\mathrm{e4}}{+}{\mathrm{e5}}{-}{2}{}{\mathrm{e6}}{-}{4}{}{\mathrm{e7}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}\frac{{\mathrm{e5}}}{{3}}{-}\frac{{2}{}{\mathrm{e6}}}{{3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{-}{4}{}{\mathrm{e4}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e4}}{+}\frac{{\mathrm{e5}}}{{3}}{-}\frac{{2}{}{\mathrm{e6}}}{{3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{2}{}{\mathrm{e1}}{+}{\mathrm{e2}}{+}{3}{}{\mathrm{e3}}{+}{3}{}{\mathrm{e4}}{+}\frac{{2}{}{\mathrm{e5}}}{{3}}{-}\frac{{4}{}{\mathrm{e6}}}{{3}}{-}{4}{}{\mathrm{e7}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{4}{}{\mathrm{e1}}{+}{4}{}{\mathrm{e3}}{+}{4}{}{\mathrm{e4}}{+}{\mathrm{e5}}{-}{2}{}{\mathrm{e6}}{-}{4}{}{\mathrm{e7}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e1}}{-}{\mathrm{e3}}{-}{\mathrm{e4}}{+}{\mathrm{e7}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e1}}{+}{\mathrm{e3}}{+}{\mathrm{e4}}{-}{\mathrm{e7}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{4}{}{\mathrm{e1}}{+}{4}{}{\mathrm{e3}}{+}{4}{}{\mathrm{e4}}{+}{\mathrm{e5}}{-}{2}{}{\mathrm{e6}}{-}{4}{}{\mathrm{e7}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e1}}{-}{\mathrm{e3}}{-}{\mathrm{e4}}{+}{\mathrm{e7}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e1}}{+}{\mathrm{e3}}{+}{\mathrm{e4}}{-}{\mathrm{e7}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e6}}\right]{=}{-}{3}{}{\mathrm{e4}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e7}}\right]{=}{-}{4}{}{\mathrm{e1}}{-}{4}{}{\mathrm{e3}}{-}{\mathrm{e4}}{-}{\mathrm{e5}}{+}{2}{}{\mathrm{e6}}{+}{4}{}{\mathrm{e7}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e1}}{+}{\mathrm{e3}}{+}{\mathrm{e4}}{-}{\mathrm{e7}}\right]$ (2.1)
 > DGsetup(L):

We can use the Query command to determine that the algebra is decomposable.

 Alg1 > Query("Indecomposable");
 ${\mathrm{false}}$ (2.2)

To actually find the direct sum decomposition, we use the Decompose command. With the optional argument factoralgebras = true, the Decompose procedure will return the Lie algebra data structures for each Lie algebra summand in the direct sum decomposition.

 Alg1 > decomposition := Decompose(factoralgebras = true);
 ${\mathrm{decomposition}}{≔}\left[\left[\left[\begin{array}{ccccccc}{2}& {0}& {0}& {0}& {-}{2}& {-}{1}& {1}\\ {2}& {1}& {0}& {1}& {-}{2}& {-}{1}& {2}\\ \frac{{1}}{{3}}& {0}& {0}& {0}& \frac{{1}}{{3}}& {-}\frac{{1}}{{3}}& \frac{{1}}{{3}}\\ {-}{1}& {1}& {1}& {0}& {0}& {0}& {0}\\ {0}& {1}& {0}& {0}& {0}& {0}& {0}\\ {0}& {-}{1}& {-}{1}& {0}& {2}& {1}& {-}{1}\\ {0}& {-}{2}& {-}{2}& {0}& {2}& {1}& {-}{2}\end{array}\right]{,}\left[\left[{\mathrm{e1}}{+}{\mathrm{e3}}{-}{\mathrm{e7}}{,}{\mathrm{e4}}{,}{\mathrm{e5}}{-}{2}{}{\mathrm{e6}}{,}{\mathrm{e1}}{+}{2}{}{\mathrm{e3}}{+}{2}{}{\mathrm{e4}}{+}\frac{{\mathrm{e5}}}{{3}}{-}\frac{{2}{}{\mathrm{e6}}}{{3}}{-}{2}{}{\mathrm{e7}}{,}{\mathrm{e2}}{-}{\mathrm{e3}}{-}{\mathrm{e4}}{,}{\mathrm{e1}}{+}{\mathrm{e3}}{+}\frac{{\mathrm{e5}}}{{3}}{+}\frac{{4}{}{\mathrm{e6}}}{{3}}{,}{\mathrm{e4}}{-}{\mathrm{e6}}{-}{\mathrm{e7}}\right]\right]{,}\left[\left[\left[\left[\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]\right]{=}{\mathrm{e1}}{+}{\mathrm{e2}}{+}\frac{{\mathrm{e3}}}{{3}}{,}\left[\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]\right]{=}{-}{6}{}{\mathrm{e1}}{-}{9}{}{\mathrm{e2}}{-}{\mathrm{e3}}{,}\left[\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]\right]{=}{6}{}{\mathrm{e1}}{+}{6}{}{\mathrm{e2}}{+}{\mathrm{e3}}\right]\right]{,}\left[\left[\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]\right]{=}{-}{2}{}{\mathrm{e1}}{-}{\mathrm{e2}}\right]\right]{,}\left[\left[{}\right]\right]\right]\right]\right]\right]$ (2.3)

The output here is a list consisting of three entries. The first entry in the list is a matrix which defines a Lie algebra isomorphism from Alg1 to the Lie algebra defined by the basis given in the second entry. It is in this basis that the Lie algebra decomposes into a direct sum.  The third entry is the list of Lie algebra data structures for each of the summands.  There are a variety of ways one can verify that this decomposition is correct.

One approach is to calculate the structure equations for the Lie algebra with respect to the basis decomposition[2], initialize the result and verify that the Lie algebra is a direct sum in this new basis.

 Alg1 > L2 := LieAlgebraData(decomposition[2], Alg2);
 ${\mathrm{L2}}{≔}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{\mathrm{e1}}{+}{\mathrm{e2}}{+}\frac{{\mathrm{e3}}}{{3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{-}{6}{}{\mathrm{e1}}{-}{9}{}{\mathrm{e2}}{-}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{6}{}{\mathrm{e1}}{+}{6}{}{\mathrm{e2}}{+}{\mathrm{e3}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{-}{2}{}{\mathrm{e4}}{-}{\mathrm{e5}}\right]$ (2.4)
 Alg1 > DGsetup(L2, [f], [omega]);
 ${\mathrm{Lie algebra: Alg2}}$ (2.5)
 Alg2 > MultiplicationTable("LieTable");
 $\left[\begin{array}{ccccccccc}{}& {\mathrm{|}}& {\mathrm{f1}}& {\mathrm{f2}}& {\mathrm{f3}}& {\mathrm{f4}}& {\mathrm{f5}}& {\mathrm{f6}}& {\mathrm{f7}}\\ {}& {\mathrm{---}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}\\ {\mathrm{f1}}& {\mathrm{|}}& {0}& {\mathrm{f1}}{+}{\mathrm{f2}}{+}\frac{{\mathrm{f3}}}{{3}}& {-}{6}{}{\mathrm{f1}}{-}{9}{}{\mathrm{f2}}{-}{\mathrm{f3}}& {0}& {0}& {0}& {0}\\ {\mathrm{f2}}& {\mathrm{|}}& {-}{\mathrm{f1}}{-}{\mathrm{f2}}{-}\frac{{\mathrm{f3}}}{{3}}& {0}& {6}{}{\mathrm{f1}}{+}{6}{}{\mathrm{f2}}{+}{\mathrm{f3}}& {0}& {0}& {0}& {0}\\ {\mathrm{f3}}& {\mathrm{|}}& {6}{}{\mathrm{f1}}{+}{9}{}{\mathrm{f2}}{+}{\mathrm{f3}}& {-}{6}{}{\mathrm{f1}}{-}{6}{}{\mathrm{f2}}{-}{\mathrm{f3}}& {0}& {0}& {0}& {0}& {0}\\ {\mathrm{f4}}& {\mathrm{|}}& {0}& {0}& {0}& {0}& {-}{2}{}{\mathrm{f4}}{-}{\mathrm{f5}}& {0}& {0}\\ {\mathrm{f5}}& {\mathrm{|}}& {0}& {0}& {0}& {2}{}{\mathrm{f4}}{+}{\mathrm{f5}}& {0}& {0}& {0}\\ {\mathrm{f6}}& {\mathrm{|}}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {\mathrm{f7}}& {\mathrm{|}}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\end{array}\right]$ (2.6)

From the multiplication table we see that Alg2, which is same abstract Lie algebra as Alg1(just given in a different basis) is the direct sum of the

subalgebras S1, S2, S3 defined by:

 Alg2 > S1 := [f1, f2, f3];
 ${\mathrm{S1}}{≔}\left[{\mathrm{f1}}{,}{\mathrm{f2}}{,}{\mathrm{f3}}\right]$ (2.7)
 Alg2 > S2 := [f4 ,f5];
 ${\mathrm{S2}}{≔}\left[{\mathrm{f4}}{,}{\mathrm{f5}}\right]$ (2.8)
 Alg2 > S3 := [f6, f7];
 ${\mathrm{S3}}{≔}\left[{\mathrm{f6}}{,}{\mathrm{f7}}\right]$ (2.9)

We check that each of S1, S2, S3 is a subalgebra and that [Si, Sj] = 0 for i <>j.

 Alg2 > Query(S1, "Subalgebra"), Query(S2, "Subalgebra"), Query(S3, "Subalgebra");
 ${\mathrm{true}}{,}{\mathrm{true}}{,}{\mathrm{true}}$ (2.10)
 Alg2 > BracketOfSubspaces(S1,S2), BracketOfSubspaces(S1,S3),BracketOfSubspaces(S2,S3);
 $\left[{}\right]{,}\left[{}\right]{,}\left[{}\right]$ (2.11)

This can be all be checked with the Query command.

 Alg2 > Query([S1,S2,S3], "DirectSumDecomposition");
 ${\mathrm{true}}$ (2.12)

We can also use the Query command to verify that the matrix A =decomposition[1] is a Lie algebra homomorphism from Alg1 to Alg2

 Alg2 > Query(Alg1, Alg2, decomposition[1], "Homomorphism");
 ${\mathrm{true}}$ (2.13)
 Alg2 >

Note that this check can actually be performed in the original basis for Alg1:

 > B:= decomposition[2];
 ${B}{≔}\left[{\mathrm{e1}}{+}{\mathrm{e3}}{-}{\mathrm{e7}}{,}{\mathrm{e4}}{,}{\mathrm{e5}}{-}{2}{}{\mathrm{e6}}{,}{\mathrm{e1}}{+}{2}{}{\mathrm{e3}}{+}{2}{}{\mathrm{e4}}{+}\frac{{\mathrm{e5}}}{{3}}{-}\frac{{2}{}{\mathrm{e6}}}{{3}}{-}{2}{}{\mathrm{e7}}{,}{\mathrm{e2}}{-}{\mathrm{e3}}{-}{\mathrm{e4}}{,}{\mathrm{e1}}{+}{\mathrm{e3}}{+}\frac{{\mathrm{e5}}}{{3}}{+}\frac{{4}{}{\mathrm{e6}}}{{3}}{,}{\mathrm{e4}}{-}{\mathrm{e6}}{-}{\mathrm{e7}}\right]$ (2.14)
 Alg1 > T1 := B[1..3];
 ${\mathrm{T1}}{≔}\left[{\mathrm{e1}}{+}{\mathrm{e3}}{-}{\mathrm{e7}}{,}{\mathrm{e4}}{,}{\mathrm{e5}}{-}{2}{}{\mathrm{e6}}\right]$ (2.15)
 Alg1 > T2 := B[4..5];
 ${\mathrm{T2}}{≔}\left[{\mathrm{e1}}{+}{2}{}{\mathrm{e3}}{+}{2}{}{\mathrm{e4}}{+}\frac{{\mathrm{e5}}}{{3}}{-}\frac{{2}{}{\mathrm{e6}}}{{3}}{-}{2}{}{\mathrm{e7}}{,}{\mathrm{e2}}{-}{\mathrm{e3}}{-}{\mathrm{e4}}\right]$ (2.16)
 Alg1 > T3 := B[6..7];
 ${\mathrm{T3}}{≔}\left[{\mathrm{e1}}{+}{\mathrm{e3}}{+}\frac{{\mathrm{e5}}}{{3}}{+}\frac{{4}{}{\mathrm{e6}}}{{3}}{,}{\mathrm{e4}}{-}{\mathrm{e6}}{-}{\mathrm{e7}}\right]$ (2.17)
 Alg1 > Query([T1, T2, T3], "DirectSumDecomposition");
 ${\mathrm{true}}$ (2.18)

We can also use the third entry in the output of the Decompose command to construct the direct sum of the 3 given algebras. The resulting structure equations are

identical to the structure equations L2 for the basis decomposition[2].

 Alg1 > L3 := DirectSum(decomposition[3], Alg3);
 ${\mathrm{L3}}{≔}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{\mathrm{e1}}{+}{\mathrm{e2}}{+}\frac{{\mathrm{e3}}}{{3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{-}{6}{}{\mathrm{e1}}{-}{9}{}{\mathrm{e2}}{-}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{6}{}{\mathrm{e1}}{+}{6}{}{\mathrm{e2}}{+}{\mathrm{e3}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{-}{2}{}{\mathrm{e4}}{-}{\mathrm{e5}}\right]$ (2.19)
 Alg1 > L2;
 $\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{\mathrm{e1}}{+}{\mathrm{e2}}{+}\frac{{\mathrm{e3}}}{{3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{-}{6}{}{\mathrm{e1}}{-}{9}{}{\mathrm{e2}}{-}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{6}{}{\mathrm{e1}}{+}{6}{}{\mathrm{e2}}{+}{\mathrm{e3}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{-}{2}{}{\mathrm{e4}}{-}{\mathrm{e5}}\right]$ (2.20)

Finally, we can initialize any one of the summands in the third entry of the output and study its properties. For example, we find that the first summand is seni-simple.

 Alg1 > DGsetup(decomposition[3][1]);
 ${\mathrm{Lie algebra: Alg1:Factor1}}$ (2.21)
 Alg1:Factor1 > Query("Semisimple");
 ${\mathrm{true}}$ (2.22)
 Alg1:Factor1 >

Calculate the Levi decomposition of a Lie algebra

Every Lie algebra g admits a decomposition into the semi-direct sum g = r + s, where r is the radical of g and s is a semi-simple subalgebra.  Such a decomposition is called a Levi decomposition.  Since the radical is an ideal we have [r, r] in r, [r, s] in r, and [s, s] in s.  The radical r is uniquely defined but the semi-simple subalgebra s is not.

We demonstrate the calculation of the Levi decomposition using the LeviDecomposition command.

 > with(DifferentialGeometry): with(LieAlgebras):
 > L := _DG([["LieAlgebra", Alg1, [7]], [[[1, 2, 2], 2], [[1, 3, 2], -1], [[1, 3, 3], -2], [[1, 3, 4], 1], [[1, 3, 5], -1], [[1, 3, 7], 2], [[1, 4, 4], 2], [[1, 6, 6], -2], [[1, 7, 2], -1], [[1, 7, 4], 1], [[2, 3, 1], 1], [[2, 3, 2], 1], [[2, 3, 4], -1], [[2, 5, 2], -2], [[2, 5, 4], 2], [[2, 6, 5], 1], [[3, 4, 1], -1], [[3, 4, 5], 1], [[3, 5, 5], -1], [[3, 5, 6], 2], [[3, 6, 6], -1], [[4, 5, 2], -2], [[4, 5, 4], 2], [[4, 6, 5], 1], [[4, 7, 2], -1], [[4, 7, 4], 1], [[5, 7, 5], 1], [[6, 7, 6], 1]]]);
 Alg1:Factor1 > DGsetup(L);
 ${L}{≔}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{2}{}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{-}{\mathrm{e2}}{-}{2}{}{\mathrm{e3}}{+}{\mathrm{e4}}{-}{\mathrm{e5}}{+}{2}{}{\mathrm{e7}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{2}{}{\mathrm{e4}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e6}}\right]{=}{-}{2}{}{\mathrm{e6}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e2}}{+}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{+}{\mathrm{e2}}{-}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{-}{2}{}{\mathrm{e2}}{+}{2}{}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e1}}{+}{\mathrm{e5}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e5}}{+}{2}{}{\mathrm{e6}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{-}{2}{}{\mathrm{e2}}{+}{2}{}{\mathrm{e4}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e2}}{+}{\mathrm{e4}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e6}}\right]$ ${\mathrm{Lie algebra: Alg1}}$ (3.1)
 Alg1 > LD := LeviDecomposition();
 ${\mathrm{LD}}{≔}\left[\left[{\mathrm{e2}}{-}{\mathrm{e4}}{,}{\mathrm{e5}}{,}{\mathrm{e6}}{,}{\mathrm{e7}}\right]{,}\left[{\mathrm{e1}}{-}{\mathrm{e4}}{,}{\mathrm{e2}}{,}{\mathrm{e3}}{-}{\mathrm{e7}}\right]\right]$ (3.2)

The radical is given by the first list, the semisimple subalgebra by the second:

 Alg1 > R := LD[1];
 ${R}{≔}\left[{\mathrm{e2}}{-}{\mathrm{e4}}{,}{\mathrm{e5}}{,}{\mathrm{e6}}{,}{\mathrm{e7}}\right]$ (3.3)
 Alg1 > S := LD[2];
 ${S}{≔}\left[{\mathrm{e1}}{-}{\mathrm{e4}}{,}{\mathrm{e2}}{,}{\mathrm{e3}}{-}{\mathrm{e7}}\right]$ (3.4)

We use the Query command to check that that R and S have the required properties.

 Alg1 > Query(R, "Ideal"), Query(R, "Solvable"),Query(S,"Semisimple");
 ${\mathrm{true}}{,}{\mathrm{true}}{,}{\mathrm{true}}$ (3.5)

Let us change to a basis adapted to the Levi decomposition.

 Alg1 > B := [op(R), op(S)];
 ${B}{≔}\left[{\mathrm{e2}}{-}{\mathrm{e4}}{,}{\mathrm{e5}}{,}{\mathrm{e6}}{,}{\mathrm{e7}}{,}{\mathrm{e1}}{-}{\mathrm{e4}}{,}{\mathrm{e2}}{,}{\mathrm{e3}}{-}{\mathrm{e7}}\right]$ (3.6)
 Alg1 > L1 := LieAlgebraData(B, Alg2);
 ${\mathrm{L1}}{≔}\left[\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{-}{2}{}{\mathrm{e1}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{-}{2}{}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e6}}\right]{=}{2}{}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e7}}\right]{=}{-}{2}{}{\mathrm{e3}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e2}}{+}{2}{}{\mathrm{e3}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e6}}\right]{=}{2}{}{\mathrm{e6}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e5}}{-}{\mathrm{e6}}{-}{2}{}{\mathrm{e7}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e5}}{+}{\mathrm{e6}}\right]$ (3.7)
 Alg1 > DGsetup(L1);
 ${\mathrm{Lie algebra: Alg2}}$ (3.8)

The structure equations in this adapted basis are much simpler. From the multiplication table, we can see clearly that [e1, e2, e3, e4] (the radical in the new basis) is an ideal and [e5, e6, e7] (the semisimple part) is a subalgebra.

 Alg2 > MultiplicationTable("LieTable");
 $\left[\begin{array}{ccccccccc}{}& {\mathrm{|}}& {\mathrm{e1}}& {\mathrm{e2}}& {\mathrm{e3}}& {\mathrm{e4}}& {\mathrm{e5}}& {\mathrm{e6}}& {\mathrm{e7}}\\ {}& {\mathrm{---}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}\\ {\mathrm{e1}}& {\mathrm{|}}& {0}& {0}& {0}& {\mathrm{e1}}& {-}{2}{}{\mathrm{e1}}& {0}& {\mathrm{e2}}\\ {\mathrm{e2}}& {\mathrm{|}}& {0}& {0}& {0}& {\mathrm{e2}}& {-}{2}{}{\mathrm{e1}}& {2}{}{\mathrm{e1}}& {-}{2}{}{\mathrm{e3}}\\ {\mathrm{e3}}& {\mathrm{|}}& {0}& {0}& {0}& {\mathrm{e3}}& {\mathrm{e2}}{+}{2}{}{\mathrm{e3}}& {-}{\mathrm{e2}}& {0}\\ {\mathrm{e4}}& {\mathrm{|}}& {-}{\mathrm{e1}}& {-}{\mathrm{e2}}& {-}{\mathrm{e3}}& {0}& {0}& {0}& {0}\\ {\mathrm{e5}}& {\mathrm{|}}& {2}{}{\mathrm{e1}}& {2}{}{\mathrm{e1}}& {-}{\mathrm{e2}}{-}{2}{}{\mathrm{e3}}& {0}& {0}& {2}{}{\mathrm{e6}}& {-}{\mathrm{e5}}{-}{\mathrm{e6}}{-}{2}{}{\mathrm{e7}}\\ {\mathrm{e6}}& {\mathrm{|}}& {0}& {-}{2}{}{\mathrm{e1}}& {\mathrm{e2}}& {0}& {-}{2}{}{\mathrm{e6}}& {0}& {\mathrm{e5}}{+}{\mathrm{e6}}\\ {\mathrm{e7}}& {\mathrm{|}}& {-}{\mathrm{e2}}& {2}{}{\mathrm{e3}}& {0}& {0}& {\mathrm{e5}}{+}{\mathrm{e6}}{+}{2}{}{\mathrm{e7}}& {-}{\mathrm{e5}}{-}{\mathrm{e6}}& {0}\end{array}\right]$ (3.9)

To show that the semisimple subalgebra is S is not unique, we find an automorphism of the original Lie algebra Alg and apply it to the vectors in our Levi decomposition.

 Alg2 > ChangeFrame(Alg1);
 ${\mathrm{Alg2}}$ (3.10)

Define a matrix A and use it to define a linear transformation from Alg1 to Alg1.

 Alg1 > A := Matrix([[1, 0, -1, 0, 0, 0, 0], [0, 1, 1, 0, 2, -1, 1], [0, 0, 1, 0, 0, 0, 0], [2, 0, -2, 1, -2, 1, -1], [0, 0, 1, 0, 1, -1, 0], [0, 0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0, 1]]);
 ${A}{≔}\left[\begin{array}{rrrrrrr}{1}& {0}& {-}{1}& {0}& {0}& {0}& {0}\\ {0}& {1}& {1}& {0}& {2}& {-}{1}& {1}\\ {0}& {0}& {1}& {0}& {0}& {0}& {0}\\ {2}& {0}& {-}{2}& {1}& {-}{2}& {1}& {-}{1}\\ {0}& {0}& {1}& {0}& {1}& {-}{1}& {0}\\ {0}& {0}& {0}& {0}& {0}& {1}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {1}\end{array}\right]$ (3.11)
 Alg1 > phi := Transformation(Alg1, Alg1, A);
 ${\mathrm{φ}}{≔}\left[\left[{\mathrm{e1}}{,}{\mathrm{e1}}{+}{2}{}{\mathrm{e4}}\right]{,}\left[{\mathrm{e2}}{,}{\mathrm{e2}}\right]{,}\left[{\mathrm{e3}}{,}{-}{\mathrm{e1}}{+}{\mathrm{e2}}{+}{\mathrm{e3}}{-}{2}{}{\mathrm{e4}}{+}{\mathrm{e5}}\right]{,}\left[{\mathrm{e4}}{,}{\mathrm{e4}}\right]{,}\left[{\mathrm{e5}}{,}{2}{}{\mathrm{e2}}{-}{2}{}{\mathrm{e4}}{+}{\mathrm{e5}}\right]{,}\left[{\mathrm{e6}}{,}{-}{\mathrm{e2}}{+}{\mathrm{e4}}{-}{\mathrm{e5}}{+}{\mathrm{e6}}\right]{,}\left[{\mathrm{e7}}{,}{\mathrm{e2}}{-}{\mathrm{e4}}{+}{\mathrm{e7}}\right]\right]$ (3.12)

Check that phi is a Lie algebra homomorphism.

 Alg1 > Query(phi,"Homomorphism");
 ${\mathrm{true}}$ (3.13)

Apply the automorphism phi to the vectors in the radical.

 Alg1 > R;
 $\left[{\mathrm{e2}}{-}{\mathrm{e4}}{,}{\mathrm{e5}}{,}{\mathrm{e6}}{,}{\mathrm{e7}}\right]$ (3.14)
 Alg1 > newR := map2(ApplyHomomorphism, phi, R);
 ${\mathrm{newR}}{≔}\left[{\mathrm{e2}}{-}{\mathrm{e4}}{,}{2}{}{\mathrm{e2}}{-}{2}{}{\mathrm{e4}}{+}{\mathrm{e5}}{,}{-}{\mathrm{e2}}{+}{\mathrm{e4}}{-}{\mathrm{e5}}{+}{\mathrm{e6}}{,}{\mathrm{e2}}{-}{\mathrm{e4}}{+}{\mathrm{e7}}\right]$ (3.15)
 Alg1 > newR := Tools:-CanonicalBasis(newR, [e1,e2,e3,e4,e5,e6,e7]);
 ${\mathrm{newR}}{≔}\left[{\mathrm{e2}}{-}{\mathrm{e4}}{,}{\mathrm{e5}}{,}{\mathrm{e6}}{,}{\mathrm{e7}}\right]$ (3.16)

This is exactly the same same as R, illustrating the fact that the radical is an automorphism invariant.

Apply the automorphism phi to the vectors in the semisimple algebra.

 Alg1 > S;
 $\left[{\mathrm{e1}}{-}{\mathrm{e4}}{,}{\mathrm{e2}}{,}{\mathrm{e3}}{-}{\mathrm{e7}}\right]$ (3.17)
 Alg1 > newS := map2(ApplyHomomorphism, phi, S);
 ${\mathrm{newS}}{≔}\left[{\mathrm{e1}}{+}{\mathrm{e4}}{,}{\mathrm{e2}}{,}{-}{\mathrm{e1}}{+}{\mathrm{e3}}{-}{\mathrm{e4}}{+}{\mathrm{e5}}{-}{\mathrm{e7}}\right]$ (3.18)
 Alg1 > newS := Tools:-CanonicalBasis(newS, [e1,e2,e3,e4,e5,e6,e7]);
 ${\mathrm{newS}}{≔}\left[{\mathrm{e1}}{+}{\mathrm{e4}}{,}{\mathrm{e2}}{,}{\mathrm{e3}}{+}{\mathrm{e5}}{-}{\mathrm{e7}}\right]$ (3.19)
 Alg1 > Tools:-DGequal(S,newS);
 ${\mathrm{false}}$ (3.20)

Thus the semisimple algebra S is changed by the automorphism phi. This gives us a second Levi-decomposition:

 Alg1 > newLD := [newR, newS];
 ${\mathrm{newLD}}{≔}\left[\left[{\mathrm{e2}}{-}{\mathrm{e4}}{,}{\mathrm{e5}}{,}{\mathrm{e6}}{,}{\mathrm{e7}}\right]{,}\left[{\mathrm{e1}}{+}{\mathrm{e4}}{,}{\mathrm{e2}}{,}{\mathrm{e3}}{+}{\mathrm{e5}}{-}{\mathrm{e7}}\right]\right]$ (3.21)
 Alg1 >
 >