LieAlgebra Lessons
Lesson 6: Decompositions
Overview
Decompose a Lie into a direct sum of Lie algebras
Calculate the Levi decomposition of a Lie algebra
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.
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 ≔ e1,e2=2e1+e2+3e3+3e4+e5−2e6−4e7,e1,e4=e53−2e63,e1,e5=−4e4,e1,e6=e4,e1,e7=−e4+e53−2e63,e2,e3=2e1+e2+3e3+3e4+2e53−4e63−4e7,e2,e5=4e1+4e3+4e4+e5−2e6−4e7,e2,e6=−e1−e3−e4+e7,e2,e7=e1+e3+e4−e7,e4,e5=4e1+4e3+4e4+e5−2e6−4e7,e4,e6=−e1−e3−e4+e7,e4,e7=e1+e3+e4−e7,e5,e6=−3e4,e5,e7=−4e1−4e3−e4−e5+2e6+4e7,e6,e7=e1+e3+e4−e7
DGsetup(L):
We can use the Query command to determine that the algebra is decomposable.
Query("Indecomposable");
false
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.
decomposition := Decompose(factoralgebras = true);
decomposition ≔ 2000−2−112101−2−121300013−1313−111000001000000−1−1021−10−2−2021−2,e1+e3−e7,e4,e5−2e6,e1+2e3+2e4+e53−2e63−2e7,e2−e3−e4,e1+e3+e53+4e63,e4−e6−e7,e1,e2=e1+e2+e33,e1,e3=−6e1−9e2−e3,e2,e3=6e1+6e2+e3,e1,e2=−2e1−e2,
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.
L2 := LieAlgebraData(decomposition[2], Alg2);
L2 ≔ e1,e2=e1+e2+e33,e1,e3=−6e1−9e2−e3,e2,e3=6e1+6e2+e3,e4,e5=−2e4−e5
DGsetup(L2, [f], [omega]);
Lie algebra: Alg2
MultiplicationTable("LieTable");
`|`f1f2f3f4f5f6f7-------------------------------f1`|`0f1+f2+f33−6f1−9f2−f30000f2`|`−f1−f2−f3306f1+6f2+f30000f3`|`6f1+9f2+f3−6f1−6f2−f300000f4`|`0000−2f4−f500f5`|`0002f4+f5000f6`|`0000000f7`|`0000000
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:
S1 := [f1, f2, f3];
S1 ≔ f1,f2,f3
S2 := [f4 ,f5];
S2 ≔ f4,f5
S3 := [f6, f7];
S3 ≔ f6,f7
We check that each of S1, S2, S3 is a subalgebra and that [Si, Sj] = 0 for i <>j.
Query(S1, "Subalgebra"), Query(S2, "Subalgebra"), Query(S3, "Subalgebra");
true,true,true
BracketOfSubspaces(S1,S2), BracketOfSubspaces(S1,S3),BracketOfSubspaces(S2,S3);
,,
This can be all be checked with the Query command.
Query([S1,S2,S3], "DirectSumDecomposition");
true
We can also use the Query command to verify that the matrix A =decomposition[1] is a Lie algebra homomorphism from Alg1 to Alg2
Query(Alg1, Alg2, decomposition[1], "Homomorphism");
Note that this check can actually be performed in the original basis for Alg1:
B:= decomposition[2];
B ≔ e1+e3−e7,e4,e5−2e6,e1+2e3+2e4+e53−2e63−2e7,e2−e3−e4,e1+e3+e53+4e63,e4−e6−e7
T1 := B[1..3];
T1 ≔ e1+e3−e7,e4,e5−2e6
T2 := B[4..5];
T2 ≔ e1+2e3+2e4+e53−2e63−2e7,e2−e3−e4
T3 := B[6..7];
T3 ≔ e1+e3+e53+4e63,e4−e6−e7
Query([T1, T2, T3], "DirectSumDecomposition");
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].
L3 := DirectSum(decomposition[3], Alg3);
L3 ≔ e1,e2=e1+e2+e33,e1,e3=−6e1−9e2−e3,e2,e3=6e1+6e2+e3,e4,e5=−2e4−e5
L2;
e1,e2=e1+e2+e33,e1,e3=−6e1−9e2−e3,e2,e3=6e1+6e2+e3,e4,e5=−2e4−e5
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.
DGsetup(decomposition[3][1]);
Lie algebra: Alg1:Factor1
Query("Semisimple");
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.
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]]]);
DGsetup(L);
L ≔ e1,e2=2e2,e1,e3=−e2−2e3+e4−e5+2e7,e1,e4=2e4,e1,e6=−2e6,e1,e7=−e2+e4,e2,e3=e1+e2−e4,e2,e5=−2e2+2e4,e2,e6=e5,e3,e4=−e1+e5,e3,e5=−e5+2e6,e3,e6=−e6,e4,e5=−2e2+2e4,e4,e6=e5,e4,e7=−e2+e4,e5,e7=e5,e6,e7=e6
Lie algebra: Alg1
LD := LeviDecomposition();
LD ≔ e2−e4,e5,e6,e7,e1−e4,e2,e3−e7
The radical is given by the first list, the semisimple subalgebra by the second:
R := LD[1];
R ≔ e2−e4,e5,e6,e7
S := LD[2];
S ≔ e1−e4,e2,e3−e7
We use the Query command to check that that R and S have the required properties.
Query(R, "Ideal"), Query(R, "Solvable"),Query(S,"Semisimple");
Let us change to a basis adapted to the Levi decomposition.
B := [op(R), op(S)];
B ≔ e2−e4,e5,e6,e7,e1−e4,e2,e3−e7
L1 := LieAlgebraData(B, Alg2);
L1 ≔ e1,e4=e1,e1,e5=−2e1,e1,e7=e2,e2,e4=e2,e2,e5=−2e1,e2,e6=2e1,e2,e7=−2e3,e3,e4=e3,e3,e5=e2+2e3,e3,e6=−e2,e5,e6=2e6,e5,e7=−e5−e6−2e7,e6,e7=e5+e6
DGsetup(L1);
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.
`|`e1e2e3e4e5e6e7-------------------------------e1`|`000e1−2e10e2e2`|`000e2−2e12e1−2e3e3`|`000e3e2+2e3−e20e4`|`−e1−e2−e30000e5`|`2e12e1−e2−2e3002e6−e5−e6−2e7e6`|`0−2e1e20−2e60e5+e6e7`|`−e22e300e5+e6+2e7−e5−e60
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.
ChangeFrame(Alg1);
Alg2
Define a matrix A and use it to define a linear transformation from Alg1 to 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 ≔ 10−1000001102−11001000020−21−21−100101−1000000100000001
phi := Transformation(Alg1, Alg1, A);
φ ≔ e1,e1+2e4,e2,e2,e3,−e1+e2+e3−2e4+e5,e4,e4,e5,2e2−2e4+e5,e6,−e2+e4−e5+e6,e7,e2−e4+e7
Check that phi is a Lie algebra homomorphism.
Query(phi,"Homomorphism");
Apply the automorphism phi to the vectors in the radical.
R;
e2−e4,e5,e6,e7
newR := map2(ApplyHomomorphism, phi, R);
newR ≔ e2−e4,2e2−2e4+e5,−e2+e4−e5+e6,e2−e4+e7
newR := Tools:-CanonicalBasis(newR, [e1,e2,e3,e4,e5,e6,e7]);
newR ≔ e2−e4,e5,e6,e7
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.
S;
e1−e4,e2,e3−e7
newS := map2(ApplyHomomorphism, phi, S);
newS ≔ e1+e4,e2,−e1+e3−e4+e5−e7
newS := Tools:-CanonicalBasis(newS, [e1,e2,e3,e4,e5,e6,e7]);
newS ≔ e1+e4,e2,e3+e5−e7
Tools:-DGequal(S,newS);
Thus the semisimple algebra S is changed by the automorphism phi. This gives us a second Levi-decomposition:
newLD := [newR, newS];
newLD ≔ e2−e4,e5,e6,e7,e1+e4,e2,e3+e5−e7
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