Query[Indecomposable] - check if a Lie algebra is decomposable as a direct sum of Lie algebras over the real numbers
Query[AbsolutelyIndecomposable] - check if a Lie algebra is decomposable as a direct sum of Lie algebras over the complex numbers
Calling Sequences
Query(Alg, "Indecomposable")
Query(Alg, "AbsolutelyIndecomposable")
Parameters
Alg - (optional) the name of an initialized Lie algebra or a Lie algebra data structure
Description
Examples
A collection of subalgebras S1 ,S2, ... of a Lie algebra 𝔤 defines a direct sum decomposition of 𝔤 if 𝔤 = S1⊕S2 ⊕ ⋅⋅⋅ (vector space direct sum) and Si, Sj = 0 for i≠ j.
Query(Alg, "Indecomposable") returns false if the Lie algebra Alg is decomposable as a direct sum of Lie algebras over the real numbers, otherwise true is returned.
Query(Alg, "AbsolutelyIndecomposable") returns false if the Lie algebra Alg is decomposable as a direct sum of Lie algebras over the complex numbers, otherwise true is returned.
The command Query is part of the DifferentialGeometry:-LieAlgebras package. It can be used in the form Query(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:-LieAlgebras:-Query(...).
withDifferentialGeometry:withLieAlgebras:
Example 1.
In this example we illustrate the fact that the result of Inquiry("Indecomposable") does not depend upon the choice of basis for the Lie algebra. First we initialize a Lie algebra.
L1 ≔ _DGLieAlgebra,Alg1,4,1,2,1,1,3,4,3,1
L1:=e1,e2=e1,e3,e4=e3
DGsetupL1:
Now we make a change of basis in the Lie algebra. In this basis it is not possible to see that the Lie algebra is decomposable by examining the multiplication table.
L2 ≔ LieAlgebraDatae1+e4,e2−e3,e2+e4,e1,Alg2
L2:=e1,e2=−e1−e2+e3+2e4,e1,e3=e4,e2,e3=e1+e2−e3−e4,e2,e4=−e4,e3,e4=−e4
DGsetupL2:
Both Alg1 and Alg2 are seen to be decomposable.
QueryAlg1,Indecomposable
false
QueryAlg2,Indecomposable
Example 2
Here is the simplest example of a solvable Lie algebra which is absolutely decomposable but not decomposable. First we initialize the Lie algebra and display the multiplication table.
L ≔ _DGLieAlgebra,Alg3,4,1,3,1,1,2,4,1,1,1,4,2,−1,2,3,2,1
L:=e1,e3=e1,e1,e4=−e2,e2,e3=e2,e2,e4=e1
DGsetupL:
The algebra is indecomposable over the real numbers.
QueryL,Indecomposable
true
The algebra is decomposable over the complex numbers.
QueryL,AbsolutelyIndecomposable
The explicit decomposition of this Lie algebra is given in the help page for the command Decompose.
See Also
DifferentialGeometry
LieAlgebras
Decompose
LieAlgebraData
Query
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