Query[CartanSubalgebra] - check if a list of vectors defines a Cartan subalgebra
Calling Sequences
Query(A, options , CartanSubalgebra)
Parameters
A - a list of vectors, defining a subspace of a Lie algebra
options - one or more of the keyword arguments rank = n (where n is a positive integer), algebratype = "Semisimple" or algebratype = "Simple"
Description
Examples
Let 𝔤 be a Lie algebra. A Cartan subalgebra h is a nilpotent subalgebra whose normalizer in g is itself, that is, nor𝔥 = 𝔥 .
If the Lie algebra 𝔤 is semi-simple and the rank of the Lie algebra is m, then any Cartan subalgebra is of dimension m and is Abelian. This simplifies checking if a given subspace of vectors is a Cartan subalgebra ( the nilpotent character of h need not be verified).
withDifferentialGeometry:withLieAlgebras:
Example 1.
We test if certain subalgebras of sl3 are Cartan subalgebras. First define the standard matrix representation for sl3 as the space of 3 × 3 trace-free matrices.
A≔mapconvert,1,0,0,0,−1,0,0,0,0,0,0,0,0,1,0,0,0,−1,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,Matrix
Calculate the structure equations for these matrices and initialize the resulting Lie algebra.
LD≔LieAlgebraDataA,sl3
LD:=e1,e3=2e3,e1,e4=e4,e1,e5=−2e5,e1,e6=−e6,e1,e7=−e7,e1,e8=e8,e2,e3=−e3,e2,e4=e4,e2,e5=e5,e2,e6=2e6,e2,e7=−e7,e2,e8=−2e8,e3,e5=e1,e3,e6=e4,e3,e7=−e8,e4,e5=−e6,e4,e7=e1+e2,e4,e8=e3,e5,e8=−e7,e6,e7=e5,e6,e8=e2
DGsetupLD
Lie algebra: sl3
Let's check that sl3 is semi-simple.
Querysl3,Semisimple
true
Test to see if a list of vectors defines a Cartan subalgebra.
A≔e1,e2
A:=e1,e2
QueryA,CartanSubalgebra
Since A has 2 elements, this implies that the rank of sl3 is 2. We can use this information to simplify checking that other subalgebras are Cartan subalgebras
A≔evalDGe1+e6,e2−2e6
A:=e1+e6,e2−2e6
QueryA,rank=2,algebratype=Semisimple,CartanSubalgebra
Here is a 2-dimensional Abelian subalgebra which is not self-normalizing and therefore not a Cartan subalgebra.
A≔e3,e4
A:=e3,e4
false
SubalgebraNormalizerA
e8,e6,e4,e3,e2,e1
Example 2.
The notion of a Cartan subalgebra is not restricted to semi-simple Lie algebras. We define a solvable Lie algebra and test to see if some subalgebras are Cartan subalgebras.
LD≔_DGLieAlgebra,alg,5,1,4,1,a,2,4,2,1,3,4,3,1,1,5,1,1,3,5,2,1
LD:=e1,e4=ae1,e1,e5=e1,e2,e4=e2,e3,e4=e3,e3,e5=e2
Lie algebra: alg
QuerySolvable
A≔e4,e5
A:=e4,e5
Any subalgebra which is an ideal cannot be a Cartan subalgebra.
QueryA,Ideal
See Also
DifferentialGeometry
CartanSubalgebra
LieAlgebraData
Query[Ideal]
Query[Solvable]
Query[Subalgebra]
Query[Semisimple]
Download Help Document
