Query[CartanSubalgebra] - check if a list of vectors defines a Cartan subalgebra
Calling Sequences
Query( )
Parameters
A - a list of vectors, defining a subspace of a Lie algebra
options - one or more of the keyword arguments rank = n (where is a positive integer), algebratype = "Semisimple" or algebratype = "Simple"
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Description
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Let be a Lie algebra. A Cartan subalgebra h is a nilpotent subalgebra whose normalizer in g is itself, that is, .
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Examples
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Example 1.
We test if certain subalgebras of are Cartan subalgebras. First define the standard matrix representation for as the space of trace-free matrices.
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![A := map(convert, [[[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)](/support/helpjp/helpview.aspx?si=7182/file07645/math83.png)
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| (2.1) |
Calculate the structure equations for these matrices and initialize the resulting Lie algebra.
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![_DG([["LieAlgebra", sl3, [8, table( [ ] )]], [[[1, 3, 3], 2], [[1, 4, 4], 1], [[1, 5, 5], -2], [[1, 6, 6], -1], [[1, 7, 7], -1], [[1, 8, 8], 1], [[2, 3, 3], -1], [[2, 4, 4], 1], [[2, 5, 5], 1], [[2, 6, 6], 2], [[2, 7, 7], -1], [[2, 8, 8], -2], [[3, 5, 1], 1], [[3, 6, 4], 1], [[3, 7, 8], -1], [[4, 5, 6], -1], [[4, 7, 1], 1], [[4, 7, 2], 1], [[4, 8, 3], 1], [[5, 8, 7], -1], [[6, 7, 5], 1], [[6, 8, 2], 1]]])](/support/helpjp/helpview.aspx?si=7182/file07645/math103.png)
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| (2.3) |
Let's check that is semi-simple.
sl3 >
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Test to see if a list of vectors defines a Cartan subalgebra.
sl3 >
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| (2.5) |
sl3 >
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| (2.6) |
Since has 2 elements, this implies that the rank of is 2. We can use this information to simplify checking that other subalgebras are Cartan subalgebras
sl3 >
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| (2.7) |
sl3 >
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| (2.8) |
Here is a 2-dimensional Abelian subalgebra which is not self-normalizing and therefore not a Cartan subalgebra.
sl3 >
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| (2.9) |
sl3 >
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| (2.10) |
sl3 >
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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.
sl3 >
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sl3 >
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alg >
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| (2.14) |
alg >
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| (2.15) |
alg >
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| (2.16) |
Any subalgebra which is an ideal cannot be a Cartan subalgebra.
alg >
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alg >
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| (2.18) |
alg >
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| (2.19) |
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