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Example 1.
In this example we initialize the simple Lie algebra
(of
trace-free matrices), calculate a Cartan subalgebra and a root space decomposition. We then illustrate the above properties of the root space decomposition.
First, we use the program SimpleLieAlgebraData to generate the Lie algebra data for s
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| (2.1) |
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| (2.2) |
The program CartanSubalgebra calculates a Cartan subalgebra for
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| (2.3) |
Now compute the root space decomposition. We see that each root space is 1-dimensional (Property 1).
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| (2.4) |
The roots are the indices for this table, given as column vectors. It is easy to see that the negative of any root is a root (Property 2).
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| (2.5) |
Here are the eigenvectors or root spaces.
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| (2.6) |
The 2nd and 6th roots add to give the 4th root. This means that the Lie bracket of the 3rd and 4th vectors in (2.6) should be a multiple of the 1st vector (Property 3).
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The 1st and 2nd roots are negatives of each other so the Lie bracket of the 1st and 2nd vectors in (2.6)
should belong to the Cartan subalgebra (Property 4).
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| (2.8) |
The vectors
form a 3-dimensional Lie algebra (Property 4).
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| (2.9) |
The Killing form restricted to the root spaces of the 2nd, 4rd and 6th roots is diagonal (Property 5).
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| (2.10) |
Example 2.
We repeat the analysis of Example 1 using the Lie algebra
This is a 21-dimensional Lie algebra of 7×7 matrices which preserve the quadratic form
First, we use the program SimpleLieAlgebraData to generate the Lie algebra data for s
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| (2.11) |
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| (2.12) |
The program CartanSubalgebra calculates a Cartan subalgebra for
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| (2.13) |
Now compute the root space decomposition. We see that each root space is 1-dimensional (Property 1).
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| (2.14) |

| (2.15) |
The roots, given as column vectors, are obtained using LieAlgebraRoots. It is easy to see that the negative of any root is a root (Property 2).
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![RT := map(Vector, [[0, 1, -1], [1, 0, 0], [1, 0, -1], [1, 1, 0], [0, -1, 0], [0, 0, -1], [0, 1, 1], [1, -1, 0], [-1, 0, -1], [-1, -1, 0], [-1, 0, 0], [0, 0, 1], [1, 0, 1], [-1, 0, 1], [-1, 1, 0], [0, -1, 1], [0, 1, 0], [0, -1, -1]])](/support/helpjp/helpview.aspx?si=6589/file05833/math342.png)
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| (2.16) |
Here are the eigenvectors or root spaces.
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| (2.17) |
The 2nd and 12th roots add to give the 13th root. This means that the Lie bracket of the 1st and 7th vectors should be a multiple of the 10th vector (Property 3).
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| (2.18) |
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| (2.19) |
The Lie bracket of any root and its negative belongs to the Cartan subalgebra (Property 4).
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| (2.20) |
The Killing form restricted to the positive root space is diagonal (Property 5).
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| (2.21) |
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| (2.22) |
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| (2.23) |