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LieAlgebras[CartanMatrixToStandardForm] - transform a Cartan matrix to standard form
Calling Sequences
CartanMatrixToStandardForm( , )
Parameters
 C - a square matrix
SR - (optional) a list of vectors, the simple roots used to determine the Cartan matrix for a simple Lie algebra
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Description
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Let ![Delta[0] = `subset`({alpha[1], alpha[2], alpha[m], () .. ()}, Delta)](/support/helpjp/helpview.aspx?si=6613/file05788/math30.png) be a set of simple roots for g. Then the associated
Cartan matrix is the  matrix with entries
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![C[ij] = 2*`<,>`(alpha[i], alpha[j])/`<,>`(alpha[ij], alpha[j])](/support/helpjp/helpview.aspx?si=6613/file05788/math40.png)  ![`<,>`(H[alpha[i]], H[alpha[j]])/`<,>`(H[alpha[i]], H[alpha[i]])](/support/helpjp/helpview.aspx?si=6613/file05788/math42.png) .
(See CartanMatrix for the definition of the vectors ![H[alpha[i]]](/support/helpjp/helpview.aspx?si=6613/file05788/math49.png) )
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A permutation of the roots leads to a different but equivalent Cartan matrix.
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The command CartanMatrixToStandardForm transforms a Cartan matrix to the standard form for each root type.
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The command returns the Cartan matrix in standard form, a permutation matrix, and a string denoting the root type. The permutation matrix will transform the given Cartan matrix to its standard form by a similarity transformation.
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If the second calling is invoked, then the second element of the output is the permuted set of simple roots which will generate the standard form of the Cartan matrix.
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Examples
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Example 1.
We define 4 different Cartan matrices and calculate their standard forms and root type.
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![CM1 := Matrix([[2, -1, 0, -1, -1, 0], [-1, 2, 0, 0, 0, 0], [0, 0, 2, 0, 0, -1], [-1, 0, 0, 2, 0, -1], [-1, 0, 0, 0, 2, 0], [0, 0, -1, -1, 0, 2]])](/support/helpjp/helpview.aspx?si=6613/file05788/math79.png)
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| (2.1) |
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![CM2 := Matrix([[2, 0, 0, 0, -1, 0], [0, 2, -1, 0, 0, 0], [0, -1, 2, 0, -1, -1], [0, 0, 0, 2, 0, -1], [-1, 0, -1, 0, 2, 0], [0, 0, -1, -1, 0, 2]])](/support/helpjp/helpview.aspx?si=6613/file05788/math86.png)
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| (2.2) |
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![CM3 := Matrix([[2, 0, 0, -1, -1, 0], [0, 2, 0, -1, 0, -1], [0, 0, 2, 0, -2, 0], [-1, -1, 0, 2, 0, 0], [-1, 0, -1, 0, 2, 0], [0, -1, 0, 0, 0, 2]])](/support/helpjp/helpview.aspx?si=6613/file05788/math93.png)
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| (2.3) |
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![CM4 := Matrix([[2, -2, 0, 0, -1, 0], [-1, 2, 0, 0, 0, 0], [0, 0, 2, -1, 0, 0], [0, 0, -1, 2, 0, -1], [-1, 0, 0, 0, 2, -1], [0, 0, 0, -1, -1, 2]])](/support/helpjp/helpview.aspx?si=6613/file05788/math100.png)
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| (2.4) |
Here are the standard forms, permutation matrices and root types.
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| (2.5) |
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| (2.6) |
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| (2.7) |
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| (2.8) |
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alg >
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| (2.9) |
For each example the second output is a permutation matrix which transforms the given input Cartan matrix to its standard form.
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| (2.10) |
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| (2.11) |
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| (2.12) |
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| (2.13) |
Example 2.
We define a 21-dimensional simple Lie algebra and calculate its root type.
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Initialize this Lie algebra.
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Find a Cartan subalgebra.
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alg >
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![[_DG([["vector", alg, []], [[[1], 1]]]), _DG([["vector", alg, []], [[[16], 1], [[19], 1]]]), _DG([["vector", alg, []], [[[21], 1]]])]](/support/helpjp/helpview.aspx?si=6613/file05788/math210.png)
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Find the root space decomposition.
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alg >
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| (2.15) |
Find the roots, positive roots and a choice of simple roots.
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alg >
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| (2.16) |
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alg >
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![PR := [Vector(3, {(1) = 0, (2) = 0, (3) = -2*I}), Vector(3, {(1) = -2*I, (2) = 2*I, (3) = 0}), Vector(3, {(1) = -I, (2) = -I, (3) = -I}), Vector(3, {(1) = -I, (2) = I, (3) = -I}), Vector(3, {(1) = -2*I, (2) = 0, (3) = 0}), Vector(3, {(1) = -I, (2) = -I, (3) = I}), Vector(3, {(1) = 0, (2) = -2*I, (3) = 0}), Vector(3, {(1) = -I, (2) = I, (3) = I}), Vector(3, {(1) = -2*I, (2) = -2*I, (3) = 0})]](/support/helpjp/helpview.aspx?si=6613/file05788/math251.png)
| (2.17) |
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alg >
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Find the Cartan matrix.
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alg >
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| (2.18) |
Transform the Cartan matrix to standard form. Here we use the second calling sequence. The command CartanMatrixToStandardForm now returns a permuted set of simple roots for which the Cartan matrix will be in standard form.
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alg >
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![C1, S1, T1 := Matrix(3, 3, {(1, 1) = 2, (1, 2) = -1, (1, 3) = 0, (2, 1) = -1, (2, 2) = 2, (2, 3) = -1, (3, 1) = 0, (3, 2) = -2, (3, 3) = 2}), [Vector(3, {(1) = 0, (2) = -2*I, (3) = 0}), Vector(3, {(1) = -I, (2) = I, (3) = I}), Vector(3, {(1) = 0, (2) = 0, (3) = -2*I})], "C"](/support/helpjp/helpview.aspx?si=6613/file05788/math282.png)
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Check the result by re-calculating the Cartan matrix with respect to the permuted set of roots. We get the standard form immediately.
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alg >
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| (2.20) |
The root type of our 21-dimensional Lie algebra is ![C[3] .](/support/helpjp/helpview.aspx?si=6613/file05788/math296.png)
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