CartanMatrixToStandardForm

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LieAlgebras[CartanMatrixToStandardForm] - transform a Cartan matrix to standard form

Calling Sequences

CartanMatrixToStandardForm(C,SR)

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

Description

Examples

Description

•	Let $Δ_{0} = \{α_{1}, α_{2}, .. &period;, α_{m}\} \subseteq Δ$ be a set of simple roots for g. Then the associated Cartan matrix is the $m \times m$ matrix with entries

$C_{ij} = 2 \frac{⟨α_{i}, α_{j}⟩}{⟨α_{ij}, α_{j}⟩}$ $= 2$ $\frac{⟨H_{α_{i}}, H_{α_{j}}⟩}{⟨H_{α_{i}}, H_{α_{i}}⟩}$ .

(See CartanMatrix for the definition of the vectors $H_{α_{i}}$ )

•	A permutation of the roots leads to a different but equivalent Cartan matrix.

•	The command CartanMatrixToStandardForm transforms a Cartan matrix to the standard form for each root type.

•	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.

•	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.

Examples

>	$with (DifferentialGeometry) &colon;$ $with (LieAlgebras) &colon;$

Example 1.

We define 4 different Cartan matrices and calculate their standard forms and root type.

>	$CM1 ≔ Matrix ([[2, - 1, 0, - 1, - 1, 0], [- 1, 2, 0, 0, 0, 0], [0, 0, 2, 0, 0, - 1], [- 1, 0, 0, 2, 0, - 1], [- 1, 0, 0, 0, 2, 0], [0, 0, - 1, - 1, 0, 2]])$

>	$CM2 ≔ Matrix ([[2, 0, 0, 0, - 1, 0], [0, 2, - 1, 0, 0, 0], [0, - 1, 2, 0, - 1, - 1], [0, 0, 0, 2, 0, - 1], [- 1, 0, - 1, 0, 2, 0], [0, 0, - 1, - 1, 0, 2]])$

>	$CM3 ≔ Matrix ([[2, 0, 0, - 1, - 1, 0], [0, 2, 0, - 1, 0, - 1], [0, 0, 2, 0, - 2, 0], [- 1, - 1, 0, 2, 0, 0], [- 1, 0, - 1, 0, 2, 0], [0, - 1, 0, 0, 0, 2]])$

>	$CM4 ≔ Matrix ([[2, - 2, 0, 0, - 1, 0], [- 1, 2, 0, 0, 0, 0], [0, 0, 2, - 1, 0, 0], [0, 0, - 1, 2, 0, - 1], [- 1, 0, 0, 0, 2, - 1], [0, 0, 0, - 1, - 1, 2]])$

Here are the standard forms, permutation matrices and root types.

>	$C1, P1, T1 ≔ CartanMatrixToStandardForm (CM1)$

>	$C2, P2, T2 ≔ CartanMatrixToStandardForm (CM2)$

>	$C3, P3, T3 ≔ CartanMatrixToStandardForm (CM3)$

>	$C4, P4, T4 ≔ CartanMatrixToStandardForm (CM4)$

alg >

$C1, P1, T1 ≔ CartanMatrixToStandardForm (CM1)$

For each example the second output is a permutation matrix which transforms the given input Cartan matrix to its standard form.

>	$LinearAlgebra :- Equal ({P1}^{- 1} \cdot CM1 \cdot P1, C1)$

$true$

(2.1)

>	$LinearAlgebra :- Equal ({P2}^{- 1} \cdot CM2 \cdot P2, C2)$

$true$

(2.2)

>	$LinearAlgebra :- Equal ({P3}^{- 1} \cdot CM3 \cdot P3, C3)$

$true$

(2.3)

>	$LinearAlgebra :- Equal ({P4}^{- 1} \cdot CM4 \cdot P4, C4)$

$true$

(2.4)

Example 2.

We define a 21-dimensional simple Lie algebra and calculate its root type.

$LD ≔_DG ([[LieAlgebra, alg, [21]], [[[1, 2, 3], 1], [[1, 3, 2], - 1], [[1, 4, 5], 1], [[1, 5, 4], - 2], [[1, 5, 7], 2], [[1, 6, 8], 1], [[1, 7, 5], - 1], [[1, 8, 6], - 1], [[1, 10, 11], 1], [[1, 11, 10], - 2], [[1, 11, 13], 2], [[1, 12, 14], 1], [[1, 13, 11], - 1], [[1, 14, 12], - 1], [[1, 16, 17], 1], [[1, 17, 16], - 2], [[1, 17, 19], 2], [[1, 18, 20], 1], [[1, 19, 17], - 1], [[1, 20, 18], - 1], [[2, 3, 1], 1], [[2, 4, 6], 1], [[2, 5, 8], 1], [[2, 6, 4], - 2], [[2, 6, 9], 2], [[2, 8, 5], - 1], [[2, 9, 6], - 1], [[2, 10, 12], 1], [[2, 11, 14], 1], [[2, 12, 10], - 2], [[2, 12, 15], 2], [[2, 14, 11], - 1], [[2, 15, 12], - 1], [[2, 16, 18], 1], [[2, 17, 20], 1], [[2, 18, 16], - 2], [[2, 18, 21], 2], [[2, 20, 17], - 1], [[2, 21, 18], - 1], [[3, 5, 6], 1], [[3, 6, 5], - 1], [[3, 7, 8], 1], [[3, 8, 7], - 2], [[3, 8, 9], 2], [[3, 9, 8], - 1], [[3, 11, 12], 1], [[3, 12, 11], - 1], [[3, 13, 14], 1], [[3, 14, 13], - 2], [[3, 14, 15], 2], [[3, 15, 14], - 1], [[3, 17, 18], 1], [[3, 18, 17], - 1], [[3, 19, 20], 1], [[3, 20, 19], - 2], [[3, 20, 21], 2], [[3, 21, 20], - 1], [[4, 5, 1], 1], [[4, 6, 2], 1], [[4, 10, 16], 2], [[4, 11, 17], 1], [[4, 12, 18], 1], [[4, 16, 10], - 2], [[4, 17, 11], - 1], [[4, 18, 12], - 1], [[5, 6, 3], 1], [[5, 7, 1], 1], [[5, 8, 2], 1], [[5, 10, 17], 1], [[5, 11, 16], 2], [[5, 11, 19], 2], [[5, 12, 20], 1], [[5, 13, 17], 1], [[5, 14, 18], 1], [[5, 16, 11], - 1], [[5, 17, 10], - 2], [[5, 17, 13], - 2], [[5, 18, 14], - 1], [[5, 19, 11], - 1], [[5, 20, 12], - 1], [[6, 8, 1], 1], [[6, 9, 2], 1], [[6, 10, 18], 1], [[6, 11, 20], 1], [[6, 12, 16], 2], [[6, 12, 21], 2], [[6, 14, 17], 1], [[6, 15, 18], 1], [[6, 16, 12], - 1], [[6, 17, 14], - 1], [[6, 18, 10], - 2], [[6, 18, 15], - 2], [[6, 20, 11], - 1], [[6, 21, 12], - 1], [[7, 8, 3], 1], [[7, 11, 17], 1], [[7, 13, 19], 2], [[7, 14, 20], 1], [[7, 17, 11], - 1], [[7, 19, 13], - 2], [[7, 20, 14], - 1], [[8, 9, 3], 1], [[8, 11, 18], 1], [[8, 12, 17], 1], [[8, 13, 20], 1], [[8, 14, 19], 2], [[8, 14, 21], 2], [[8, 15, 20], 1], [[8, 17, 12], - 1], [[8, 18, 11], - 1], [[8, 19, 14], - 1], [[8, 20, 13], - 2], [[8, 20, 15], - 2], [[8, 21, 14], - 1], [[9, 12, 18], 1], [[9, 14, 20], 1], [[9, 15, 21], 2], [[9, 18, 12], - 1], [[9, 20, 14], - 1], [[9, 21, 15], - 2], [[10, 11, 1], 1], [[10, 12, 2], 1], [[10, 16, 4], 2], [[10, 17, 5], 1], [[10, 18, 6], 1], [[11, 12, 3], 1], [[11, 13, 1], 1], [[11, 14, 2], 1], [[11, 16, 5], 1], [[11, 17, 4], 2], [[11, 17, 7], 2], [[11, 18, 8], 1], [[11, 19, 5], 1], [[11, 20, 6], 1], [[12, 14, 1], 1], [[12, 15, 2], 1], [[12, 16, 6], 1], [[12, 17, 8], 1], [[12, 18, 4], 2], [[12, 18, 9], 2], [[12, 20, 5], 1], [[12, 21, 6], 1], [[13, 14, 3], 1], [[13, 17, 5], 1], [[13, 19, 7], 2], [[13, 20, 8], 1], [[14, 15, 3], 1], [[14, 17, 6], 1], [[14, 18, 5], 1], [[14, 19, 8], 1], [[14, 20, 7], 2], [[14, 20, 9], 2], [[14, 21, 8], 1], [[15, 18, 6], 1], [[15, 20, 8], 1], [[15, 21, 9], 2], [[16, 17, 1], 1], [[16, 18, 2], 1], [[17, 18, 3], 1], [[17, 19, 1], 1], [[17, 20, 2], 1], [[18, 20, 1], 1], [[18, 21, 2], 1], [[19, 20, 3], 1], [[20, 21, 3], 1]]]) &colon;$

Initialize this Lie algebra.

>	$DGsetup (LD) &colon;$

Find a Cartan subalgebra.

alg >

$CSA ≔ CartanSubalgebra ()$

$CSA := [e1, e16 + e19, e21]$

(2.5)

Find the root space decomposition.

alg >

$RSD ≔ RootSpaceDecomposition (CSA)$

$RSD := table ([[0, 0, - 2 I] = e9 + I e15, [- 2 I, 2 I, 0] = e4 + I e5 - e7 - I e10 + e11 + I e13, [I, I, I] = e6 - I e8 - I e12 - e14, [- I, - I, - I] = e6 + I e8 + I e12 - e14, [I, - I, I] = e2 - I e3 - I e18 - e20, [- I, I, - I] = e2 + I e3 + I e18 - e20, [2 I, 2 I, 0] = e4 - I e5 - e7 - I e10 - e11 + I e13, [2 I, - 2 I, 0] = e4 - I e5 - e7 + I e10 + e11 - I e13, [- 2 I, 0, 0] = e16 + I e17 - e19, [I, I, - I] = e2 - I e3 + I e18 + e20, [0, 0, 2 I] = e9 - I e15, [- I, - I, I] = e2 + I e3 - I e18 + e20, [I, - I, - I] = e6 - I e8 + I e12 + e14, [2 I, 0, 0] = e16 - I e17 - e19, [0, - 2 I, 0] = e4 + e7 + I e10 + I e13, [- I, I, I] = e6 + I e8 - I e12 + e14, [0, 2 I, 0] = e4 + e7 - I e10 - I e13, [- 2 I, - 2 I, 0] = e4 + I e5 - e7 + I e10 - e11 - I e13])$

(2.6)

Find the roots, positive roots and a choice of simple roots.

alg >

$RT ≔ LieAlgebraRoots (RSD)$

alg >

$PR ≔ PositiveRoots (RT, ⟨7 I, 3 I, I⟩)$

alg >

$SR ≔ SimpleRoots (PR)$

Find the Cartan matrix.

alg >

$CM ≔ CartanMatrix (SR, RSD)$

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.

alg >

$C1, S1, T1 ≔ CartanMatrixToStandardForm (CM, SR)$

Check the result by re-calculating the Cartan matrix with respect to the permuted set of roots. We get the standard form immediately.

alg >

$CartanMatrix (S1, RSD)$

The root type of our 21-dimensional Lie algebra is $C_{3} &period;$

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