transform a Cartan matrix to standard form

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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

Description

•	Let be a set of simple roots for g. Then the associated Cartan matrix is the matrix with entries

C[ij] = 2*`<,>`(alpha[i], alpha[j])/`<,>`(alpha[ij], alpha[j]) `<,>`(H[alpha[i]], H[alpha[j]])/`<,>`(H[alpha[i]], H[alpha[i]]) .

(See CartanMatrix for the definition of the vectors )

•	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

Example 1.

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

(2.1)

(2.2)

(2.3)

(2.4)

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

(2.5)

(2.6)

(2.7)

(2.8)

alg >

(2.9)

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

(2.10)

(2.11)

(2.12)

(2.13)

Example 2.

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

Initialize this Lie algebra.

Find a Cartan subalgebra.

alg >

(2.14)

Find the root space decomposition.

alg >

(2.15)

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

alg >

(2.16)

alg >

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})]

(2.17)

alg >

Find the Cartan matrix.

alg >

CM := Matrix(3, 3, {(1, 1) = 2, (1, 2) = 0, (1, 3) = -2, (2, 1) = 0, (2, 2) = 2, (2, 3) = -1, (3, 1) = -1, (3, 2) = -1, (3, 3) = 2})

(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.

alg >

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"