find the Cartan matrix for a simple Lie algebra from a root space decomposition, display the Cartan matrix for a given root type - Maple Programming Help

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LieAlgebras[CartanMatrix] - find the Cartan matrix for a simple Lie algebra from a root space decomposition, display the Cartan matrix for a given root type

Calling Sequences

     CartanMatrix(SimRts, RSD)

     CartanMatrix(RT, m)

Parameters

     SimRts   - a list of column vectors, defining the simple roots of a simple Lie algebra

     RSD      - a table, defining the root space decomposition of an initialized Lie algebra

     RT       - a string, the root type of a simple Lie algebra "A", "B", "C", "D", "E", "F", "G"

     m        - a positive integer, the dimension of the Cartan matrix

  

 

Description

Examples

Description

• 

 Let g be a simple Lie algebra, h a Cartan subalgebra, and &gfr; &equals; &hfr; &alpha;  &Delta; R&alpha; the root space decomposition of g with respect to h. Let <⋅,⋅> be the Killing form of g. For each root &alpha; &Delta;, there are vectors X&alpha; R&alpha; &comma; X&alpha; R&alpha; and H&alpha; &hfr; such that  &lsqb;Hα &comma; X&alpha;&rsqb;  &equals; 2 X&alpha;&comma;  &lsqb;Hα &comma;  X&alpha;&rsqb;  &equals; 2 X&alpha; and  X&alpha; &comma; X&alpha; &equals; H&alpha; &period;  These conditions uniquely determine H&period;  The vector Hα can be computed using the command RootToCartanSubalgebraElementH.

• 

Let &Delta;0 &equals; &alpha;1 &comma; &alpha;2&comma; .... &alpha;m &Delta; be a set of simple roots for g. Then the associated Cartan matrix is the m×m matrix with entries Cij&equals; 2<H&alpha;i , H&alpha;j&gt;/ <H&alpha;i, H&alpha;i >. The entries of the Cartan matrix are 0, 1, -1 or 2. The Cartan matrix is independent of the choice of Cartan subalgebra h but is dependent upon the ordering of the simple roots in &Delta;0 &period;

• 

The Cartan matrix is the fundamental invariant for semi-simple Lie algebras over C -- two complex semi-simple Lie algebras are isomorphic if and only if their Cartan matrices are the same, modulo a permutation of the vectors in the Cartan subalgebra. The command CartanMatrixToStandardForm will transform a given Cartan matrix to a standard form.

• 

The Cartan matrix encodes the re-construction of the root system of the Lie algebra from its simple roots. See PositiveRoots .

• 

The information contained in the Cartan matrix is also encoded in the Dynkin diagram of the Lie algebra.

• 

The first calling sequence calculates the Cartan matrix of a Lie algebra from a set of simple roots and a root space decomposition.

• 

The second calling sequence displays the standard form of the Cartan matrix for each possible root type of a simple Lie algebra.

Examples

withDifferentialGeometry&colon;withLieAlgebras&colon;

 

Example 1.

We use the command SimpleLieAlgebraData to obtain the Lie algebra data for the Lie algebra su4. This is the 15-dimensional Lie algebra of trace-free, skew-Hermitian matrices

We suppress the output of this command which is a lengthy list of structure equations.

 

LDSimpleLieAlgebraDatasu(4)&comma;su&colon;

 

Initialize this Lie algebra -- the basis elements are given the default labels e1&comma; e2&comma; ..&period; &comma;e15 &period;

DGsetupLD

Lie algebra: su

(2.1)

 

We remark that the command StandardRepresentation can be used to explicitly display the matrices defining su4.

su > 

StandardRepresentationsu

I00000000000000I&comma;00000I000000000I&comma;0000000000I0000I&comma;0100100000000000&comma;0000001001000000&comma;0000000000010010&comma;0010000010000000&comma;0000000100000100&comma;0001000000001000&comma;0I00I00000000000&comma;000000I00I000000&comma;00000000000I00I0&comma;00I00000I0000000&comma;0000000I00000I00&comma;000I00000000I000

(2.2)

I0000I0000000000&comma;00000I0000I00000&comma;0000000000I0000I&comma;0100100000000000&comma;0000001001000000&comma;0000000000010010&comma;0010000010000000&comma;0000000100000100&comma;0001000000001000&comma;0I00I00000000000&comma;000000I00I000000&comma;00000000000I00I0&comma;00I00000I0000000&comma;0000000I00000I00&comma;000I00000000I000

(2.3)

 

The first 3 matrices define a Cartan subalgebra. We can use the Query command to check this

su > 

CSAe1&comma;e2&comma;e3

CSA:=e1&comma;e2&comma;e3

(2.4)
su > 

QueryCSA&comma;CartanSubalgebra

true

(2.5)

 

We use the command RootSpaceDecomposition to find the root space decomposition for su4 with respect to this Cartan subalgebra.

su > 

RSDRootSpaceDecompositionCSA

RSD:=tableI&comma;2I&comma;I&equals;e8Ie14&comma;2I&comma;I&comma;I&equals;e9Ie15&comma;I&comma;I&comma;2I&equals;e6Ie12&comma;I&comma;2I&comma;I&equals;e8&plus;Ie14&comma;2I&comma;I&comma;I&equals;e9&plus;Ie15&comma;0&comma;I&comma;I&equals;e5Ie11&comma;I&comma;I&comma;0&equals;e4&plus;Ie10&comma;I&comma;0&comma;I&equals;e7Ie13&comma;I&comma;0&comma;I&equals;e7&plus;Ie13&comma;I&comma;I&comma;2I&equals;e6&plus;Ie12&comma;0&comma;I&comma;I&equals;e5&plus;Ie11&comma;I&comma;I&comma;0&equals;e4Ie10

(2.6)

 

A choice of simple roots for this root space decomposition is:

su > 

&Delta;0I&comma;I&comma;2I&comma;0&comma;I&comma;I&comma;I&comma;I&comma;0

&Delta;0:=II2I&comma;0II&comma;II0

(2.7)

 

This set of simple roots can be determined by the command SimpleRoots. The Cartan matrix for this root space decomposition and choice of simple roots is :

su > 

CMCartanMatrix&Delta;0&comma;RSD

CM:=210121012

(2.8)

 

We easily identify this as the standard Cartan matrix for A3 &period;

su > 

CartanMatrixA&comma;3

210121012

(2.9)

 

Notice that a permutation of the simple roots gives a permuted Cartan matrix.

su > 

&Delta;1&Delta;03&comma;&Delta;01&comma;&Delta;02

&Delta;1:=II0&comma;II2I&comma;0II

(2.10)
su > 

CartanMatrix&Delta;1&comma;RSD

201021112

(2.11)

 

 

Example 2.

For the exceptional Lie algebras E6, E7 and E8 there are two different conventions for the Cartan matrix. For E6 these are:

su > 

CartanMatrixE&comma;6&comma;version&equals;I&comma;CartanMatrixE&comma;6&comma;version&equals;II

201000020100102100011210000121000012&comma;210000121000012101001210000120001002

(2.12)

 

See Also

DifferentialGeometry

DynkinDiagram

CartanSubalgebra

LieAlgebras

RootSpaceDecomposition

SimpleRoots

 


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