find the simple roots for a set of positive roots - Maple Programming Help

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LieAlgebras[SimpleRoots] - find the simple roots for a set of positive roots

Calling Sequences

    SimpleRoots(PR) 

Parameters     

     PR    - a list of  vectors, giving the positive roots of a simple Lie algebra

    

 

Description

Examples

Description

• 

Let Δ ℂm be a list of roots for either an abstract root system or for a simple Lie algebra. In particular, Δ must have an even number of elements and if X Δ, then X Δ. Write Δ = Δ  + Δ where, if X Δ+then X Δ and X Δthen X Δ+. The set Δ+is called the set of positive roots .The choice of positive roots is not unique. If Δ+ is set of positive roots,then a root α  Δ+is called a simple root if it is not a sum of any other 2 positive roots. If Δ0 is a set of simple roots for Δ+, then every root in Δ+ is a linear combination of the roots in Δ0 with positive integer coefficients.The number of simple roots equals the rank of the Lie algebra.

• 

The command SimpleRoots(PR) returns a list of vectors defining a set of simple roots.

Examples

with(DifferentialGeometry): with(LieAlgebras):

 

Example 1.

We calculate the simple roots for the Lie algebra sp8, R. This is the 36-dimensional Lie algebra of 8×8 matrices A which are skew-symmetric with respect to the skew form Q = 0I4I40.

We use the command SimpleLieAlgebraData to obtain the structure equations for this Lie algebra.

LD := SimpleLieAlgebraData("sp(8, R))", sp8R, labelformat = "gl", labels = ['E', 'omega']):

DGsetup(LD);

Lie algebra: sp8R

(2.1)

 

The following diagonal elements define a Cartan subalgebra. (This can be calculated using the command CartanSubalgebra).

sp8R > 

CSA_sp8R := [E11, E22, E33, E44];

CSA_sp8R:=E11,E22,E33,E44

(2.2)

 

Here is the corresponding root space decomposition.

sp8R > 

RSD_sp8R := RootSpaceDecomposition(CSA_sp8R);

RSD_sp8R:=table1,0,1,0=E17,0,1,0,1=E42,0,0,1,1=E38,1,0,0,1=E14,2,0,0,0=E51,0,0,0,2=E48,0,0,0,2=E84,0,0,2,0=E37,0,1,1,0=E27,0,0,1,1=E34,1,0,1,0=E31,0,0,2,0=E73,0,2,0,0=E62,1,0,0,1=E54,0,1,1,0=E32,0,1,0,1=E24,0,0,1,1=E74,1,0,0,1=E18,0,1,1,0=E23,2,0,0,0=E15,1,0,1,0=E13,0,1,0,1=E64,0,2,0,0=E26,1,1,0,0=E12,1,1,0,0=E16,0,1,0,1=E28,1,1,0,0=E52,0,1,1,0=E63,1,1,0,0=E21,1,0,1,0=E53,1,0,0,1=E41,0,0,1,1=E43

(2.3)

 

We calculate the positive roots for sp8, R.

sp8R > 

PR_sp8R := PositiveRoots(RSD_sp8R, <1, 2, 3, 4>);

PR_sp8R:=1010&comma;0101&comma;0011&comma;0002&comma;0020&comma;0110&comma;1010&comma;0110&comma;1001&comma;2000&comma;0200&comma;1100&comma;0101&comma;1100&comma;1001&comma;0011

(2.4)

 

The rank of sp8&comma; R is 4 so we should find 4 positive roots.

sp8R > 

SR_sp8R := SimpleRoots(PR_sp8R);

SR_sp8R:=0110&comma;2000&comma;1100&comma;0011

(2.5)

 

We check that the positive roots are positive integer linear combinations of the simple roots with the GetComponents command.

sp8R > 

GetComponents(PR_sp8R, SR_sp8R);

1&comma;1&comma;1&comma;0&comma;1&comma;0&comma;0&comma;1&comma;2&comma;1&comma;2&comma;1&comma;2&comma;1&comma;2&comma;2&comma;2&comma;1&comma;2&comma;0&comma;1&comma;1&comma;2&comma;0&comma;1&comma;0&comma;1&comma;0&comma;1&comma;0&comma;0&comma;0&comma;1&comma;1&comma;1&comma;1&comma;0&comma;1&comma;0&comma;0&comma;0&comma;1&comma;2&comma;0&comma;0&comma;1&comma;1&comma;0&comma;1&comma;1&comma;2&comma;1&comma;0&comma;0&comma;1&comma;0&comma;1&comma;0&comma;1&comma;1&comma;0&comma;0&comma;0&comma;1

(2.6)

See Also

DifferentialGeometry

DGzip

GetComponents

LieAlgebra

RootSpaceDecomposition

PositiveRoots

SimpleLieAlgebraData