SimpleRoots - Maple Help
For the best experience, we recommend viewing online help using Google Chrome or Microsoft Edge.
Our website is currently undergoing maintenance, which may result in occasional errors while browsing. We apologize for any inconvenience this may cause and are working swiftly to restore full functionality. Thank you for your patience.

Online Help

All Products    Maple    MapleSim


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

withDifferentialGeometry:withLieAlgebras:

 

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.

LDSimpleLieAlgebraDatasp(8, R)),sp8R,labelformat=gl,labels=E,ω:

DGsetupLD

Lie algebra: sp8R

(2.1)

 

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

sp8R > 

CSA_sp8RE11,E22,E33,E44

CSA_sp8R:=E11,E22,E33,E44

(2.2)

 

Here is the corresponding root space decomposition.

sp8R > 

RSD_sp8RRootSpaceDecompositionCSA_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_sp8RPositiveRootsRSD_sp8R,1,2,3,4

 

The rank of sp8, R is 4 so we should find 4 positive roots.

sp8R > 

SR_sp8RSimpleRootsPR_sp8R

 

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

sp8R > 

GetComponentsPR_sp8R,SR_sp8R

1,1,1,0,1,0,0,1,2,1,2,1,2,1,2,2,2,1,2,0,1,1,2,0,1,0,1,0,1,0,0,0,1,1,1,1,0,1,0,0,0,1,2,0,0,1,1,0,1,1,2,1,0,0,1,0,1,0,1,1,0,0,0,1

(2.4)

See Also

DifferentialGeometry

DGzip

GetComponents

LieAlgebra

RootSpaceDecomposition

PositiveRoots

SimpleLieAlgebraData