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Query[RootSpaceDecomposition] - check that a table of roots and root spaces gives a root space decomposition for a semi-simple Lie algebra with respect to a given Cartan subalgebra

Calling Sequences

     Query(CSA, RSD, RootSpaceDecomposition)

Parameters

     CSA     - a list of vectors in a Lie algebra, defining the Cartan subalgebra of a semi-simple Lie algebra

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

      options - the keyword argument output = "root"

      

 

Description

Examples

Description

• 

 Let g be a semi-simple Lie algebra and h a Cartan subalgebra.  Let h1, h2, ... ,  hm  be a basis for 𝔥 . The linear transformations adhi are simultaneously diagonalizable over C -- if  x g is a common eigenvector for all these transformations, then adhix = hi , x =  αi  x . The m-tuples α = α1, α2, ... , αm ∈ ℂm are called the roots Δ of 𝔤  with respect to the Cartan subalgebra 𝔥 . The eigenspace decomposition 𝔤 = 𝔥  α  Δ  Rα  is called the root space decomposition of g with respect to h.  

• 

For each root α and corresponding root space x, this query checks that hi , x = αi  x .  It also checks that the span of the Cartan subalgebra and the root spaces is the full Lie algebra 𝔤 .

• 

With output = "root", this query will return the root α  if the equations hi , x =  αi  x fail.

Examples

withDifferentialGeometry:withLieAlgebras:

 

Example 1.

Check the root space decomposition for a 10-dimensional Lie algebra.

 

Here is the Lie algebra data structure.

LD_DGLieAlgebra,alg,10,1,2,3,1,1,3,2,2,1,3,4,2,1,4,3,1,1,5,6,1,1,6,5,2,1,6,7,2,1,7,6,1,1,8,9,1,1,9,8,2,1,9,10,2,1,10,9,1,2,3,1,1,2,5,8,2,2,6,9,1,2,8,5,2,2,9,6,1,3,4,1,1,3,5,9,1,3,6,8,2,3,6,10,2,3,7,9,1,3,8,6,1,3,9,5,2,3,9,7,2,3,10,6,1,4,6,9,1,4,7,10,2,4,9,6,1,4,10,7,2,5,6,1,1,5,8,2,2,5,9,3,1,6,7,1,1,6,8,3,1,6,9,2,2,6,9,4,2,6,10,3,1,7,9,3,1,7,10,4,2,8,9,1,1,9,10,1,1

LD:=e1,e2=e3,e1,e3=2e42e2,e1,e4=e3,e1,e5=e6,e1,e6=2e72e5,e1,e7=e6,e1,e8=e9,e1,e9=2e102e8,e1,e10=e9,e2,e3=e1,e2,e5=2e8,e2,e6=e9,e2,e8=2e5,e2,e9=e6,e3,e4=e1,e3,e5=e9,e3,e6=2e10+2e8,e3,e7=e9,e3,e8=e6,e3,e9=2e72e5,e3,e10=e6,e4,e6=e9,e4,e7=2e10,e4,e9=e6,e4,e10=2e7,e5,e6=e1,e5,e8=2e2,e5,e9=e3,e6,e7=e1,e6,e8=e3,e6,e9=2e4+2e2,e6,e10=e3,e7,e9=e3,e7,e10=2e4,e8,e9=e1,e9,e10=e1

(2.1)

 

Initialize the Lie algebra.

alg > 

DGsetupLD

Lie algebra: alg

(2.2)

 

Define a subalgebra and check that it is a Cartan subalgebra.

alg > 

CSAevalDGe1,e8+e10

CSA:=e1,e8+e10

(2.3)

QueryCSA,CartanSubalgebra

true

(2.4)

 

Define a table of roots and root spaces and check that it gives a root space decomposition.

alg > 

RSDmapevalDG,table2I,0=e8Ie9e10,2I,2I=e2Ie3e4Ie5e6+Ie7,2I,2I=e2+Ie3e4Ie5+e6+Ie7,0,2I=e2+e4Ie5Ie7,2I,0=e8+Ie9e10,2I,2I=e2Ie3e4+Ie5+e6Ie7,0,2I=e2+e4+Ie5+Ie7,2I,2I=e2+Ie3e4+Ie5e6Ie7

RSD:=table0,2I=e2+e4Ie5Ie7,2I,2I=e2+Ie3e4Ie5+e6+Ie7,2I,2I=e2Ie3e4Ie5e6+Ie7,2I,0=e8Ie9e10,2I,2I=e2+Ie3e4+Ie5e6Ie7,2I,2I=e2Ie3e4+Ie5+e6Ie7,2I,0=e8+Ie9e10,0,2I=e2+e4+Ie5+Ie7

(2.5)
alg > 

QueryCSA,RSD,RootSpaceDecomposition

true

(2.6)

See Also

DifferentialGeometry

CartanSubalgebra

LieAlgebras

Query

Query[CartanSubalgebra]

RootSpaceDecomposition