DifferentialGeometry/LieAlgebras/Query/ParabolicSubalgebra - Maple Help

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Query[ParabolicSubalgebra] - check if a list of vectors defines a parabolic subalgebra of a semi-simple Lie algebra

Calling Sequences

     Query(P, Parabolic)

Parameters

     P     - a list of vectors, defining a subalgebra of a semi-simple Lie algebra

 

Description

Examples

Description

• 

Let g be a semi-simple Lie algebra. A Borel subalgebra  b is any maximal solvable subalgebra. A parabolic subalgebra p is any subalgebra containing a Borel subalgebra. Alternatively, a subalgebra p is parabolic if its nilradical is the orthogonal complement of p with respect to the Killing form B.

• 

This Query command returns true if the subalgebra p defined by the vectors P satisfies nil𝔭 = 𝔭.

Examples

withDifferentialGeometry:withLieAlgebras:

 

 

We check to see if 3 subalgebras of sl3are parabolic.  We construct the Lie algebra sl3directly from its standard matrix representation.

MmapMatrix,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,1,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0

M:=100000001,000010001,010000000,001000000,000100000,000001000,000000100,000000010

(2.1)

LDLieAlgebraDataM,sl3

LD:=e1,e3=e3,e1,e4=2e4,e1,e5=e5,e1,e6=e6,e1,e7=2e7,e1,e8=e8,e2,e3=e3,e2,e4=e4,e2,e5=e5,e2,e6=2e6,e2,e7=e7,e2,e8=2e8,e3,e5=e2+e1,e3,e6=e4,e3,e7=e8,e4,e5=e6,e4,e7=e1,e4,e8=e3,e5,e8=e7,e6,e7=e5,e6,e8=e2

(2.2)

 

Initialize the Lie algebra. We label the basis elements for sl3 in a manner consistent with its matrix representation.

DGsetupLD,E11,E22,E12,E13,E21,E23,E31,E32,ω11,ω22,ω12,ω13,ω21,ω23,ω31,ω32

Lie algebra: sl3

(2.3)

 

Subalgebra 1.

sl3 > 

P1E11,E22,E12,E23,E13,E21

P1:=E11,E22,E12,E23,E13,E21

(2.4)
sl3 > 

QueryP1,Parabolic

true

(2.5)

 

Subalgebra 2.

sl3 > 

P2E11,E22,E12,E23,E13,E32

P2:=E11,E22,E12,E23,E13,E32

(2.6)
sl3 > 

QueryP2,Parabolic

true

(2.7)

 

Subalgebra 3.

sl3 > 

P3E11,E32,E23,E22

P3:=E11,E32,E23,E22

(2.8)
sl3 > 

QueryP3,Subalgebra

true

(2.9)
sl3 > 

QueryP3,Parabolic

false

(2.10)

 

See Also

DifferentialGeometry

LieAlgebras

ComplementaryBasis

Nilradical

ParabolicSubalgebra