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LieAlgebras[ParabolicSubalgebra] - find the parabolic subalgebra defined by a set of simple roots or a set of restricted simple roots

LieAlgebras[ParabolicSubalgebraRoots] - find the simple roots which generate a parabolic subalgebra

Calling Sequences

     ParabolicSubalgebra(Σ, T1)

     ParabolicSubalgebra(Σ , T2, method="non-compact")

     ParabolicSubalgebraRoots(ParAlg, T2)

     ParabolicSubalgebraRoots(ParAlg, T2, method="non-compact")

Parameters

     Σ       - a list or set of column vectors, defining a subset of simple roots

   T1      - a table, with indices that include "RootSpaceDecomposition", "CartanSubalgebra", "SimpleRoots", and "PositiveRoots"

     T2      - a table, with indices that include "RestrictedRootSpaceDecomposition", "CartanSubalgebraDecomposition", "RestrictedSimpleRoots" and "RestrictedPositiveRoots"

     ParAlg  - a list of vectors in a Lie algebra, defining a parabolic subalgebra

 

 

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 b. Alternatively, a subalgebra p is parabolic if its nilradical is the orthogonal complement of p with respect to the Killing form B.

• 

Let h be an Cartan subalgebra and 𝔤 = 𝔥 α Δ+Rα the associated root space decomposition. Let Δ+ be a choice of positive roots and let Δ0  Δ+ be a set of simple roots. The subalgebra 𝔟 = 𝔥 α ΔRα  is called the standard Borel subalgebra associated to h and any parabolic subalgebra containing it is called a standard parabolic subalgebra. (One could replace the summation over the positive roots by one over the negative roots.)α  Δ+

• 

 Given a standard parabolic subalgebra p , let Φ𝔭 0 = {α Δ0 | Rα  𝔭 }. This set of simple roots completely specifies the parabolic subalgebra p. Conversely, given a set of simple roots Φ0, let Φ ={ α Δ+ | α is a linear combination of the roots in Φ0 } and set 𝔭 Φ0= 𝔥 α ΦRα  . Then 𝔭Φ0 is a standard parabolic subalgebra.

• 

For the parabolic subalgebras of a real semi-simple Lie algebra the situation is essentially the same except that one must consider the restricted root space decomposition  𝔤 = Z𝔞 α ΔSα relative to a maximal Abelian subalgebra a on which the Killing form is positive-definite.

• 

Let Σ be a subset of the simple roots Δ0 and set Φ0 = Δ0/Σ. The command ParabolicSubalgebra returns the standard parabolic subalgebra 𝔭Φ0. The command ParabolicSubalgebraRoots returns the list of simple roots Σ .

• 

With the keyword argument method = "non-compact", a real parabolic subalgebra is calculated.

• 

With  Σ = Δ0,the standard Borel subalgebra is returned.

• 

If the Lie algebra is created from the command SimpleLieAlgebraData , then the table obtained from the command SimpleLieAlgebraProperties can be used as the second argument T1 or T2. 

• 

The command Query/"ParabolicSubalgebra" will test if a given subalgebra of a semi-simple Lie algebra is parabolic.

Examples

with(DifferentialGeometry): with(LieAlgebras):

 

Example 1.

We calculate the parabolic subalgebras for sl4. We use the command SimpleLieAlgebraData to initialize the Lie algebra.

LD := SimpleLieAlgebraData("sl(4)", sl4, labelformat = "gl", labels = [E, omega]):

DGsetup(LD);

Lie algebra: sl4

(2.1)

 

We use the command SimpleLieAlgebraProperties to obtain the Cartan subalgebra, root space decomposition etc.

sl4 > 

P := SimpleLieAlgebraProperties(sl4):

 

Here are the properties we need:

sl4 > 

CSA := P["CartanSubalgebra"];

CSA:=E11,E22,E33

(2.2)
sl4 > 

RSD := eval(P["RootSpaceDecomposition"]);

RSD:=table1,1,0=E12,1,1,0=E21,2,1,1=E41,1,2,1=E24,1,0,1=E13,1,1,2=E43,0,1,1=E23,1,1,2=E34,1,0,1=E31,1,2,1=E42,2,1,1=E14,0,1,1=E32

(2.3)
sl4 > 

SR := P["SimpleRoots"];

SR:=110,011,112

(2.4)
sl4 > 

PR := P["PositiveRoots"];

PR:=110,011,112,101,121,211

(2.5)

 

The possible subsets of the simple roots are:

sl4 > 

Sigma := [[], SR[1..1], SR[2..2], SR[3..3], SR[1..2], SR[2..3], [SR[1], SR[3]], SR];

Σ:=,110,011,112,110,011,011,112,110,112,110,011,112

(2.6)

 

The possible parabolic subalgebras of sl4 are therefore:

sl > 

Sigma[1], ParabolicSubalgebra(Sigma[1], P);

,E11,E22,E33,E12,E13,E14,E21,E23,E24,E31,E32,E34,E41,E42,E43

(2.7)
sl4 > 

Sigma[2], ParabolicSubalgebra(Sigma[2], P);

110,E11,E22,E33,E12,E13,E14,E23,E24,E32,E34,E42,E43

(2.8)
sl4 > 

Sigma[3], ParabolicSubalgebra(Sigma[3], P);

011,E11,E22,E33,E12,E13,E14,E21,E23,E24,E34,E43

(2.9)
sl4 > 

Sigma[4], ParabolicSubalgebra(Sigma[4], P);

112,E11,E22,E33,E12,E13,E14,E21,E23,E24,E31,E32,E34

(2.10)
sl4 > 

Sigma[5], ParabolicSubalgebra(Sigma[5], P);

110,011,E11,E22,E33,E12,E13,E14,E23,E24,E34,E43

(2.11)
sl4 > 

Sigma[6], ParabolicSubalgebra(Sigma[6], P);

011,112,E11,E22,E33,E12,E13,E14,E21,E23,E24,E34

(2.12)
sl4 > 

Sigma[7], ParabolicSubalgebra(Sigma[7], P);

110,112,E11,E22,E33,E12,E13,E14,E23,E24,E32,E34

(2.13)
sl4 > 

Sigma[8], ParabolicSubalgebra(Sigma[8], P);

110,011,112,E11,E22,E33,E12,E13,E14,E23,E24,E34

(2.14)

 

The Query command can be used to check that these subalgebras are parabolic subalgebra.

sl4 > 

PS7 := ParabolicSubalgebra(Sigma[7], P);

PS7:=E11,E22,E33,E12,E13,E14,E23,E24,E32,E34

(2.15)
sl4 > 

Query(PS7, "Parabolic");

true

(2.16)

 

With the command ParabolicSubalgebraRoots, we can find the simple roots used to create the parabolic algebra PS7.

sl4 > 

ParabolicSubalgebraRoots(PS7, P);

110,112

(2.17)

 

Example 2.

We calculate (real) parabolic subalgebras for so6,3. We use the command SimpleLieAlgebraData to initialize the Lie algebra.

sl4 > 

LD2 := SimpleLieAlgebraData("so(5,3)", so53, labelformat = "gl", labels = [R, theta]):

sl4 > 

DGsetup(LD2);

Lie algebra: so53

(2.18)

 

We use the command SimpleLieAlgebraProperties to calculate the restricted root space decomposition and the restricted simple roots.

so53 > 

P := SimpleLieAlgebraProperties(so53):

so53 > 

RRSD := eval(P["RestrictedRootSpaceDecomposition"]);

RRSD:=table0,0,1=R37,R38,1,1,0=R12,0,1,0=R57,R58,1,0,1=R16,1,1,0=R21,0,0,1=R67,R68,1,1,0=R15,0,1,1=R26,1,0,0=R47,R48,1,0,1=R13,1,0,0=R17,R18,1,1,0=R42,0,1,1=R23,1,0,1=R43,0,1,0=R27,R28,1,0,1=R31,0,1,1=R32,0,1,1=R53

(2.19)
sl4 > 

RSR := P["RestrictedSimpleRoots"];

RSR:=110,011,001

(2.20)

 

The possible subsets of restricted simple roots are:

so53 > 

Sigma := [RSR, RSR[1..2], RSR[2..3], [RSR[1], RSR[3]], RSR[1..1], RSR[2..2], RSR[3..3], []];

Σ:=110,011,001,110,011,011,001,110,001,110,011,001,

(2.21)

 

The parabolic subalgebras defined by these sets of restricted roots are:

so53 > 

Sigma[1], ParabolicSubalgebra(Sigma[1], P, method = "non-compact");

110,011,001,R11,R12,R13,R22,R23,R33,R15,R16,R26,R17,R18,R27,R28,R37,R38,R78

(2.22)
so53 > 

Sigma[2], ParabolicSubalgebra(Sigma[2], P, method = "non-compact");

110,011,R11,R12,R13,R22,R23,R33,R15,R16,R26,R17,R18,R27,R28,R37,R38,R67,R68,R78

(2.23)
so53 > 

Sigma[3], ParabolicSubalgebra(Sigma[3], P, method = "non-compact");

011,001,R11,R12,R13,R21,R22,R23,R33,R15,R16,R26,R17,R18,R27,R28,R37,R38,R78

(2.24)
so53 > 

Sigma[4], ParabolicSubalgebra(Sigma[4], P, method = "non-compact");

110,001,R11,R12,R13,R22,R23,R32,R33,R15,R16,R26,R17,R18,R27,R28,R37,R38,R78

(2.25)
so53 > 

Sigma[5], ParabolicSubalgebra(Sigma[5], P, method = "non-compact");

110,R11,R12,R13,R22,R23,R32,R33,R15,R16,R26,R53,R17,R18,R27,R28,R37,R38,R57,R58,R67,R68,R78

(2.26)
so53 > 

Sigma[6], ParabolicSubalgebra(Sigma[6], P, method = "non-compact");

011,R11,R12,R13,R21,R22,R23,R33,R15,R16,R26,R17,R18,R27,R28,R37,R38,R67,R68,R78

(2.27)
so53 > 

Sigma[7], ParabolicSubalgebra(Sigma[7], P, method = "non-compact");

001,R11,R12,R13,R21,R22,R23,R31,R32,R33,R15,R16,R26,R17,R18,R27,R28,R37,R38,R78

(2.28)
so53 > 

Sigma[8], ParabolicSubalgebra(Sigma[8], P, method = "non-compact");

,R11,R12,R13,R21,R22,R23,R31,R32,R33,R15,R16,R26,R42,R43,R53,R17,R18,R27,R28,R37,R38,R47,R48,R57,R58,R67,R68,R78

(2.29)

Check that the subalgebra defined by (2.26) is parabolic.

so53 > 

PS5 := ParabolicSubalgebra(Sigma[5], P, method = "non-compact");

PS5:=R11,R12,R13,R22,R23,R32,R33,R15,R16,R26,R53,R17,R18,R27,R28,R37,R38,R57,R58,R67,R68,R78

(2.30)
so53 > 

Query(PS5, "Parabolic");

true

(2.31)

 

Find the restricted roots used to define PS5 .

so53 > 

ParabolicSubalgebraRoots(PS5, P, method = "non-compact");

110

(2.32)

See Also

DifferentialGeometry

CartanSubalgebra

Killing

LieAlgebras

PositiveRoots,

SimpleRoots

RootSpaceDecomposition

RestrictedRootSpaceDecomposition

QuadraticFormSignature

SimpleLieAlgebraData

SimpleLieAlgebraProperties