find the grading of a semi-simple Lie algebra defined by a set of simple roots or restricted simple roots

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LieAlgebras[GradeSemiSimpleLieAlgebra] - find the grading of a semi-simple Lie algebra defined by a set of simple roots or restricted simple roots

Calling Sequences

GradeSemiSimpleLieAlgebra(1)

GradeSemiSimpleLieAlgebra(2, method = "non-compact")

Parameters

Sigma - a list or set of column vectors, defining a subset of the simple roots or a subset of the restricted simple roots

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

T2 - a table, with indices that include "RestrictedRoot SpaceDecomposition", "CartanSubalgebraDecomposition", "RestrictedSimpleRoots", "RestrictedPositiveRoots"

Description

•

Let g be a Lie algebra. A grading of g is a (vector space) direct sum decomposition g =where Gradings of semi-simple Lie algebras can easily be constructed from the root space decomposition. Let h be a Cartan subalgebra and the associated root space decomposition. Let be a choice of positive roots and let be a set of simple roots. Every root α is a sum of simple roots, say and one defines the height of the root as ht.

•	Now let be a collection of simple roots and define the Σ height of as ht where the sum is taken over those such that . Then the subspaces

`&gfr;`[t] = (&bigoplus;)[alpha[] : ht[Sigma](alpha[]) =t] R[alpha[]] and `&gfr;`[0] = `&hfr;` (&bigoplus;)[alpha[] : ht[Sigma](alpha[]) =0] R[alpha[]]

define a (symmetric) grading g =

•	For real Lie algebras, real gradings can be similarly constructed using the restricted root space decomposition.

•	The command Query/"Gradation" will test if a given decomposition of a Lie algebra is graded.

Examples

Example 1.

We calculate the various gradations for We use the command SimpleLieAlgebraData to initialize the Lie algebra.

(2.1)

sl4 >

We use the command SimpleLieAlgebraProperties to create a table containing the structure properties of .

sl4 >

SR := [Vector(3, {(1) = 1, (2) = -1, (3) = 0}), Vector(3, {(1) = 0, (2) = 1, (3) = -1}), Vector(3, {(1) = 1, (2) = 1, (3) = 2})]

(2.2)

Here are the possible subsets of the set of simple roots.

sl4 >

(2.3)

Here are the gradings defined by each subset of the simple roots.

sl4 >

(2.4)

sl4 >

[Vector(3, {(1) = 1, (2) = -1, (3) = 0})], table([0 = [E11, E22, E33, E23, E34, E24, E32, E43, E42], 1 = [E12, E13, E14], -1 = [E21, E31, E41]])

(2.5)

sl4 >

[Vector(3, {(1) = 0, (2) = 1, (3) = -1})], table([0 = [E11, E22, E33, E12, E34, E21, E43], 1 = [E23, E13, E24, E14], -1 = [E32, E31, E42, E41]])

(2.6)

sl4 >

[Vector(3, {(1) = 1, (2) = 1, (3) = 2})], table([0 = [E11, E22, E33, E12, E23, E13, E21, E32, E31], 1 = [E34, E24, E14], -1 = [E43, E42, E41]])

(2.7)

sl4 >

[Vector(3, {(1) = 1, (2) = -1, (3) = 0}), Vector(3, {(1) = 0, (2) = 1, (3) = -1})], table([0 = [E11, E22, E33, E34, E43], 1 = [E12, E23, E24], 2 = [E13, E14], -2 = [E31, E41], -1 = [E21, E32, E42]])

(2.8)

sl4 >

[Vector(3, {(1) = 0, (2) = 1, (3) = -1}), Vector(3, {(1) = 1, (2) = 1, (3) = 2})], table([0 = [E11, E22, E33, E12, E21], 1 = [E23, E34, E13], 2 = [E24, E14], -2 = [E42, E41], -1 = [E32, E43, E31]])

(2.9)

sl4 >

[Vector(3, {(1) = 1, (2) = -1, (3) = 0}), Vector(3, {(1) = 1, (2) = 1, (3) = 2})], table([0 = [E11, E22, E33, E23, E32], 1 = [E12, E34, E13, E24], 2 = [E14], -2 = [E41], -1 = [E21, E43, E31, E42]])

(2.10)

sl4 >

[Vector(3, {(1) = 1, (2) = -1, (3) = 0}), Vector(3, {(1) = 0, (2) = 1, (3) = -1}), Vector(3, {(1) = 1, (2) = 1, (3) = 2})], table([0 = [E11, E22, E33], 1 = [E12, E23, E34], 2 = [E13, E24], 3 = [E14], -3 = [E41], -2 = [E31, E42], -1 = [E21, E32, E43]])

(2.11)

sl4 >

[Vector(3, {(1) = 1, (2) = -1, (3) = 0})], table([0 = [E11, E22, E33, E23, E34, E24, E32, E43, E42], 1 = [E12, E13, E14], -1 = [E21, E31, E41]])

(2.12)

The Query command can be used to check that each of these define a grading of .

sl4 >

(2.13)

sl4 >

(2.14)

Example 2.

We calculate the various gradings for We use the command SimpleLieAlgebraData to initialize the Lie algebra.

sl4 >

(2.15)

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

so53 >

RSR := [Vector(3, {(1) = 1, (2) = -1, (3) = 0}), Vector(3, {(1) = 0, (2) = 1, (3) = -1}), Vector(3, {(1) = 0, (2) = 0, (3) = 1})]

(2.16)

The subsets of the restricted simple roots are:

so53 >

Here are the possible gradings for

so53 >

[Vector(3, {(1) = 1, (2) = -1, (3) = 0}), Vector(3, {(1) = 0, (2) = 1, (3) = -1}), Vector(3, {(1) = 0, (2) = 0, (3) = 1})], table([0 = [R22, R11, R78, R33], 1 = [R12, R23, R37, R38], 2 = [R13, R27, R28], 3 = [R17, R18, R26], 5 = [R15], 4 = [R16], -5 = [R42], -4 = [R43], -3 = [R47, R48, R53], -2 = [R31, R57, R58], -1 = [R21, R32, R67, R68]])

(2.17)

so53 >

[Vector(3, {(1) = 1, (2) = -1, (3) = 0}), Vector(3, {(1) = 0, (2) = 1, (3) = -1})], table([0 = [R78, R33, R22, R11, R37, R38, R67, R68], 1 = [R26, R27, R28, R12, R23], 2 = [R16, R13, R17, R18], 3 = [R15], -3 = [R42], -2 = [R43, R31, R47, R48], -1 = [R53, R57, R58, R21, R32]])

(2.18)

so53 >

[Vector(3, {(1) = 0, (2) = 1, (3) = -1}), Vector(3, {(1) = 0, (2) = 0, (3) = 1})], table([0 = [R78, R33, R22, R11, R12, R21], 1 = [R13, R23, R37, R38], 2 = [R17, R18, R27, R28], 3 = [R16, R26], 4 = [R15], -4 = [R42], -3 = [R43, R53], -2 = [R47, R48, R57, R58], -1 = [R31, R32, R67, R68]])

(2.19)

so53 >

[Vector(3, {(1) = 1, (2) = -1, (3) = 0}), Vector(3, {(1) = 0, (2) = 0, (3) = 1})], table([0 = [R78, R33, R22, R11, R23, R32], 1 = [R13, R27, R28, R12, R37, R38], 2 = [R17, R18, R26], 3 = [R16, R15], -3 = [R43, R42], -2 = [R47, R48, R53], -1 = [R31, R57, R58, R21, R67, R68]])

(2.20)

so53 >

[Vector(3, {(1) = 1, (2) = -1, (3) = 0})], table([0 = [R78, R33, R22, R11, R26, R27, R28, R23, R37, R38, R53, R57, R58, R32, R67, R68], 1 = [R16, R13, R17, R18, R12, R15], -1 = [R43, R31, R47, R48, R21, R42]])

(2.21)

so53 >

[Vector(3, {(1) = 0, (2) = 1, (3) = -1})], table([0 = [R78, R33, R22, R11, R12, R37, R38, R21, R67, R68], 1 = [R16, R13, R17, R18, R26, R27, R28, R23], 2 = [R15], -2 = [R42], -1 = [R43, R31, R47, R48, R53, R57, R58, R32]])

(2.22)

so53 >

[Vector(3, {(1) = 0, (2) = 0, (3) = 1})], table([0 = [R78, R33, R22, R11, R13, R12, R23, R31, R21, R32], 1 = [R17, R18, R27, R28, R37, R38], 2 = [R16, R26, R15], -2 = [R43, R53, R42], -1 = [R47, R48, R57, R58, R67, R68]])

(2.23)

so53 >

(2.24)

The Query command can be used to check that each of these define a grading of .

so53 >

(2.25)

so53 >

(2.26)

	See Also
	DifferentialGeometry, CartanSubalgebra, KillingForm, LieAlgebras, PositiveRoots, Query, SimpleRoots, RootSpaceDecomposition, RestrictedRootSpaceDecomposition, SimpleLieAlgebraData, SimpleLieAlgebraProperties