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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(T2, method = "non-compact")

Parameters

$\mathrm{Σ}$       - 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 "RestrictedRootSpaceDecomposition", "CartanSubalgebra", "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 ${}_{\mathrm{Σ}}$where the sum is taken over those such that . Then the subspaces

  and  

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

 > $\mathrm{with}\left(\mathrm{DifferentialGeometry}\right):$$\mathrm{with}\left(\mathrm{LieAlgebras}\right):$

Example 1.

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

 > $\mathrm{LD}â‰”\mathrm{SimpleLieAlgebraData}\left("sl\left(4\right)",\mathrm{sl4},\mathrm{labelformat}="gl",\mathrm{labels}=\left[E,\mathrm{ω}\right]\right):$
 > $\mathrm{DGsetup}\left(\mathrm{LD}\right)$
 ${\mathrm{Lie algebra: sl4}}$ (2.1)
 sl4 > $Pâ‰”\mathrm{SimpleLieAlgebraProperties}\left(\mathrm{sl4}\right):$

We use the command SimpleLieAlgebraProperties to create a table containing the structure properties of $\mathrm{sl}\left(4\right)$.

 > $Tâ‰”\mathrm{SimpleLieAlgebraProperties}\left(\mathrm{sl4}\right):$
 sl4 > $\mathrm{SR}â‰”{T}_{"SimpleRoots"}$

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

 sl4 > $\mathrm{Σ}â‰”\left[\left[\right],{\mathrm{SR}}_{1..1},{\mathrm{SR}}_{2..2},{\mathrm{SR}}_{3..3},{\mathrm{SR}}_{1..2},{\mathrm{SR}}_{2..3},\left[{\mathrm{SR}}_{1},{\mathrm{SR}}_{3}\right],\mathrm{SR}\right]$

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

 sl4 > ${\mathrm{Σ}}_{1},\mathrm{GradeSemiSimpleLieAlgebra}\left({\mathrm{Σ}}_{1},P\right)$
 $\left[{}\right]{,}{\mathrm{table}}\left(\left[{0}{=}\left[{\mathrm{E11}}{,}{\mathrm{E22}}{,}{\mathrm{E33}}{,}{\mathrm{E12}}{,}{\mathrm{E23}}{,}{\mathrm{E34}}{,}{\mathrm{E13}}{,}{\mathrm{E24}}{,}{\mathrm{E14}}{,}{\mathrm{E21}}{,}{\mathrm{E32}}{,}{\mathrm{E43}}{,}{\mathrm{E31}}{,}{\mathrm{E42}}{,}{\mathrm{E41}}\right]\right]\right)$ (2.2)
 sl4 > ${\mathrm{Σ}}_{2},\mathrm{GradeSemiSimpleLieAlgebra}\left({\mathrm{Σ}}_{2},P\right)$
 sl4 > ${\mathrm{Σ}}_{3},\mathrm{GradeSemiSimpleLieAlgebra}\left({\mathrm{Σ}}_{3},P\right)$
 sl4 > ${\mathrm{Σ}}_{4},\mathrm{GradeSemiSimpleLieAlgebra}\left({\mathrm{Σ}}_{4},P\right)$
 sl4 > ${\mathrm{Σ}}_{5},\mathrm{GradeSemiSimpleLieAlgebra}\left({\mathrm{Σ}}_{5},P\right)$
 sl4 > ${\mathrm{Σ}}_{6},\mathrm{GradeSemiSimpleLieAlgebra}\left({\mathrm{Σ}}_{6},P\right)$
 sl4 > ${\mathrm{Σ}}_{7},\mathrm{GradeSemiSimpleLieAlgebra}\left({\mathrm{Σ}}_{7},P\right)$
 sl4 > ${\mathrm{Σ}}_{8},\mathrm{GradeSemiSimpleLieAlgebra}\left({\mathrm{Σ}}_{8},P\right)$
 sl4 > ${\mathrm{Σ}}_{2},\mathrm{GradeSemiSimpleLieAlgebra}\left({\mathrm{Σ}}_{2},P\right)$

The Query command can be used to check that each of these define a grading of $\mathrm{sl}\left(4\right)$.

 sl4 > $\mathrm{G7}â‰”\mathrm{GradeSemiSimpleLieAlgebra}\left({\mathrm{Σ}}_{7},P\right)$
 ${\mathrm{G7}}{:=}{\mathrm{table}}\left(\left[{0}{=}\left[{\mathrm{E11}}{,}{\mathrm{E22}}{,}{\mathrm{E33}}{,}{\mathrm{E23}}{,}{\mathrm{E32}}\right]{,}{1}{=}\left[{\mathrm{E12}}{,}{\mathrm{E34}}{,}{\mathrm{E13}}{,}{\mathrm{E24}}\right]{,}{2}{=}\left[{\mathrm{E14}}\right]{,}{-}{2}{=}\left[{\mathrm{E41}}\right]{,}{-}{1}{=}\left[{\mathrm{E21}}{,}{\mathrm{E43}}{,}{\mathrm{E31}}{,}{\mathrm{E42}}\right]\right]\right)$ (2.3)
 sl4 > $\mathrm{Query}\left(\mathrm{G7},"Gradation"\right)$
 ${\mathrm{true}}$ (2.4)

Example 2.

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

 sl4 > $\mathrm{LD2}â‰”\mathrm{SimpleLieAlgebraData}\left("so\left(5,3\right)",\mathrm{so53},\mathrm{labelformat}="gl",\mathrm{labels}=\left[R,\mathrm{θ}\right]\right):$
 sl4 > $\mathrm{DGsetup}\left(\mathrm{LD2}\right)$
 ${\mathrm{Lie algebra: so53}}$ (2.5)



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

 so53 > $Tâ‰”\mathrm{SimpleLieAlgebraProperties}\left(\mathrm{so53}\right):$
 so53 > $\mathrm{RSR}â‰”{T}_{"RestrictedSimpleRoots"}$

The subsets of the restricted simple roots are:

 so53 > $\mathrm{Σ}â‰”\left[\mathrm{RSR},{\mathrm{RSR}}_{1..2},{\mathrm{RSR}}_{2..3},\left[{\mathrm{RSR}}_{1},{\mathrm{RSR}}_{3}\right],{\mathrm{RSR}}_{1..1},{\mathrm{RSR}}_{2..2},{\mathrm{RSR}}_{3..3},\left[\right]\right]$

Here are the possible gradings for

 so53 > ${\mathrm{Σ}}_{1},\mathrm{GradeSemiSimpleLieAlgebra}\left({\mathrm{Σ}}_{1},T,\mathrm{method}="non-compact"\right)$
 so53 > ${\mathrm{Σ}}_{2},\mathrm{GradeSemiSimpleLieAlgebra}\left({\mathrm{Σ}}_{2},T,\mathrm{method}="non-compact"\right)$
 so53 > ${\mathrm{Σ}}_{3},\mathrm{GradeSemiSimpleLieAlgebra}\left({\mathrm{Σ}}_{3},T,\mathrm{method}="non-compact"\right)$
 so53 > ${\mathrm{Σ}}_{4},\mathrm{GradeSemiSimpleLieAlgebra}\left({\mathrm{Σ}}_{4},T,\mathrm{method}="non-compact"\right)$
 so53 > ${\mathrm{Σ}}_{5},\mathrm{GradeSemiSimpleLieAlgebra}\left({\mathrm{Σ}}_{5},T,\mathrm{method}="non-compact"\right)$
 so53 > ${\mathrm{Σ}}_{6},\mathrm{GradeSemiSimpleLieAlgebra}\left({\mathrm{Σ}}_{6},T,\mathrm{method}="non-compact"\right)$
 so53 > ${\mathrm{Σ}}_{7},\mathrm{GradeSemiSimpleLieAlgebra}\left({\mathrm{Σ}}_{7},T,\mathrm{method}="non-compact"\right)$
 so53 > ${\mathrm{Σ}}_{8},\mathrm{GradeSemiSimpleLieAlgebra}\left({\mathrm{Σ}}_{8},T,\mathrm{method}="non-compact"\right)$
 $\left[{}\right]{,}{\mathrm{table}}\left(\left[{0}{=}\left[{\mathrm{R78}}{,}{\mathrm{R33}}{,}{\mathrm{R22}}{,}{\mathrm{R11}}{,}{\mathrm{R16}}{,}{\mathrm{R13}}{,}{\mathrm{R17}}{,}{\mathrm{R18}}{,}{\mathrm{R26}}{,}{\mathrm{R27}}{,}{\mathrm{R28}}{,}{\mathrm{R12}}{,}{\mathrm{R23}}{,}{\mathrm{R15}}{,}{\mathrm{R37}}{,}{\mathrm{R38}}{,}{\mathrm{R43}}{,}{\mathrm{R31}}{,}{\mathrm{R47}}{,}{\mathrm{R48}}{,}{\mathrm{R53}}{,}{\mathrm{R57}}{,}{\mathrm{R58}}{,}{\mathrm{R21}}{,}{\mathrm{R32}}{,}{\mathrm{R42}}{,}{\mathrm{R67}}{,}{\mathrm{R68}}\right]\right]\right)$ (2.6)

The Query command can be used to check that each of these define a grading of $\mathrm{so}\left(5,3\right)$.

 so53 > $\mathrm{G1}â‰”\mathrm{GradeSemiSimpleLieAlgebra}\left({\mathrm{Σ}}_{1},T,\mathrm{method}="non-compact"\right)$
 ${\mathrm{G1}}{:=}{\mathrm{table}}\left(\left[{0}{=}\left[{\mathrm{R78}}{,}{\mathrm{R33}}{,}{\mathrm{R22}}{,}{\mathrm{R11}}\right]{,}{1}{=}\left[{\mathrm{R12}}{,}{\mathrm{R23}}{,}{\mathrm{R37}}{,}{\mathrm{R38}}\right]{,}{2}{=}\left[{\mathrm{R13}}{,}{\mathrm{R27}}{,}{\mathrm{R28}}\right]{,}{3}{=}\left[{\mathrm{R17}}{,}{\mathrm{R18}}{,}{\mathrm{R26}}\right]{,}{5}{=}\left[{\mathrm{R15}}\right]{,}{4}{=}\left[{\mathrm{R16}}\right]{,}{-}{5}{=}\left[{\mathrm{R42}}\right]{,}{-}{4}{=}\left[{\mathrm{R43}}\right]{,}{-}{3}{=}\left[{\mathrm{R47}}{,}{\mathrm{R48}}{,}{\mathrm{R53}}\right]{,}{-}{2}{=}\left[{\mathrm{R31}}{,}{\mathrm{R57}}{,}{\mathrm{R58}}\right]{,}{-}{1}{=}\left[{\mathrm{R21}}{,}{\mathrm{R32}}{,}{\mathrm{R67}}{,}{\mathrm{R68}}\right]\right]\right)$ (2.7)
 so53 > $\mathrm{Query}\left(\mathrm{G1},"Gradation"\right)$
 ${\mathrm{true}}$ (2.8)