GradeSemiSimpleLieAlgebra

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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( $Σ$ $,$ T1)

GradeSemiSimpleLieAlgebra( $Σ,$ T2, method = "non-compact")

Parameters

$Σ$ - 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

Examples

See Also

Description

•

Let g be a Lie algebra. A grading of g is a (vector space) direct sum decomposition g = $⨁_{k \in &integers;}^{}$ ${&gfr;}_{k}$ where $[{&gfr;}_{k}, {&gfr;}_{l}] \subseteq {&gfr;}_{k + l} &period;$ Gradings of semi-simple Lie algebras can easily be constructed from the root space decomposition. Let h be a Cartan subalgebra and $&gfr; = &hfr; ⨁_{α \in Δ}^{} R_{α}$ the associated root space decomposition Let $Δ^{+}$ be a choice of positive roots and let $Δ^{0} \subseteq Δ^{+}$ be a set of simple roots. Every root α is a sum of simple roots, say $α_{} = Σ_{i = 1}^{m} a_{i} α_{i},$ and one defines the height of the root $α_{}$ as ht $(α) = Σ_{i = 1}^{m} a_{i}$ .

•	Now let $Σ \subseteq Δ^{0}$ be a collection of simple roots and define the Σ height of $α_{}$ as ht $_{Σ}$ $(α) = Σ_{i}^{} a_{i},$ where the sum is taken over those $i$ such that $α \in Σ_{}$ . Then the subspaces

${&gfr;}_{t} = ⨁_{α_{} &colon; {ht}_{Σ} (α_{}) = t} R_{α_{}}$ and ${&gfr;}_{0} = &hfr; \oplus ⨁_{α_{} &colon; {ht}_{Σ} (α_{}) = 0} R_{α_{}}$

define a (symmetric) grading g = $⨁_{t = - k}^{k} 𝔤_{t} &period;$

•	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

>	$with (DifferentialGeometry) &colon;$ $with (LieAlgebras) &colon;$

Example 1.

We calculate the various gradations for $sl (4) &period;$ We use the command SimpleLieAlgebraData to initialize the Lie algebra.

>	$LD ≔ SimpleLieAlgebraData (sl(4), sl4, labelformat = gl, labels = [E, ω]) &colon;$

>	$DGsetup (LD)$

$Lie algebra: sl4$

(2.1)

sl4 >

$P ≔ SimpleLieAlgebraProperties (sl4) &colon;$

We use the command SimpleLieAlgebraProperties to create a table $T$ containing the structure properties of $sl (4)$ .

>	$T ≔ SimpleLieAlgebraProperties (sl4) &colon;$

sl4 >

$SR ≔ T [SimpleRoots]$

$SR := [[\begin{array}{r} 1 \\ - 1 \\ 0 \end{array}], [\begin{array}{r} 0 \\ 1 \\ - 1 \end{array}], [\begin{array}{r} 1 \\ 1 \\ 2 \end{array}]]$

(2.2)

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

sl4 >

$Σ ≔ [[], SR [1 .. 1], SR [2 .. 2], SR [3 .. 3], SR [1 .. 2], SR [2 .. 3], [SR [1], SR [3]], SR]$

$Σ := [[], [[\begin{array}{r} 1 \\ - 1 \\ 0 \end{array}]], [[\begin{array}{r} 0 \\ 1 \\ - 1 \end{array}]], [[\begin{array}{r} 1 \\ 1 \\ 2 \end{array}]], [[\begin{array}{r} 1 \\ - 1 \\ 0 \end{array}], [\begin{array}{r} 0 \\ 1 \\ - 1 \end{array}]], [[\begin{array}{r} 0 \\ 1 \\ - 1 \end{array}], [\begin{array}{r} 1 \\ 1 \\ 2 \end{array}]], [[\begin{array}{r} 1 \\ - 1 \\ 0 \end{array}], [\begin{array}{r} 1 \\ 1 \\ 2 \end{array}]], [[\begin{array}{r} 1 \\ - 1 \\ 0 \end{array}], [\begin{array}{r} 0 \\ 1 \\ - 1 \end{array}], [\begin{array}{r} 1 \\ 1 \\ 2 \end{array}]]]$

(2.3)

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

sl4 >

$Σ [1], GradeSemiSimpleLieAlgebra (Σ [1], P)$

$[], table ([0 = [E11, E22, E33, E12, E23, E34, E13, E24, E14, E21, E32, E43, E31, E42, E41]])$

(2.4)

sl4 >

$Σ [2], GradeSemiSimpleLieAlgebra (Σ [2], P)$

$[[\begin{array}{r} 1 \\ - 1 \\ 0 \end{array}]], table ([0 = [E11, E22, E33, E23, E34, E24, E32, E43, E42], 1 = [E12, E13, E14], - 1 = [E21, E31, E41]])$

(2.5)

sl4 >

$Σ [3], GradeSemiSimpleLieAlgebra (Σ [3], P)$

$[[\begin{array}{r} 0 \\ 1 \\ - 1 \end{array}]], table ([0 = [E11, E22, E33, E12, E34, E21, E43], 1 = [E23, E13, E24, E14], - 1 = [E32, E31, E42, E41]])$

(2.6)

sl4 >

$Σ [4], GradeSemiSimpleLieAlgebra (Σ [4], P)$

$[[\begin{array}{r} 1 \\ 1 \\ 2 \end{array}]], table ([0 = [E11, E22, E33, E12, E23, E13, E21, E32, E31], 1 = [E34, E24, E14], - 1 = [E43, E42, E41]])$

(2.7)

sl4 >

$Σ [5], GradeSemiSimpleLieAlgebra (Σ [5], P)$

$[[\begin{array}{r} 1 \\ - 1 \\ 0 \end{array}], [\begin{array}{r} 0 \\ 1 \\ - 1 \end{array}]], table ([0 = [E11, E22, E33, E34, E43], 1 = [E12, E23, E24], 2 = [E13, E14], - 2 = [E31, E41], - 1 = [E21, E32, E42]])$

(2.8)

sl4 >

$Σ [6], GradeSemiSimpleLieAlgebra (Σ [6], P)$

$[[\begin{array}{r} 0 \\ 1 \\ - 1 \end{array}], [\begin{array}{r} 1 \\ 1 \\ 2 \end{array}]], table ([0 = [E11, E22, E33, E12, E21], 1 = [E23, E34, E13], 2 = [E24, E14], - 2 = [E42, E41], - 1 = [E32, E43, E31]])$

(2.9)

sl4 >

$Σ [7], GradeSemiSimpleLieAlgebra (Σ [7], P)$

$[[\begin{array}{r} 1 \\ - 1 \\ 0 \end{array}], [\begin{array}{r} 1 \\ 1 \\ 2 \end{array}]], table ([0 = [E11, E22, E33, E23, E32], 1 = [E12, E34, E13, E24], 2 = [E14], - 2 = [E41], - 1 = [E21, E43, E31, E42]])$

(2.10)

sl4 >

$Σ [8], GradeSemiSimpleLieAlgebra (Σ [8], P)$

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

(2.11)

sl4 >

$Σ [2], GradeSemiSimpleLieAlgebra (Σ [2], P)$

$[[\begin{array}{r} 1 \\ - 1 \\ 0 \end{array}]], table ([0 = [E11, E22, E33, E23, E34, E24, E32, E43, E42], 1 = [E12, E13, E14], - 1 = [E21, E31, E41]])$

(2.12)

The Query command can be used to check that each of these define a grading of $sl (4)$ .

sl4 >

$G7 ≔ GradeSemiSimpleLieAlgebra (Σ [7], P)$

$G7 := table ([0 = [E11, E22, E33, E23, E32], 1 = [E12, E34, E13, E24], 2 = [E14], - 2 = [E41], - 1 = [E21, E43, E31, E42]])$

(2.13)

sl4 >

$Query (G7, Gradation)$

$true$

(2.14)

Example 2.

We calculate the various gradings for $so (5, 3) &period;$ We use the command SimpleLieAlgebraData to initialize the Lie algebra.

sl4 >

$LD2 ≔ SimpleLieAlgebraData (so(5,3), so53, labelformat = gl, labels = [R, θ]) &colon;$

sl4 >

$DGsetup (LD2)$

$Lie algebra: so53$

(2.15)

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

so53 >

$T ≔ SimpleLieAlgebraProperties (so53) &colon;$

so53 >

$RSR ≔ T [RestrictedSimpleRoots]$

$RSR := [[\begin{array}{r} 1 \\ - 1 \\ 0 \end{array}], [\begin{array}{r} 0 \\ 1 \\ - 1 \end{array}], [\begin{array}{r} 0 \\ 0 \\ 1 \end{array}]]$

(2.16)

The subsets of the restricted simple roots are:

so53 >

$Σ ≔ [RSR, RSR [1 .. 2], RSR [2 .. 3], [RSR [1], RSR [3]], RSR [1 .. 1], RSR [2 .. 2], RSR [3 .. 3], []]$

Here are the possible gradings for $so (5, 3) &period;$

so53 >

$Σ [1], GradeSemiSimpleLieAlgebra (Σ [1], T, method = non-compact)$

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

(2.17)

so53 >

$Σ [2], GradeSemiSimpleLieAlgebra (Σ [2], T, method = non-compact)$

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

(2.18)

so53 >

$Σ [3], GradeSemiSimpleLieAlgebra (Σ [3], T, method = non-compact)$

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

(2.19)

so53 >

$Σ [4], GradeSemiSimpleLieAlgebra (Σ [4], T, method = non-compact)$

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

(2.20)

so53 >

$Σ [5], GradeSemiSimpleLieAlgebra (Σ [5], T, method = non-compact)$

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

(2.21)

so53 >

$Σ [6], GradeSemiSimpleLieAlgebra (Σ [6], T, method = non-compact)$

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

(2.22)

so53 >

$Σ [7], GradeSemiSimpleLieAlgebra (Σ [7], T, method = non-compact)$

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

(2.23)

so53 >

$Σ [8], GradeSemiSimpleLieAlgebra (Σ [8], T, method = non-compact)$

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

(2.24)

The Query command can be used to check that each of these define a grading of $so (5, 3)$ .

so53 >

$G1 ≔ GradeSemiSimpleLieAlgebra (Σ [1], T, method = non-compact)$

$G1 := table ([0 = [R78, R33, R22, R11], 1 = [R12, R23, R37, R38], 2 = [R13, R27, R28], 3 = [R17, R18, R26], 5 = [R15], 4 = [R16], - 5 = [R42], - 4 = [R43], - 3 = [R47, R48, R53], - 2 = [R31, R57, R58], - 1 = [R21, R32, R67, R68]])$

(2.25)

so53 >

$Query (G1, Gradation)$

$true$

(2.26)

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

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