SimpleLieAlgebraProperties

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LieAlgebras[SimpleLieAlgebraProperties] - provide a table of properties for any real simple Lie algebra

Calling Sequences

SimpleLieAlgebraProperties(alg)

Parameters

alg - a name or string, the name of a real simple Lie algebra created using SimpleLieAlgebraData

Description

Examples

Description

•

The DifferentialGeometry package provides two different approaches for studying simple and semi-simple Lie algebras. If a semi-simple Lie algebra arises, for example, as the symmetries of some geometric objects (metric, connections, differential equations) then there is an extensive set of commands for analyzing the structure of this algebra. These commands include CartanSubalgebra, CartanMatrix, CartanMatrixToStandardForm, Killing, PositiveRoots, SimpleRoots, RootSpaceDecomposition, Cartan Decomposition. If one wishes to work with a specific real simple Lie algebra, then the structure equations are available with the command SimpleLieAlgebraData. Many structural properties of the simple Lie algebras has been tabulated and are available using the procedure SimpleLieAlgebraProperties.

•

The procedure SimpleLieAlgebraProperties returns a table whose indices specify the properties that are available for the given algebra. Let $&gfr;$ be a simple Lie algebra of dimension $n$ and rank $m$ with Cartan subalgebra h, root space decomposition $&gfr; = &hfr; \oplus ⨁_{α \in Δ_{r}}^{}_{} R_{α}$ , and restricted root space decomposition $&gfr; = {&gfr;}_{0} \oplus ⨁_{α \in Δ}^{} S_{α} &period;$

•	The following properties are computed.

"CartanDecomposition" : 2 lists of vectors k and p; $&gfr; = &kfr; \oplus &pfr;,$ where t is a subalgebra, p a subspace, [t, p] ⊆ p and [p, p] ⊆ t. The Killing form is negative-definite on t and positive-definite on p. ) Cartan Decomposition	"CartanInvolution" : a transformation Θ : g → g with $Θ^{2} = Id$ and such that the symmetry bilinear form $B_{Θ_{}} (x, y) = - ⟨x, Θ (y)⟩$ is positive-definite.) CartanInvolution
"CartanMatrix" : an $m \times m$ matrix, the standard Cartan matrix for the root type of the Lie algebra. CartanMatrix	$"CartanSubalgebra$ " : a list of $m$ vectors. A Cartan subalgebra h is a nilpotent subalgebra whose normalizer in g is itself, that is, $nor (&hfr;) = &hfr;$ . The Cartan subalgebras for the classic matrix algebras are diagonal matrices. A basis is chosen such that the roots consist of integers or pure imaginary numbers (with integer coefficients). CartanSubalgebra,
"CartanSubalgebraDecomposition" : 2 lists of vectors spanning the Cartan subalgebra h. The Killing form is negative-definite on the first list and positive-definite on the second list. $&clubs;$	"IwasawaDecomposition" : 3 lists of vectors $&kfr;, &afr;, &nfr;$ such that $&gfr; = &kfr; \oplus &afr; \oplus &nfr;$ , where k is a compact semi-simple subalgebra; $&afr;$ is abelian, and n nilpotent. The Killing form is positive-definite on $&afr; &period; &clubs;$
"KillingMatrix" : $n \times n$ symmetric, non-degenerate matrix. Killing	"NegativeRootSpaces" : a table, the indices are the roots α ∈ $Δ^{-}$ (as lists) and the entries are vectors defining the root spaces $R_{α}$ . RootSpaceDecomposition
"NegativeRoots" : a list of column vectors defining the negative roots $Δ^{-} &period;$ A root is negative if its first non-zero component is $k$ or $kI$ , where $k$ is a negative integer.	"PositiveRootSpaces" : a table, the indices are the roots α ∈ $Δ^{+}$ (as lists) and the entries are vectors defining the root spaces $R_{α}$ . RootSpaceDecomposition
"PositiveRoots" : a list of column vectors defining the positive roots $Δ^{+} &period;$ A root is positive if its first non-zero component is $k$ or $kI$ , where $k$ is a positive integer. PositiveRoots	"RestrictedPositiveRoots" : a table, the indices are the restricted positive roots α ∈ $Δ_{r}^{+}$ (as lists) and the entries are lists of vectors defining the root spaces $S_{α}$ .% $♣$
"RestrictedRootSpaceDecomposition" : a table, the indices are the roots α ∈ $Δ_{r}$ (as lists) and the entries are lists of vectors defining the restricted root spaces $S_{α}$ .% $♣$	"RestrictedSimpleRoots" : a list of column vectors, giving the restricted simple roots $Δ_{r}^{0} &period; &diamondsuit; ♣$
"RootSpaceDecomposition" : a table, the indices are the roots α ∈Δ (as lists) and the entries are vectors defining the root spaces $R_{α}$ . RootSpaceDecomposition	"SimpleRootSpaces" : a table, the indices are the simple roots α ∈ $Δ^{0}$ (as lists) and the entries are vectors defining the root spaces $R_{α}$ .
"SimpleRoots" : a list of column vectors $Δ^{0}$ defining the simple roots. Every positive root is a linear combination of the simple roots with positive integer coefficients. SimpleRoots	$&clubs; &colon;$ Not computed for compact Lie algebras, that is, if the Killing form is negative-definite. % : Not computed for split-real forms, that is, if the root space decomposition is real.

•	For split-real forms the standard Borel sub-algebra (see ParabolicSubalgebra) is also given.

•	Many of these properties of simple and semi-simple Lie algebras can be checked with the Query command.

Examples

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

Example 1

We obtain the structural properties for the Lie algebra $sl (4)$ . This is the split real form for Lie algebras of root type A. First, we use the command SimpleLieAlgebraData to initialize this Lie algebra.

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

>	$DGsetup (LD)$

$Lie algebra: sl4$

(2.1)

We use the command SimpleLieAlgebraProperties to obtain the properties of the Lie algebra $sl (4)$ .

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

Here are the indices for the table Properties.

sl4 >

$Ind ≔ indices (Properties)$

$Ind := [CartanInvolution], [CartanDecomposition], [SimpleRoots], [NegativeRootSpaces], [CartanSubalgebra], [BorelSubalgebra], [NegativeRoots], [PositiveRootSpaces], [CartanMatrix], [RootSpaceDecomposition], [PositiveRoots], [SimpleRootSpaces], [KillingForm]$

(2.2)

It is convenient to use the map and op commands to display the indices as a list of strings.

sl3 >

$Ind ≔ sort (map (op, [indices (Properties)]))$

$Ind := [BorelSubalgebra, CartanDecomposition, CartanInvolution, CartanMatrix, CartanSubalgebra, KillingForm, NegativeRootSpaces, NegativeRoots, PositiveRootSpaces, PositiveRoots, RootSpaceDecomposition, SimpleRootSpaces, SimpleRoots]$

(2.3)

Here are some of the individual properties for the Lie algebra $sl (3)$ .

sl3 >

$CSA ≔ Properties [CartanSubalgebra]$

$CSA := [E11, E22, E33]$

(2.4)

sl3 >

$RT ≔ eval (Properties [RootSpaceDecomposition])$

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

(2.5)

The command LieAlgebraRoots lists the roots associated to this root space decomposition. Note that the roots are all real.

sl4 >

$LieAlgebraRoots (RT)$

Note that the first non-zero component of each positive root is positive and that the first non-zero component of each negative root is negative.

sl4 >

$PR ≔ Properties [PositiveRoots]$

sl4 >

$NR ≔ Properties [NegativeRoots]$

It is easy to check that positive roots are positive linear combinations of the simple roots.

sl4 >

$ST ≔ Properties [SimpleRoots]$

sl4 >

$CD ≔ Properties [CartanDecomposition]$

$CD := [E12 - E21, E23 - E32, E34 - E43, E13 - E31, E24 - E42, E14 - E41], [E11, E22, E33, E12 + E21, E23 + E32, E34 + E43, E13 + E31, E24 + E42, E14 + E41]$

(2.6)

We check that the Killing form is positive-definite on the first list of vectors CD[1] and negative-definitive on the second list of vectors.

sl4 >

$Killing (CD [1])$

sl4 >

$Killing (CD [2])$

Example 2

We obtain the structural properties for the Lie algebra $su (4)$ . This is the compact form for Lie algebras of root type A. First, we use the command SimpleLieAlgebraData to initialize this Lie algebra.

>	$LD ≔ SimpleLieAlgebraData (su(4), su4, labelformat = gl, labels = ['U','θ']) &colon;$

>	$DGsetup (LD)$

$Lie algebra: su4$

(2.7)

We use the command SimpleLieAlgebraProperties to obtain the properties of the Lie algebra $su (4)$ .

su4 >

$Properties ≔ SimpleLieAlgebraProperties (su4) &colon;$

It is convenient to use the map and op commands to display the indices as a list of strings.

su4 >

$Ind ≔ sort (map (op, [indices (Properties)]))$

$Ind := [CartanMatrix, CartanSubalgebra, CartanSubalgebraDecomposition, NegativeRootSpaces, NegativeRoots, PositiveRootSpaces, PositiveRoots, RootSpaceDecomposition, SimpleRootSpaces, SimpleRoots]$

(2.8)

Here are some of the individual properties for the Lie algebra $su (4)$ .

su4 >

$CSA ≔ Properties [CartanSubalgebra]$

$CSA := [Ui11, Ui22, Ui33]$

(2.9)

su4 >

$RT ≔ eval (Properties [RootSpaceDecomposition])$

$RT := table ([[- I, 0, I] = U13 + I Ui13, [I, 2 I, I] = U24 - I Ui24, [0, - I, I] = U23 + I Ui23, [0, I, - I] = U23 - I Ui23, [- 2 I, - I, - I] = U14 + I Ui14, [I, - I, 0] = U12 - I Ui12, [2 I, I, I] = U14 - I Ui14, [I, I, 2 I] = U34 - I Ui34, [- I, - I, - 2 I] = U34 + I Ui34, [- I, - 2 I, - I] = U24 + I Ui24, [- I, I, 0] = U12 + I Ui12, [I, 0, - I] = U13 - I Ui13])$

(2.10)

The roots are all pure imaginary numbers so that this is indeed the compact form.

su4 >

$LieAlgebraRoots (RT)$

The first non-zero coefficient of $I$ in each positive root is positive.

su4 >

$PR ≔ Properties [PositiveRoots]$

su4 >

$ST ≔ Properties [SimpleRoots]$

Example 3

We obtain the structural properties for the Lie algebra $su (2, 2)$ . First, we use the command SimpleLieAlgebraData to initialize this Lie algebra.

>	$LD ≔ SimpleLieAlgebraData (su(2, 2), su22, labelformat = gl, labels = ['V','σ']) &colon;$

>	$DGsetup (LD)$

$Lie algebra: su22$

(2.11)

We use the command SimpleLieAlgebraProperties to obtain the properties of the Lie algebra $su (2, 2)$ .

su22 >

$Properties ≔ SimpleLieAlgebraProperties (su22) &colon;$

It is convenient to use the map and op commands to display the indices as a list of strings.

su22 >

$Ind ≔ sort (map (op, [indices (Properties)]))$

$Ind := [CartanDecomposition, CartanInvolution, CartanMatrix, CartanSubalgebra, CartanSubalgebraDecomposition, IwasawaDecomposition, NegativeRootSpaces, NegativeRoots, PositiveRootSpaces, PositiveRoots, RestrictedPositiveRoots, RestrictedRootSpaceDecomposition, RestrictedSimpleRoots, RootSpaceDecomposition, SimpleRootSpaces, SimpleRoots]$

(2.12)

Here are some of the individual properties for the Lie algebra $su (2, 2)$ .

su22 >

$Properties [CartanSubalgebra]$

$[V11, V22, Vi11]$

(2.13)

su22 >

$eval (Properties [RootSpaceDecomposition])$

$table ([[1, - 1, - 2 I] = V12 - I Vi12, [0, 2, 0] = Vi24, [1, 1, - 2 I] = V14 + I Vi14, [1, 1, 2 I] = V14 - I Vi14, [- 1, 1, 2 I] = V21 - I Vi21, [- 1, - 1, - 2 I] = V32 + I Vi32, [- 1, - 1, 2 I] = V32 - I Vi32, [- 1, 1, - 2 I] = V21 + I Vi21, [0, - 2, 0] = Vi42, [1, - 1, 2 I] = V12 + I Vi12, [2, 0, 0] = Vi13, [- 2, 0, 0] = Vi31])$

(2.14)

Note that first two components of the roots are real and the third component is pure imaginary.

su22 >

$Properties [PositiveRoots]$

su22 >

$Properties [SimpleRoots]$

Since the root vectors are neither real nor pure imaginary, we have a restricted root space decomposition.

su22 >

$RRSD ≔ eval (Properties [RestrictedRootSpaceDecomposition])$

$RRSD := table ([[1, - 1] = [V12, Vi12], [2, 0] = [Vi13], [0, - 2] = [Vi42], [0, 2] = [Vi24], [- 1, 1] = [V21, Vi21], [1, 1] = [V14, Vi14], [- 2, 0] = [Vi31], [- 1, - 1] = [V32, Vi32]])$

(2.15)

The restricted roots are the projections of the roots which yield real vectors. Since the restricted root [1,1] is the projection of the 2 roots [1, 1, 2I] and [1, 1, -2I], the restricted root space for [1,1] is 2-dimensional. Note also that while the root spaces are defined over C, the restricted root space are real subspaces of $su (2, 2)$ .

su22 >

$Properties [RestrictedPositiveRoots]$

su22 >

$Properties [RestrictedSimpleRoots]$

su22 >

$Properties [CartanSubalgebraDecomposition]$

$[Vi11], [V11, V22]$

(2.16)

su4 >

$K, P ≔ Properties [CartanDecomposition]$

$K, P := [- V21 + V12, Vi21 + Vi12, V32 + V14, Vi32 + Vi14, Vi42 + Vi24, Vi31 + Vi13, Vi11], [V21 + V12, - Vi21 + Vi12, - V32 + V14, - Vi32 + Vi14, - Vi42 + Vi24, - Vi31 + Vi13, V22, V11]$

(2.17)

su22 >

$Properties [CartanInvolution]$

$[[Vi11, Vi11], [V11, - V11], [V22, - V22], [V21, - V12], [V32, V14], [V12, - V21], [V14, V32], [Vi31, Vi13], [Vi42, Vi24], [Vi21, Vi12], [Vi32, Vi14], [Vi12, Vi21], [Vi13, Vi31], [Vi24, Vi42], [Vi14, Vi32]]$

(2.18)

su22 >

$K, A, N ≔ Properties [IwasawaDecomposition]$

$K, A, N := [- V21 + V12, Vi21 + Vi12, V32 + V14, Vi32 + Vi14, Vi42 + Vi24, Vi31 + Vi13, Vi11], [V11, V22], [V12, Vi12, Vi13, Vi24, V14, Vi14]$

(2.19)

Let's us check the properties of this KAN decomposition. The first list of vectors defines a subalgebra with negative-definite Killing form.

su22 >

$Query (K, Subalgebra)$

$true$

(2.20)

su22 >

$Killing (K)$

The second list of vectors defines an abelian subalgebra.

su22 >

$Query (A, Abelian)$

$true$

(2.21)

The third list of vectors defines a nilpotent Lie algebra.

su22 >

$Query (A, Nilpotent)$

$true$

(2.22)

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