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LieAlgebras[MatrixSubalgebra] - find the subalgebra of a Lie algebra which preserves a collection of tensors or subspaces of tensors

Calling Sequences

MatrixSubalgebra(rho, Inv)

MatrixSubalgebra( $alg$ , M, Inv)

MatrixSubalgebra( $alg$ , Gamma, Inv)

MatrixSubalgebra( $alg$ 1, Inv)

Parameters

rho - a representation of a Lie algebra

Inv - a list, where each element is a tensor or a list of tensors

alg - a name or a string, the name of an initialized Lie algebra $𝔤$

M - a list of square matrices defining a Lie algebra, with the same structure equations as $𝔤$

Gamma - a list of vector fields defining a Lie algebra, with the same structure equations as $𝔤$

alg1 - a name or a string, the name of an initialized Lie algebra $𝔤$ $,$ which has been created by the command $SimpleLieAlgebraData$

Description

Examples

Description

•	Let $V$ be a vector space and $φ_{} &colon; V \to V$ a linear transformation (not necessarily invertible). Let $T$ be a type (1,1) tensor on $V &period;$ Then the (1,1) tensor $φ_{} \cdot T$ is defined by

$(φ \cdot T)$ $(X, α_{}) = T (φ_{} (X), alpha) - T (X, φ^{&ast;} (alpha)),$ where $X \in V$ and $α_{} \in V$ *.

If ${φ^{i}}_{j}^{}$ and $T_{j}^{i}$ are the components of $φ_{}$ and $T$ with respect to a basis $e_{i}$ for $V$ (and dual basis $ϵ^{i}$ for $V^{&ast;}$ ), then ${(φ \cdot T)}_{j}^{i} = φ_{j}^{k} T_{k}^{i} - φ_{k}^{i} T_{j}^{k}$ .

•	This formula extends in the natural way to define $φ_{} \cdot T$ for any tensor $T &period;$ One says that $T$ is $φ -$ invariant if $φ \cdot T_{} = 0.$

•

Let g be a Lie algebra and let $ρ &colon; &gfr; \to gl (V)$ be a representation of $V &period;$ The set a $= {x \in &gfr; | ρ (x) \cdot T = 0}$ is a subalgebra of g. Likewise, if T is a subspace of tensors, then the set b $= {x \in &gfr; | ρ (x) \cdot T \in &Tscr;$ for all $T \in &Tscr;}$ is also a subalgebra of g.

•

The command $MatrixSubalgebra$ allows one to make subalgebras via this general construction. The argument Inv is a list where each element is a tensor or a list of tensors. For example, if $Inv = [R, [S, T]]$ then $MatrixSubalgebra$ calculates the subalgebra consisting of $x \in &gfr;$ such that $ρ (x) \cdot R = 0, ρ (x) \cdot S \in$ span $\{S, T\},$ $ρ (x) \cdot T \in$ span $\{S, T\} &period;$

•

When a Lie algebra is created with the command SimpleLieAlgebraData, its standard matrix representation is encoded in the Lie algebra data structure for that algebra. For such algebras, the construction of subalgebras via invariant tensors can be performed without explicitly specifying a representation.

Examples

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

Example 1.

We construct the Lie algebras $so (3)$ and $so$ (3)⊕ $so$ (2) as subalgebras of $so$ (5). First, here are the 5×5 skew-symmetric matrices which define $so (5) &period;$

$A ≔ map (Matrix, [[[0, - 1, 0, 0, 0], [1, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0]], [[0, 0, - 1, 0, 0], [0, 0, 0, 0, 0], [1, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0]], [[0, 0, 0, - 1, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [1, 0, 0, 0, 0], [0, 0, 0, 0, 0]], [[0, 0, 0, 0, - 1], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [1, 0, 0, 0, 0]], [[0, 0, 0, 0, 0], [0, 0, - 1, 0, 0], [0, 1, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0]], [[0, 0, 0, 0, 0], [0, 0, 0, - 1, 0], [0, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 0, 0, 0]], [[0, 0, 0, 0, 0], [0, 0, 0, 0, - 1], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 1, 0, 0, 0]], [[0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 0, - 1, 0], [0, 0, 1, 0, 0], [0, 0, 0, 0, 0]], [[0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, - 1], [0, 0, 0, 0, 0], [0, 0, 1, 0, 0]], [[0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, - 1], [0, 0, 0, 1, 0]]])$

Calculate the structure equations and initialize.

>	$LD ≔ LieAlgebraData (A, so5)$

$LD := [[e1, e2] = e5, [e1, e3] = e6, [e1, e4] = e7, [e1, e5] = - e2, [e1, e6] = - e3, [e1, e7] = - e4, [e2, e3] = e8, [e2, e4] = e9, [e2, e5] = e1, [e2, e8] = - e3, [e2, e9] = - e4, [e3, e4] = e10, [e3, e6] = e1, [e3, e8] = e2, [e3, e10] = - e4, [e4, e7] = e1, [e4, e9] = e2, [e4, e10] = e3, [e5, e6] = e8, [e5, e7] = e9, [e5, e8] = - e6, [e5, e9] = - e7, [e6, e7] = e10, [e6, e8] = e5, [e6, e10] = - e7, [e7, e9] = e5, [e7, e10] = e6, [e8, e9] = e10, [e8, e10] = - e9, [e9, e10] = e8]$

(2.1)

>	$lprint (%)$

_DG([["LieAlgebra", so5, [10, table( [ ] )]], [[[1, 2, 5], 1], [[1, 3, 6], 1], [[1, 4, 7], 1], [[1, 5, 2], -1], [[1, 6, 3], -1], [[1, 7, 4], -1], [[2, 3, 8], 1], [[2, 4, 9], 1], [[2, 5, 1], 1], [[2, 8, 3], -1], [[2, 9, 4], -1], [[3, 4, 10], 1], [[3, 6, 1], 1], [[3, 8, 2], 1], [[3, 10, 4], -1], [[4, 7, 1], 1], [[4, 9, 2], 1], [[4, 10, 3], 1], [[5, 6, 8], 1], [[5, 7, 9], 1], [[5, 8, 6], -1], [[5, 9, 7], -1], [[6, 7, 10], 1], [[6, 8, 5], 1], [[6, 10, 7], -1], [[7, 9, 5], 1], [[7, 10, 6], 1], [[8, 9, 10], 1], [[8, 10, 9], -1], [[9, 10, 8], 1]]])

>	$DGsetup (LD)$

$Lie algebra: so5$

(2.2)

Define the representation space $V$ . We shall define the invariant tensors we need on $V &period;$

so5 >

$DGsetup ([x1, x2, x3, x4, x5], V)$

$frame name: V$

(2.3)

The standard inclusion of $so (3)$ in $so (5)$ is given as the subalgebra of matrices which fix the vectors $D_{x4}$ and $D_{x5}$ .

V >	$Inv1 ≔ [D_x4, D_x5]$

$Inv1 := [D_x4, D_x5]$

(2.4)

V >	$MatrixSubalgebra (so5, A, Inv1)$

$[e1, e2, e5]$

(2.5)

Comparing with the matrices in A, we see this is precisely the subalgebra $so (3)$ we want.

V >	$[A [1], A [2], A [5]]$

We can define $so (3) \oplus so (2)$ in $so (5)$ as the subalgebra which preserves the subspaces spanned by $[D_{x1}, D_{x^{2}}, D_{x^{5}}$ ] and $[D_{x4}, D_{x^{5}}]$ .

(2.6)

so5 >

$Inv2, Inv3 ≔ [D_x1, D_x2, D_x3], [D_x4, D_x5]$

$Inv2, Inv3 := [D_x1, D_x2, D_x3], [D_x4, D_x5]$

(2.7)

>	$MatrixSubalgebra (so5, A, [Inv2, Inv3])$

$[e1, e2, e5, e10]$

(2.8)

Example 2.

The computation of Example 1 can be done with the other calling sequences.

1. With a representation.

so5 >

$ρ ≔ Representation (so5, V, A)$

so5 >

$MatrixSubalgebra (ρ, Inv1)$

$[e1, e2, e5]$

(2.9)

2. With a Lie algebra of vector fields.

V >	$Gamma ≔ evalDG ([x2 D_x1 - x1 D_x2, x3 D_x1 - x1 D_x3, x4 D_x1 - x1 D_x4, x5 D_x1 - x1 D_x5, x3 D_x2 - x2 D_x3, x4 D_x2 - x2 D_x4, x5 D_x2 - x2 D_x5, x4 D_x3 - x3 D_x4, x5 D_x3 - x3 D_x5, x5 D_x4 - x4 D_x5])$

$Γ := [x2 D_x1 - x1 D_x2, x3 D_x1 - x1 D_x3, x4 D_x1 - x1 D_x4, x5 D_x1 - x1 D_x5, x3 D_x2 - x2 D_x3, x4 D_x2 - x2 D_x4, x5 D_x2 - x2 D_x5, x4 D_x3 - x3 D_x4, x5 D_x3 - x3 D_x5, x5 D_x4 - x4 D_x5]$

(2.10)

V >	$MatrixSubalgebra (so5, Gamma, Inv1)$

$[e1, e2, e5]$

(2.11)

3. With a Lie algebra constructed using the procedure SimpleLieAlgebraData .

so5 >

$LD1 ≔ SimpleLieAlgebraData (so(5), alg1)$

$LD1 := [[e1, e2] = e5, [e1, e3] = e6, [e1, e4] = e7, [e1, e5] = - e2, [e1, e6] = - e3, [e1, e7] = - e4, [e2, e3] = e8, [e2, e4] = e9, [e2, e5] = e1, [e2, e8] = - e3, [e2, e9] = - e4, [e3, e4] = e10, [e3, e6] = e1, [e3, e8] = e2, [e3, e10] = - e4, [e4, e7] = e1, [e4, e9] = e2, [e4, e10] = e3, [e5, e6] = e8, [e5, e7] = e9, [e5, e8] = - e6, [e5, e9] = - e7, [e6, e7] = e10, [e6, e8] = e5, [e6, e10] = - e7, [e7, e9] = e5, [e7, e10] = e6, [e8, e9] = e10, [e8, e10] = - e9, [e9, e10] = e8]$

(2.12)

so5 >

$DGsetup (LD1)$

$Lie algebra: alg1$

(2.13)

alg >

$MatrixSubalgebra (alg1, Inv1)$

$[e1, e2, e5]$

(2.14)

Example 3.

Calculate the subalgebra of $gl (6)$ consisting of 2×2 block upper triangular matrices. First initialize the Lie algebra of all $6 \times 6$ matrices. The labels 'E' and 'theta' must be unassigned names.

>	$LD ≔ SimpleLieAlgebraData (gl(6), gl6, labelformat = gl, labels = ['E','θ']) &colon;$

>	$DGsetup (LD) &colon;$

Define the representation space.

>	$DGsetup ([x1, x2, x3, x4, x5, x6], V6)$

$frame name: V6$

(2.15)

The matrices we want preserve the following subspaces of $V &period;$

V6 >	$Inv ≔ [[D_x1, D_x2], [D_x1, D_x2, D_x3, D_x4]]$

$Inv := [[D_x1, D_x2], [D_x1, D_x2, D_x3, D_x4]]$

(2.16)

V6 >	$A ≔ MatrixSubalgebra (gl6, Inv)$

$A := [E11, E12, E13, E14, E15, E16, E21, E22, E23, E24, E25, E26, E33, E34, E35, E36, E43, E44, E45, E46, E55, E56, E65, E66]$

(2.17)

We can see what matrices these correspond to in several ways. One method is to first form a general linear combination $X$ of the vectors in $A$ .

gl6 >

$X ≔ DGzip ([seq (a ‖ i, i = 1 .. 24)], A, plus)$

$X := a1 E11 + a2 E12 + a3 E13 + a4 E14 + a5 E15 + a6 E16 + a7 E21 + a8 E22 + a9 E23 + a10 E24 + a11 E25 + a12 E26 + a13 E33 + a14 E34 + a15 E35 + a16 E36 + a17 E43 + a18 E44 + a19 E45 + a20 E46 + a21 E55 + a22 E56 + a23 E65 + a24 E66$

(2.18)

Now calculate the matrix associated to $X$ in the standard representation.

gl6 >

$StandardRepresentation (gl6, X)$

Example 4.

In this example we calculate the intersection $so (8) ⋂ sp (8, &reals;) &period;$ These are the skew-symmetric $8 \times 8$ matrices which also preserve a non-degenerate 2-form. Then we show that this intersection is isomorphic to $u (4) &period;$ First we initialize the Lie algebra for $so (8) &period;$ The labels 'R' and 'sigma' must be unassigned names.

gl6 >

$LD ≔ SimpleLieAlgebraData (so(8), so8, labelformat = gl, labels = ['R','σ']) &colon;$

gl6 >

$DGsetup (LD)$

$Lie algebra: so8$

(2.19)

Now define an 8-dimensional representation space $V$ and a 2-form $Ω$ on $V &period;$

so8 >

$DGsetup ([x1, x2, x3, x4, x5, x6, x7, x8], V8)$

$frame name: V8$

(2.20)

V8 >	$Ω ≔ evalDG (dx1 &w dx5 + dx2 &w dx6 + dx3 &w dx7 + dx4 &w dx8)$

$Ω := dx1 ⋀ dx5 + dx2 ⋀ dx6 + dx3 ⋀ dx7 + dx4 ⋀ dx8$

(2.21)

Find the subalgebra of $so (8)$ which preserves this 2-form.

V8 >	$H ≔ MatrixSubalgebra (so8, [Ω])$

$H := [R12 + R56, R13 + R57, R14 + R58, R15, R16 + R25, R17 + R35, R18 + R45, R23 + R67, R24 + R68, R26, R27 + R36, R28 + R46, R34 + R78, R37, R38 + R47, R48]$

(2.22)

Here are the explicit matrices.

so8 >

$M ≔ map2 (StandardRepresentation, so8, H)$

Check that the matrices belong to $so (8) &period;$

so8 >

$Query (M, so(8), MatrixAlgebra)$

$true$

(2.23)

Check that the matrices belong to $sp (8, &reals;)$ .

so8 >

$Query (M, sp(8, R), MatrixAlgebra)$

$true$

(2.24)

The isomorphism to $u (4)$ is given by $Φ_{} ([\begin{array}{r} A & B \\ C & D \end{array}])$ = $A + I B,$ where $A, B, C, D$ are 4×4 matrices. We use the command SubMatrix to construct this map.

so8 >

$Φ ≔ X \mapsto LinearAlgebra :- SubMatrix (X, 1 .. 4, 1 .. 4) + I \cdot LinearAlgebra :- SubMatrix (X, 5 .. 8, 1 .. 4)$

$Φ := X \to LinearAlgebra :- SubMatrix (X, 1 .. 4, 1 .. 4) + I LinearAlgebra :- SubMatrix (X, 5 .. 8, 1 .. 4)$

(2.25)

so8 >

$N ≔ map (Φ, M)$

Check that each of these matrices belong to $u (4)$ .

so8 >

$Query (N, u(4), MatrixAlgebra)$

$true$

(2.26)

Finally, we see that the structure equations for these two matrix algebras are identical.

so8 >

$LieAlgebraData (M, alg1)$

$[[e1, e2] = e8, [e1, e3] = e9, [e1, e4] = e5, [e1, e5] = - 2 e4 + 2 e10, [e1, e6] = e11, [e1, e7] = e12, [e1, e8] = - e2, [e1, e9] = - e3, [e1, e10] = - e5, [e1, e11] = - e6, [e1, e12] = - e7, [e2, e3] = e13, [e2, e4] = e6, [e2, e5] = e11, [e2, e6] = - 2 e4 + 2 e14, [e2, e7] = e15, [e2, e8] = e1, [e2, e11] = - e5, [e2, e13] = - e3, [e2, e14] = - e6, [e2, e15] = - e7, [e3, e4] = e7, [e3, e5] = e12, [e3, e6] = e15, [e3, e7] = - 2 e4 + 2 e16, [e3, e9] = e1, [e3, e12] = - e5, [e3, e13] = e2, [e3, e15] = - e6, [e3, e16] = - e7, [e4, e5] = e1, [e4, e6] = e2, [e4, e7] = e3, [e5, e6] = e8, [e5, e7] = e9, [e5, e8] = - e6, [e5, e9] = - e7, [e5, e10] = e1, [e5, e11] = e2, [e5, e12] = e3, [e6, e7] = e13, [e6, e8] = e5, [e6, e11] = e1, [e6, e13] = - e7, [e6, e14] = e2, [e6, e15] = e3, [e7, e9] = e5, [e7, e12] = e1, [e7, e13] = e6, [e7, e15] = e2, [e7, e16] = e3, [e8, e9] = e13, [e8, e10] = e11, [e8, e11] = - 2 e10 + 2 e14, [e8, e12] = e15, [e8, e13] = - e9, [e8, e14] = - e11, [e8, e15] = - e12, [e9, e10] = e12, [e9, e11] = e15, [e9, e12] = - 2 e10 + 2 e16, [e9, e13] = e8, [e9, e15] = - e11, [e9, e16] = - e12, [e10, e11] = e8, [e10, e12] = e9, [e11, e12] = e13, [e11, e13] = - e12, [e11, e14] = e8, [e11, e15] = e9, [e12, e13] = e11, [e12, e15] = e8, [e12, e16] = e9, [e13, e14] = e15, [e13, e15] = - 2 e14 + 2 e16, [e13, e16] = - e15, [e14, e15] = e13, [e15, e16] = e13]$

(2.27)

so8 >

$LieAlgebraData (N, alg2, method = real)$

(2.28)

Example 5.

The compact real form of the exceptional Lie algebra $g_{2}$ as the subalgebra of $so (7)$ can be computed using the command MatrixAlgebras. First we initialize the Lie algebra $so (7) &period;$

so7 >

$RemoveFrame (so8) &colon;$

gl6 >

$LD ≔ SimpleLieAlgebraData (so(7), so7, labelformat = gl, labels = ['R','σ'])$

$LD := [[e1, e2] = e7, [e1, e3] = e8, [e1, e4] = e9, [e1, e5] = e10, [e1, e6] = e11, [e1, e7] = - e2, [e1, e8] = - e3, [e1, e9] = - e4, [e1, e10] = - e5, [e1, e11] = - e6, [e2, e3] = e12, [e2, e4] = e13, [e2, e5] = e14, [e2, e6] = e15, [e2, e7] = e1, [e2, e12] = - e3, [e2, e13] = - e4, [e2, e14] = - e5, [e2, e15] = - e6, [e3, e4] = e16, [e3, e5] = e17, [e3, e6] = e18, [e3, e8] = e1, [e3, e12] = e2, [e3, e16] = - e4, [e3, e17] = - e5, [e3, e18] = - e6, [e4, e5] = e19, [e4, e6] = e20, [e4, e9] = e1, [e4, e13] = e2, [e4, e16] = e3, [e4, e19] = - e5, [e4, e20] = - e6, [e5, e6] = e21, [e5, e10] = e1, [e5, e14] = e2, [e5, e17] = e3, [e5, e19] = e4, [e5, e21] = - e6, [e6, e11] = e1, [e6, e15] = e2, [e6, e18] = e3, [e6, e20] = e4, [e6, e21] = e5, [e7, e8] = e12, [e7, e9] = e13, [e7, e10] = e14, [e7, e11] = e15, [e7, e12] = - e8, [e7, e13] = - e9, [e7, e14] = - e10, [e7, e15] = - e11, [e8, e9] = e16, [e8, e10] = e17, [e8, e11] = e18, [e8, e12] = e7, [e8, e16] = - e9, [e8, e17] = - e10, [e8, e18] = - e11, [e9, e10] = e19, [e9, e11] = e20, [e9, e13] = e7, [e9, e16] = e8, [e9, e19] = - e10, [e9, e20] = - e11, [e10, e11] = e21, [e10, e14] = e7, [e10, e17] = e8, [e10, e19] = e9, [e10, e21] = - e11, [e11, e15] = e7, [e11, e18] = e8, [e11, e20] = e9, [e11, e21] = e10, [e12, e13] = e16, [e12, e14] = e17, [e12, e15] = e18, [e12, e16] = - e13, [e12, e17] = - e14, [e12, e18] = - e15, [e13, e14] = e19, [e13, e15] = e20, [e13, e16] = e12, [e13, e19] = - e14, [e13, e20] = - e15, [e14, e15] = e21, [e14, e17] = e12, [e14, e19] = e13, [e14, e21] = - e15, [e15, e18] = e12, [e15, e20] = e13, [e15, e21] = e14, [e16, e17] = e19, [e16, e18] = e20, [e16, e19] = - e17, [e16, e20] = - e18, [e17, e18] = e21, [e17, e19] = e16, [e17, e21] = - e18, [e18, e20] = e16, [e18, e21] = e17, [e19, e20] = e21, [e19, e21] = - e20, [e20, e21] = e19], [R12, R13, R14, R15, R16, R17, R23, R24, R25, R26, R27, R34, R35, R36, R37, R45, R46, R47, R56, R57, R67], [σ12, σ13, σ14, σ15, σ16, σ17, σ23, σ24, σ25, σ26, σ27, σ34, σ35, σ36, σ37, σ45, σ46, σ47, σ56, σ57, σ67]$

(2.29)

gl6 >

$DGsetup (LD)$

$Lie algebra: so7$

(2.30)

Now define a7-dimensional representation space $V$ and a 3-form $Φ_{}$ on $V &period;$

so8 >

$DGsetup ([x1, x2, x3, x4, x5, x6, x7], V7)$

$frame name: V7$

(2.31)

V7 >	$σ1 ≔ evalDG (dx1 &w dx3 - dx2 &w dx4)$

$σ1 := dx1 ⋀ dx3 - dx2 ⋀ dx4$

(2.32)

V7 >	$σ2 ≔ evalDG (dx1 &w dx4 + dx2 &w dx3)$

$σ2 := dx1 ⋀ dx4 + dx2 ⋀ dx3$

(2.33)

V7 >	$σ3 ≔ evalDG (dx1 &w dx2 + dx3 &w dx4)$

$σ3 := dx1 ⋀ dx2 + dx3 ⋀ dx4$

(2.34)

so8 >

$Φ ≔ evalDG (σ1 &w dx5 - σ2 &w dx6 + σ3 &w dx7 + (dx5 &w dx6) &w dx7)$

$Φ := dx1 ⋀ dx2 ⋀ dx7 + dx1 ⋀ dx3 ⋀ dx5 - dx1 ⋀ dx4 ⋀ dx6 - dx2 ⋀ dx3 ⋀ dx6 - dx2 ⋀ dx4 ⋀ dx5 + dx3 ⋀ dx4 ⋀ dx7 + dx5 ⋀ dx6 ⋀ dx7$

(2.35)

Calculate the subalgebra of $so (7)$ which leaves the 3-form $Φ_{}$ invariant.

V8 >	$G2 ≔ MatrixSubalgebra (so7, [Φ])$

$G2 := [R12 - R56, R13 - R67, R14 - R57, R15 + R47, R16 + R37, R17 - R45, R23 - R57, R24 + R67, R25 + R37, R26 - R47, R27 + R46, R34 - R56, R35 + R46, R36 - R45]$

(2.36)

Here are the explicit matrices.

so8 >

$M2 ≔ map2 (StandardRepresentation, so7, G2)$

The Lie algebra defined by either the vectors $G2$ or the matrices $M2$ is a 14-dimensional Lie algebra with negative-definite Killing form and 2-dimensional Cartan subalgebra.

>	$LD2 ≔ LieAlgebraData (G2, g2) &colon;$

so7 >

$DGsetup (LD2)$

$Lie algebra: g2$

(2.37)

so7 >

$LinearAlgebra :- IsDefinite (- Killing ())$

$true$

(2.38)

g2 >	$CartanSubalgebra ()$

$[e1, e4 - e10]$

(2.39)

Example 6.

The split real form of the exceptional Lie algebra $g_{2}$ as the subalgebra of $so$ (4, 3) is similarly computed.

so7 >

$RemoveFrame (so7)$

$8$

(2.40)

gl6 >

$LD ≔ SimpleLieAlgebraData (so(4, 3), so43, version = 2, labelformat = gl, labels = ['R','σ']) &colon;$

gl6 >

$DGsetup (LD)$

$Lie algebra: so43$

(2.41)

Now define a 7-dimensional representation space $V$ and a 3-form $Φ_{}$ on $V &period;$

so8 >

$DGsetup ([x1, x2, x3, x4, x5, x6, x7], V7)$

$frame name: V7$

(2.42)

V7 >	$σ1 ≔ evalDG (dx1 &w dx3 - dx2 &w dx4)$

$σ1 := dx1 ⋀ dx3 - dx2 ⋀ dx4$

(2.43)

V7 >	$σ2 ≔ evalDG (dx1 &w dx4 + dx2 &w dx3)$

$σ2 := dx1 ⋀ dx4 + dx2 ⋀ dx3$

(2.44)

V7 >	$σ3 ≔ evalDG (dx1 &w dx2 + dx3 &w dx4)$

$σ3 := dx1 ⋀ dx2 + dx3 ⋀ dx4$

(2.45)

so8 >

$Φ ≔ evalDG (σ1 &w dx5 - σ2 &w dx6 + σ3 &w dx7 - (dx5 &w dx6) &w dx7)$

$Φ := dx1 ⋀ dx2 ⋀ dx7 + dx1 ⋀ dx3 ⋀ dx5 - dx1 ⋀ dx4 ⋀ dx6 - dx2 ⋀ dx3 ⋀ dx6 - dx2 ⋀ dx4 ⋀ dx5 + dx3 ⋀ dx4 ⋀ dx7 - dx5 ⋀ dx6 ⋀ dx7$

(2.46)

Calculate the subalgebra of $so (7)$ which leaves the 3-form $Φ_{}$ invariant.

V8 >	$G2 ≔ MatrixSubalgebra (so43, [Φ])$

$G2 := [R12 - R56, R13 - R67, R14 - R57, R23 - R57, R24 + R67, R34 - R56, R15 + R47, R16 + R37, R17 - R45, R25 + R37, R26 - R47, R27 + R46, R35 + R46, R36 - R45]$

(2.47)

Here are the explicit matrices.

so8 >

$M2 ≔ map2 (StandardRepresentation, so43, G2)$

The Lie algebra defined by either the vectors $G2$ or the matrices $M2$ is a 14-dimensional Lie algebra.

V7 >	$LD6 ≔ LieAlgebraData (G2, g2S)$

$LD6 := [[e1, e2] = e4, [e1, e3] = e5, [e1, e4] = - e2, [e1, e5] = - e3, [e1, e7] = - e8 + e10, [e1, e8] = e7 + e11, [e1, e9] = e12, [e1, e10] = - e7 - e11, [e1, e11] = - e8 + e10, [e1, e12] = - e9, [e1, e13] = - e14, [e1, e14] = e13, [e2, e3] = e6, [e2, e4] = e1, [e2, e6] = - e3, [e2, e7] = e13, [e2, e8] = - 2 e9 + 2 e14, [e2, e9] = e8, [e2, e10] = - e9 + e14, [e2, e11] = - e12, [e2, e12] = e11, [e2, e13] = - e7, [e2, e14] = - e8, [e3, e5] = e1, [e3, e6] = e2, [e3, e7] = - 2 e9, [e3, e8] = e13, [e3, e9] = 2 e7, [e3, e10] = - e12 + e13, [e3, e11] = e9, [e3, e12] = - e8 + e10, [e3, e13] = - e8, [e3, e14] = e7, [e4, e5] = e6, [e4, e6] = - e5, [e4, e7] = - e9, [e4, e8] = - e12 + e13, [e4, e9] = e7, [e4, e10] = - 2 e12 + 2 e13, [e4, e11] = e14, [e4, e12] = e10, [e4, e13] = - e10, [e4, e14] = - e11, [e5, e6] = e4, [e5, e7] = - e12, [e5, e8] = e9 - e14, [e5, e9] = - e8 + e10, [e5, e10] = - e14, [e5, e11] = 2 e12, [e5, e12] = - 2 e11, [e5, e13] = - e11, [e5, e14] = e10, [e6, e7] = - e8, [e6, e8] = e7, [e6, e9] = e13, [e6, e10] = - e11, [e6, e11] = e10, [e6, e12] = - e14, [e6, e13] = - 2 e14, [e6, e14] = 2 e13, [e7, e8] = e6, [e7, e9] = 2 e3, [e7, e10] = - e1 + e6, [e7, e12] = e5, [e7, e13] = - e2, [e7, e14] = e3, [e8, e9] = e2, [e8, e11] = - e1 + e6, [e8, e12] = - e3 + e4, [e8, e13] = - e3, [e8, e14] = - e2, [e9, e10] = - e2 - e5, [e9, e11] = e3, [e9, e12] = - e1, [e9, e13] = - e6, [e10, e11] = e6, [e10, e12] = e4, [e10, e13] = - e4, [e10, e14] = e5, [e11, e12] = - 2 e5, [e11, e13] = - e5, [e11, e14] = - e4, [e12, e14] = e6, [e13, e14] = 2 e6]$

(2.48)

so7 >

$DGsetup (LD6)$

$Lie algebra: g2S$

(2.49)

g2S >

$B ≔ KillingForm ()$

$B := - 8 θ1 θ6 - 8 θ7 θ11 + 16 θ8 θ8 + 8 θ5 θ2 - 16 θ5 θ5 + 8 θ8 θ10 + 16 θ9 θ9 + 16 θ12 θ12 + 8 θ12 θ13 - 16 θ1 θ1 - 8 θ4 θ3 - 16 θ4 θ4 + 8 θ9 θ14 + 8 θ10 θ8 + 16 θ10 θ10 - 8 θ11 θ7 + 16 θ11 θ11 + 8 θ13 θ12 + 16 θ13 θ13 - 16 θ3 θ3 - 8 θ3 θ4 + 8 θ14 θ9 + 16 θ14 θ14 - 16 θ2 θ2 + 8 θ2 θ5 - 8 θ6 θ1 - 16 θ6 θ6 + 16 θ7 θ7$

(2.50)

g2S >

$Tensor :- QuadraticFormSignature (B)$

$[[e7, e7 + 2 e11, e8, e8 - 2 e10, e9, e9 - 2 e14, e12, e12 - 2 e13], [e1, e1 - 2 e6, e2, e2 + 2 e5, e3, e3 - 2 e4], []]$

(2.51)

g2S >

$map (nops, %)$

$[8, 6, 0]$

(2.52)

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