DifferentialGeometry/LieAlgebras/Query/MatrixAlgebra

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Query[MatrixAlgebra] - check if each matrix in a list of matrices belongs to a specified classical matrix algebra

Calling Sequences

Query( $A, alg, options,$ "MatrixAlgebra")

Parameters

A - a list of square matrices, or a matrix representation of a Lie algebra

alg - a string, specifying a classical matrix algebra

options - (optional) keyword arguments output, quadraticform, skewform

Description

Examples

Description

•	This query checks if a given list of matrices belongs to one of the following matrix algebras :

$sl (n), sl (n, ℂ), su (p, q), su * (n), u (n), so (n), so (n, ℂ), so (p, q), so * (n), sp (n, ℝ), sp (p, q), sp (n), sol (n), nil (n) &period;$

•	For the definitions of all these matrix algebras see, SimpleLieAlgebraData.

Examples

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

Example 1.

We check if each matrix in a list of matrices belongs to $sl (2) &period;$

>	$A1 ≔ [Matrix ([[1, 0], [0, - 1]]), Matrix ([[0, 1], [0, 0]]), Matrix ([[0, 0], [1, 0]])]$

$A1 := [[\begin{array}{r} 1 & 0 \\ 0 & - 1 \end{array}], [\begin{array}{r} 0 & 1 \\ 0 & 0 \end{array}], [\begin{array}{r} 0 & 0 \\ 1 & 0 \end{array}]]$

(2.1)

>	$Query (A1, sl(2), MatrixAlgebra)$

$true$

(2.2)

>	$A2 ≔ [Matrix ([[1, 0], [0, - 1]]), Matrix ([[1, 1], [0, 0]]), Matrix ([[0, 0], [1, 0]])]$

$A2 := [[\begin{array}{r} 1 & 0 \\ 0 & - 1 \end{array}], [\begin{array}{r} 1 & 1 \\ 0 & 0 \end{array}], [\begin{array}{r} 0 & 0 \\ 1 & 0 \end{array}]]$

(2.3)

>	$Query (A2, sl(2), MatrixAlgebra)$

$false$

(2.4)

With the keyword argument output = 'integer' , 0 is returned if all the matrices belong to the specified matrix algebra, otherwise the position of the first matrix which does not belong to the specified matrix algebra is returned.

>	$Query (A1, sl(2), output ='integer', MatrixAlgebra)$

$0$

(2.5)

>	$Query (A2, sl(2), output ='integer', MatrixAlgebra)$

$2$

(2.6)

Example 2.

We check if each matrix in list of matrices belong to $so (2, 2)$ . This is the Lie algebra of 4×4 matrices which are skew-symmetric with respect to a quadratic form of signature [2,2]. The default choice for the quadratic form is $Q_{1} = [\begin{array}{r} 0 & I_{2} \\ I_{2} & 0 \end{array}]$ . With the keyword argument version = 2, the quadratic form $Q_{2} = [\begin{array}{r} I_{2} & 0 \\ 0 & - I_{2} \end{array}]$ is used. With the keyword argument quadraticform = $M$ , the quadratic form $M$ (a 4×4 symmetric matrix with signature [2, 2]) is used.

1. Default option.

$B1 ≔ map (Matrix, [[[1, 0, 0, 0], [0, 0, 0, 0], [0, 0, - 1, 0], [0, 0, 0, 0]], [[0, 1, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, - 1, 0]], [[0, 0, 0, 0], [1, 0, 0, 0], [0, 0, 0, - 1], [0, 0, 0, 0]], [[0, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, 0], [0, 0, 0, - 1]], [[0, 0, 0, - 1], [0, 0, 1, 0], [0, 0, 0, 0], [0, 0, 0, 0]], [[0, 0, 0, 0], [0, 0, 0, 0], [0, - 1, 0, 0], [1, 0, 0, 0]]])$

$B1 := [[\begin{array}{r} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & - 1 & 0 \\ 0 & 0 & 0 & 0 \end{array}], [\begin{array}{r} 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & - 1 & 0 \end{array}], [\begin{array}{r} 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & - 1 \\ 0 & 0 & 0 & 0 \end{array}], [\begin{array}{r} 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & - 1 \end{array}], [\begin{array}{r} 0 & 0 & 0 & - 1 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}], [\begin{array}{r} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & - 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{array}]]$

(2.7)

>	$Query (B1, so(2, 2), MatrixAlgebra)$

$true$

(2.8)

2. with version = 2.

$B2 ≔ map (Matrix, [[[0, - 1, 0, 0], [1, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]], [[0, 0, 1, 0], [0, 0, 0, 0], [1, 0, 0, 0], [0, 0, 0, 0]], [[0, 0, 0, 1], [0, 0, 0, 0], [0, 0, 0, 0], [1, 0, 0, 0]], [[0, 0, 0, 0], [0, 0, 1, 0], [0, 1, 0, 0], [0, 0, 0, 0]], [[0, 0, 0, 0], [0, 0, 0, 1], [0, 0, 0, 0], [0, 1, 0, 0]], [[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, - 1], [0, 0, 1, 0]]])$

$B2 := [[\begin{array}{r} 0 & - 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}], [\begin{array}{r} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}], [\begin{array}{r} 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{array}], [\begin{array}{r} 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}], [\begin{array}{r} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{array}], [\begin{array}{r} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & - 1 \\ 0 & 0 & 1 & 0 \end{array}]]$

(2.9)

>	$Query (B2, so(2,2), version = 2, MatrixAlgebra)$

$true$

(2.10)

3. with quadraticform = M

$B3 ≔ map (Matrix, [[[1, 0, 0, 0], [0, - 1, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]], [[0, 0, 1, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, - 1, 0, 0]], [[0, 0, 0, 1], [0, 0, 0, 0], [0, - 1, 0, 0], [0, 0, 0, 0]], [[0, 0, 0, 0], [0, 0, 1, 0], [0, 0, 0, 0], [- 1, 0, 0, 0]], [[0, 0, 0, 0], [0, 0, 0, 1], [- 1, 0, 0, 0], [0, 0, 0, 0]], [[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 1, 0], [0, 0, 0, - 1]]])$

$B3 := [[\begin{array}{r} 1 & 0 & 0 & 0 \\ 0 & - 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}], [\begin{array}{r} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & - 1 & 0 & 0 \end{array}], [\begin{array}{r} 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & - 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}], [\begin{array}{r} 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ - 1 & 0 & 0 & 0 \end{array}], [\begin{array}{r} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ - 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}], [\begin{array}{r} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & - 1 \end{array}]]$

(2.11)

>	$M ≔ Matrix ([[0, 1, 0, 0], [1, 0, 0, 0], [0, 0, 0, 1], [0, 0, 1, 0]])$

$M := [\begin{array}{r} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{array}]$

(2.12)

>	$Query (B3, so(2, 2), quadraticform = M, MatrixAlgebra)$

$true$

(2.13)

Example 3.

We check if the members of a list of matrices belong to $sp (4, &reals;) &period;$ This is the real Lie algebra of matrices which are skew-symmetric with respect to a skew-symmetric matrix $J$ . The default choice is $J = [\begin{array}{r} 0 & I_{n} \\ - I_{n} & 0 \end{array}]$ . Other forms for $J$ can be specified with the keyword argument skewform = $J$ $&period;$

Here is the standard form of the matrices for $sp (4, ℝ)$ .

$C1 ≔ map (Matrix, [[[1, 0, 0, 0], [0, 0, 0, 0], [0, 0, - 1, 0], [0, 0, 0, 0]], [[0, 1, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, - 1, 0]], [[0, 0, 0, 0], [1, 0, 0, 0], [0, 0, 0, - 1], [0, 0, 0, 0]], [[0, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, 0], [0, 0, 0, - 1]], [[0, 0, 1, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]], [[0, 0, 0, 1], [0, 0, 1, 0], [0, 0, 0, 0], [0, 0, 0, 0]], [[0, 0, 0, 0], [0, 0, 0, 1], [0, 0, 0, 0], [0, 0, 0, 0]], [[0, 0, 0, 0], [0, 0, 0, 0], [1, 0, 0, 0], [0, 0, 0, 0]], [[0, 0, 0, 0], [0, 0, 0, 0], [0, 1, 0, 0], [1, 0, 0, 0]], [[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 1, 0, 0]]])$

$C1 := [[\begin{array}{r} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & - 1 & 0 \\ 0 & 0 & 0 & 0 \end{array}], [\begin{array}{r} 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & - 1 & 0 \end{array}], [\begin{array}{r} 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & - 1 \\ 0 & 0 & 0 & 0 \end{array}], [\begin{array}{r} 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & - 1 \end{array}], [\begin{array}{r} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}], [\begin{array}{r} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}], [\begin{array}{r} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}], [\begin{array}{r} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}], [\begin{array}{r} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{array}], [\begin{array}{r} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{array}]]$

(2.14)

>	$Query (C1, sp(4, R), MatrixAlgebra)$

$true$

(2.15)

Define a skew-symmetric matrix $J$ .

>	$J ≔ Matrix ([[0, - 1, 0, 0], [1, 0, 0, 0], [0, 0, 0, 1], [0, 0, - 1, 0]])$

$J := [\begin{array}{r} 0 & - 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & - 1 & 0 \end{array}]$

(2.16)

Here is the form of the matrices for $sp (4, ℝ)$ with respect to $J &period;$

$C2 ≔ map (Matrix, [[[- 1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]], [[0, 0, 0, 0], [0, 0, 1, 0], [0, 0, 0, 0], [- 1, 0, 0, 0]], [[0, 0, 0, - 1], [0, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, 0]], [[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 1, 0], [0, 0, 0, - 1]], [[0, 0, 0, 0], [1, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]], [[0, 0, 0, 0], [0, 0, 0, 1], [1, 0, 0, 0], [0, 0, 0, 0]], [[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 1], [0, 0, 0, 0]], [[0, 1, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]], [[0, 0, 1, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 1, 0, 0]], [[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 1, 0]]])$

$C2 := [[\begin{array}{r} - 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}], [\begin{array}{r} 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ - 1 & 0 & 0 & 0 \end{array}], [\begin{array}{r} 0 & 0 & 0 & - 1 \\ 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}], [\begin{array}{r} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & - 1 \end{array}], [\begin{array}{r} 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}], [\begin{array}{r} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}], [\begin{array}{r} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{array}], [\begin{array}{r} 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}], [\begin{array}{r} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{array}], [\begin{array}{r} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}]]$

(2.17)

>	$Query (C2, sp(4, R), skewform = J, MatrixAlgebra)$

$true$

(2.18)

Example 4.

Check that a list of matrices consists of upper triangular matrices.

>	$D1 ≔ map (Matrix, [[[1, 0, 0], [0, 0, 0], [0, 0, 0]], [[0, 1, 0], [0, 0, 0], [0, 0, 0]], [[0, 0, 1], [0, 0, 0], [0, 0, 0]], [[0, 0, 0], [0, 1, 0], [0, 0, 0]], [[0, 0, 0], [0, 0, 1], [0, 0, 0]], [[0, 0, 0], [0, 0, 0], [0, 0, 1]]])$

$D1 := [[\begin{array}{r} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}], [\begin{array}{r} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}], [\begin{array}{r} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}], [\begin{array}{r} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{array}], [\begin{array}{r} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{array}], [\begin{array}{r} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{array}]]$

(2.19)

>	$Query (D1, sol(3), MatrixAlgebra)$

$true$

(2.20)

Example 5.

Check that a list of matrices consists of nilpotent matrices.

alg >

$E ≔ map (Matrix, [[[- 1, 2, 1, 3], [- 1, 2, 1, 3], [1, - 2, - 1, - 3], [0, 0, 0, 0]], [[- 1, 3, 2, 4], [- 1, 2, 1, 2], [1, - 4, - 3, - 6], [0, 1, 1, 2]], [[0, 1, 1, 0], [0, 1, 1, 0], [0, - 1, - 1, 0], [0, 0, 0, 0]]])$

$E := [[\begin{array}{r} - 1 & 2 & 1 & 3 \\ - 1 & 2 & 1 & 3 \\ 1 & - 2 & - 1 & - 3 \\ 0 & 0 & 0 & 0 \end{array}], [\begin{array}{r} - 1 & 3 & 2 & 4 \\ - 1 & 2 & 1 & 2 \\ 1 & - 4 & - 3 & - 6 \\ 0 & 1 & 1 & 2 \end{array}], [\begin{array}{r} 0 & 1 & 1 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & - 1 & - 1 & 0 \\ 0 & 0 & 0 & 0 \end{array}]]$

(2.21)

>	$Query (E, nil(4), MatrixAlgebra)$

$true$

(2.22)

>	$LieAlgebraData (D1, NN)$

$[[e1, e2] = e2, [e1, e3] = e3, [e2, e4] = e2, [e2, e5] = e3, [e3, e6] = e3, [e4, e5] = e5, [e5, e6] = e5]$

(2.23)

Example 6.

Check that the following matrices define a Lie algebra and that this representation is unitary.

u3 >	$F ≔ map (Matrix, [[[0, 0, 0, 0], [0, I, 0, 0], [0, 0, - I, 0], [0, 0, 0, 0]], [[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, I, 0], [0, 0, 0, - I]], [[0, 0, 0, 0], [0, 0, 0, - 1], [0, 0, 0, 0], [0, 1, 0, 0]], [[0, 0, 0, 0], [0, 0, 0, I], [0, 0, 0, 0], [0, I, 0, 0]]])$

$F := [[\begin{array}{c} 0 & 0 & 0 & 0 \\ 0 & I & 0 & 0 \\ 0 & 0 & - I & 0 \\ 0 & 0 & 0 & 0 \end{array}], [\begin{array}{c} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & I & 0 \\ 0 & 0 & 0 & - I \end{array}], [\begin{array}{r} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & - 1 \\ 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{array}], [\begin{array}{c} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & I \\ 0 & 0 & 0 & 0 \\ 0 & I & 0 & 0 \end{array}]]$

(2.24)

u3 >	$LD ≔ LieAlgebraData (F, alg)$

$LD := [[e1, e3] = - e4, [e1, e4] = e3, [e2, e3] = - e4, [e2, e4] = e3, [e3, e4] = - 2 e2 - 2 e1]$

(2.25)

u3 >	$DGsetup (LD)$

$Lie algebra: alg$

(2.26)

u3 >	$DGsetup ([x1, x2, x3], V)$

$frame name: V$

(2.27)

alg >

$ρ ≔ Representation (alg, V, F)$

$ρ := [[e1, [\begin{array}{c} 0 & 0 & 0 & 0 \\ 0 & I & 0 & 0 \\ 0 & 0 & - I & 0 \\ 0 & 0 & 0 & 0 \end{array}]], [e2, [\begin{array}{c} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & I & 0 \\ 0 & 0 & 0 & - I \end{array}]], [e3, [\begin{array}{r} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & - 1 \\ 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{array}]], [e4, [\begin{array}{c} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & I \\ 0 & 0 & 0 & 0 \\ 0 & I & 0 & 0 \end{array}]]]$

(2.28)

alg >

$Query (ρ, u(4), MatrixAlgebra)$

$true$

(2.29)

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