DifferentialGeometry/LieAlgebras/Query/MatrixAlgebra - Maple Help
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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 :

sln,  sln,ℂ,  sup, q,  su∗n,  un,  son,  son,ℂ,  sop, q,  so∗n,  spn, ℝ,  spp, q,  spn,  soln,  niln.

• 

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

Examples

> 

with⁡DifferentialGeometry:with⁡LieAlgebras:

 

Example 1.

We check if each matrix in a list of matrices belongs to sl2.

> 

A1≔Matrix⁡1,0,0,−1,Matrix⁡0,1,0,0,Matrix⁡0,0,1,0

> 

Query⁡A1,sl(2),MatrixAlgebra

true

(2.1)
> 

A2≔Matrix⁡1,0,0,−1,Matrix⁡1,1,0,0,Matrix⁡0,0,1,0

> 

Query⁡A2,sl(2),MatrixAlgebra

false

(2.2)

 

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

(2.3)
> 

Query⁡A2,sl(2),output=integer,MatrixAlgebra

2

(2.4)

 

Example 2.

We check if each matrix in list of matrices belong to so2,2. 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 Q1 = 0I2I20.  With the keyword argument version  = 2, the quadratic form Q2  = I200−I2 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

> 

Query⁡B1,so(2, 2),MatrixAlgebra

true

(2.5)

 

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

> 

Query⁡B2,so(2,2),version=2,MatrixAlgebra

true

(2.6)

 

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

> 

M≔Matrix⁡0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0

> 

Query⁡B3,so(2, 2),quadraticform=M,MatrixAlgebra

true

(2.7)

Example 3.

We check if the members of a list of matrices belong to sp4, ℝ. 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  =0In−In0.  Other forms for J  can be specified with the keyword argument skewform = J.  

Here is the standard form of the matrices for sp4, 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

> 

Query⁡C1,sp(4, R),MatrixAlgebra

true

(2.8)

 

Define a skew-symmetric matrix J.

> 

J≔Matrix⁡0,−1,0,0,1,0,0,0,0,0,0,1,0,0,−1,0

 

Here is the form of the matrices for sp4, ℝ with respect to J.

> 

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

> 

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

true

(2.9)

 

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

> 

Query⁡D1,sol(3),MatrixAlgebra

true

(2.10)

 

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

> 

Query⁡E,nil(4),MatrixAlgebra

true

(2.11)
> 

LieAlgebraData⁡D1,NN

e1,e2=e2,e1,e3=e3,e2,e4=e2,e2,e5=e3,e3,e6=e3,e4,e5=e5,e5,e6=e5

(2.12)

 

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

u3 > 

LD≔LieAlgebraData⁡F,alg

LD:=e1,e3=−e4,e1,e4=e3,e2,e3=−e4,e2,e4=e3,e3,e4=−2⁢e2−2⁢e1

(2.13)
u3 > 

DGsetup⁡LD

Lie algebra: alg

(2.14)
u3 > 

DGsetup⁡x1,x2,x3,V

frame name: V

(2.15)
alg > 

ρ≔Representation⁡alg,V,F

alg > 

Query⁡ρ,u(4),MatrixAlgebra

true

(2.16)

See Also

DifferentialGeometry

Query

Representation

SimpleLieAlgebraData

StandardRepresentation