Query[MatrixAlgebra] - check if each matrix in a list of matrices belongs to a specified classical matrix algebra
Query(A, alg, options, "MatrixAlgebra")
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
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.
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
A2 ≔ Matrix⁡1,0,0,−1,Matrix⁡1,1,0,0,Matrix⁡0,0,1,0
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.
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
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
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
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
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
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
Check that a list of matrices consists of nilpotent matrices.
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
Check that the following matrices define a Lie algebra and that this representation is unitary.
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
LD ≔ LieAlgebraData⁡F,alg
Lie algebra: alg
frame name: V
ρ ≔ Representation⁡alg,V,F
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