Step 1: Use Query to determine indecomposability of : so4
the Lie algebra so4 is decomposable
Step 2: Performing central decomposition
Detailed calculations for central decomposition
[i] center = C = []
[ii] derived algebra = DA = [-e4, -e5, e2, e3, -e6, -e1]
[iii] intersection of center and derived algebra = M = []
[iv] central factor = complement of M in C = []
End of detailed calculations for central decomposition
central decomposition is trivial
Step 3 Start:
ToBeDecomposedFactors = [_DG([["LieAlgebra", "so4:2", [6]], [[[1, 2, 4], -1], [[1, 3, 5], -1], [[1, 4, 2], 1], [[1, 5, 3], 1], [[2, 3, 6], -1], [[2, 4, 1], -1], [[2, 6, 3], 1], [[3, 5, 1], -1], [[3, 6, 2], -1], [[4, 5, 6], -1], [[4, 6, 5], 1], [[5, 6, 4], -1]]])[1][2]]
IndecomposableFactors = []
Step 3.1 Start: Decomposing the algebra "so4:2"
Detailed Calculations :
[i] the matrix representatives S for the centralizer of Adjoint, modulo its Jacobson radical are :
S[1] = Matrix(6, 6, {NULL}, storage = empty, shape = [identity])
S[2] = Matrix(6, 6, [[0,0,0,0,0,1],[0,0,0,0,-1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,-1,0,0,0,0],[1,0,0,0,0,0]])
the algebra "so4:2" is decomposable
Detailed Calculations :
[ii] the miminal polynomials P for the matrices S are :
P = [-1+_z1, -1+_z1^2]
Detailed Calculations :
[iii] the following relatively prime factorization of the polynomial P[2] will be used:
-1+_z1^2 = [_z1+1, -1+_z1]
Detailed Calculations :
[iv] the projection matrices onto the summands of the decomposition are:
N1 = Matrix(6, 3, [[1,0,0],[0,1,0],[0,0,1],[0,0,-1],[0,1,0],[-1,0,0]])
N2 = Matrix(6, 3, [[1,0,0],[0,1,0],[0,0,1],[0,0,1],[0,-1,0],[1,0,0]])
factor 1 = "so4:3" = [e1-e6, e2+e5, e3-e4]
_DG([["LieAlgebra", "so4:3", [3]], [[[1, 2, 3], 2], [[1, 3, 2], -2], [[2, 3, 1], 2]]])
factor 2 = "so4:4" = [e1+e6, e2-e5, e3+e4]
_DG([["LieAlgebra", "so4:4", [3]], [[[1, 2, 3], -2], [[1, 3, 2], 2], [[2, 3, 1], -2]]])
Step 3.1 End:
ToBeDecomposedFactors = [_DG([["LieAlgebra", "so4:3", [3]], [[[1, 2, 3], 2], [[1, 3, 2], -2], [[2, 3, 1], 2]]])[1][2], _DG([["LieAlgebra", "so4:4", [3]], [[[1, 2, 3], -2], [[1, 3, 2], 2], [[2, 3, 1], -2]]])[1][2]]
IndecomposableFactors = []
Step 3.2 Start: Decomposing the algebra "so4:4"
Detailed Calculations :
[i] the matrix representatives S for the centralizer of Adjoint, modulo its Jacobson radical are :
S[1] = Matrix(3, 3, {NULL}, storage = empty, shape = [identity])
the algebra "so4:4" is indecomposable
Step 3.2 End:
ToBeDecomposedFactors = [_DG([["LieAlgebra", "so4:3", [3]], [[[1, 2, 3], 2], [[1, 3, 2], -2], [[2, 3, 1], 2]]])[1][2]]
IndecomposableFactors = [_DG([["LieAlgebra", "so4:4", [3]], [[[1, 2, 3], -2], [[1, 3, 2], 2], [[2, 3, 1], -2]]])[1][2]]
Step 3.3 Start: Decomposing the algebra "so4:3"
Detailed Calculations :
[i] the matrix representatives S for the centralizer of Adjoint, modulo its Jacobson radical are :
S[1] = Matrix(3, 3, {NULL}, storage = empty, shape = [identity])
the algebra "so4:3" is indecomposable
Step 3.3 End:
ToBeDecomposedFactors = []
IndecomposableFactors = [_DG([["LieAlgebra", "so4:3", [3]], [[[1, 2, 3], 2], [[1, 3, 2], -2], [[2, 3, 1], 2]]])[1][2], _DG([["LieAlgebra", "so4:4", [3]], [[[1, 2, 3], -2], [[1, 3, 2], 2], [[2, 3, 1], -2]]])[1][2]]
Step 3 End:
ToBeDecomposedFactors = []
IndecomposableFactors = [_DG([["LieAlgebra", "so4:3", [3]], [[[1, 2, 3], 2], [[1, 3, 2], -2], [[2, 3, 1], 2]]])[1][2], _DG([["LieAlgebra", "so4:4", [3]], [[[1, 2, 3], -2], [[1, 3, 2], 2], [[2, 3, 1], -2]]])[1][2]]
Step 4: Sorting the summands in the IndecomposableFactors list and appending the central abelian Factor
IndecomposableFactors = [_DG([["LieAlgebra", "so4:3", [3]], [[[1, 2, 3], 2], [[1, 3, 2], -2], [[2, 3, 1], 2]]])[1][2], _DG([["LieAlgebra", "so4:4", [3]], [[[1, 2, 3], -2], [[1, 3, 2], 2], [[2, 3, 1], -2]]])[1][2]]
| (2.16) |