An (internal) direct sum decomposition of a Lie algebra g is given by a collection of subalgebras g1, g2, ..., gN such that [i] g = g1 + g2 + ... + gN, [ii] the intersection of any two distinct summands gi and gj, i <> j is trivial, and [iii] the Lie bracket of any two distinct gi and gj is trivial, that is, [gi, gj] = 0.

Decompose() will decompose the given Lie algebra g into an internal direct sum of subalgebras g = g1 + g2 + ... + gN where each summand gi is indecomposable.  All abelian summands appear as a single factor.  If no argument is given, then the decomposition of the current Lie algebra is found.

Decompose returns a list with 2 components [A, B] where [i] A is a matrix which defines a Lie algebra isomorphism from g to the Lie algebra defined by the direct sum of the Lie algebras in the summand_list, and [ii] B is a basis for the original Lie algebra g in which g decomposes.

With saveTemporaryAlgebras = true, the Lie algebras created by the program as part of the decomposition algorithm are not erased and can be initialized for further analysis.  Use this option in conjunction with userlevel[Decompose] := 2.  The default is saveTemporaryAlgebras = false.

With method = AbsolutelyIndecomposable, the decomposition will be done over the complex numbers.  The default is method= "Indecomposable".

With factoralgebras  = true,  the program  will also return the Lie algebra structure data for each summand in the direct sum decomposition.

If Maple is unable to find a square-free factorization of certain minimal polynomials which arise, then a warning to this effect is given.  The polynomials which need to be factored can be obtained from the procedure DifferentialGeometry:-Tools:-CalculationHistory:-Get("Decompose").  The square-free factors of these polynomials can be pasted to Decompose with the hint option, for example, hint = [x, [x^2 - 1/2, [x - 1/sqrt(2), x + 1/sqrt(2)]].

This module implements the algorithm described in [i] Pate, Winternitz and Zassenhaus, On the Identification of a Lie Algebra Given by its Structure Constants I. Direct Decompositions, Levi Decompositions and Nilradicals. [ii] W. D. Rand, Pascal programs for the identification of Lie algebras I, Comput. Phys. Comm. 41: 105--125 (1986)

The command Decompose is part of the DifferentialGeometry:-LieAlgebras package.  It can be used in the form Decompose(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:-LieAlgebras:-Decompose(...).

Warning, the Lie algebra "Alg3:2" is decomposable but an explicit decomposition could not be constructed.  See CalculationHistory and rerun Decompose with optional argument hint.

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]]