An (internal) direct sum decomposition of a Lie algebra $\mathrm{\𝔤}\mathit{ }$is given by a collection of subalgebras ${\mathrm{\𝔤}}_1$, ${\mathrm{\𝔤}}_2\, ..\. \, {\mathrm{\𝔤}}_N$such that

Decompose() will decompose the given Lie algebra g into an internal direct sum of subalgebras $\mathrm{\𝔤}\mathit{ }\mathit{\=}\mathit{ }{\mathrm{\𝔤}}_1\+ {\mathrm{\𝔤}}_2\+\cdot \cdot \cdot \+{\mathrm{\𝔤}}_N\,$where each summand ${\mathrm{\𝔤}}_i$ 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 $\mathrm{\𝔤}$ in which $\mathrm{\𝔤}\mathit{ }$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 passed 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

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(...).

$\mathrm{with}\left(\mathrm{DifferentialGeometry}\right)\:$$\mathrm{with}\left(\mathrm{LieAlgebras}\right)\:$

$\mathrm{L1}≔\mathrm{\_DG}\left(\left[\left[''LieAlgebra''\,\mathrm{Alg1}\,\left[5\right]\right]\,\left[\left[\left[1\,2\,2\right]\,1\right]\,\left[\left[1\,2\,3\right]\,1\right]\,\left[\left[1\,3\,2\right]\,-1\right]\,\left[\left[1\,3\,3\right]\,-1\right]\,\left[\left[1\,4\,1\right]\,1\right]\,\left[\left[1\,4\,2\right]\,-1\right]\,\left[\left[1\,4\,3\right]\,-1\right]\,\left[\left[2\,3\,2\right]\,-1\right]\,\left[\left[2\,3\,3\right]\,-1\right]\,\left[\left[2\,5\,2\right]\,-1\right]\,\left[\left[2\,5\,3\right]\,-1\right]\,\left[\left[3\,5\,2\right]\,1\right]\,\left[\left[3\,5\,3\right]\,1\right]\,\left[\left[4\,5\,2\right]\,1\right]\,\left[\left[4\,5\,3\right]\,1\right]\,\left[\left[4\,5\,5\right]\,-1\right]\right]\right]\right)\:$

$\mathrm{DGsetup}\left(\mathrm{L1}\right)\:$

$\mathrm{MultiplicationTable}\left(''LieTable''\right)$

$\left[\begin{array}{ccccccc}& \mathrm{`|`}& \mathrm{e1}& \mathrm{e2}& \mathrm{e3}& \mathrm{e4}& \mathrm{e5}\\ & \mathrm{---}& \mathrm{----}& \mathrm{----}& \mathrm{----}& \mathrm{----}& \mathrm{----}\\ \mathrm{e1}& \mathrm{`|`}& 0& \mathrm{e2}\+\mathrm{e3}& -\mathrm{e2}-\mathrm{e3}& \mathrm{e1}-\mathrm{e2}-\mathrm{e3}& 0\\ \mathrm{e2}& \mathrm{`|`}& -\mathrm{e2}-\mathrm{e3}& 0& -\mathrm{e2}-\mathrm{e3}& 0& -\mathrm{e2}-\mathrm{e3}\\ \mathrm{e3}& \mathrm{`|`}& \mathrm{e2}\+\mathrm{e3}& \mathrm{e2}\+\mathrm{e3}& 0& 0& \mathrm{e2}\+\mathrm{e3}\\ \mathrm{e4}& \mathrm{`|`}& -\mathrm{e1}\+\mathrm{e2}\+\mathrm{e3}& 0& 0& 0& \mathrm{e2}\+\mathrm{e3}-\mathrm{e5}\\ \mathrm{e5}& \mathrm{`|`}& 0& \mathrm{e2}\+\mathrm{e3}& -\mathrm{e2}-\mathrm{e3}& -\mathrm{e2}-\mathrm{e3}\+\mathrm{e5}& 0\end{array}\right]$

$\mathrm{decomposition}≔\mathrm{Decompose}\left(\right)$

$\mathrm{decomposition}:=\left[\left[\left[\begin{array}{rrrrr}1& 0& 0& 0& 0\\ -1& 0& 0& 0& -1\\ 0& 0& 0& 1& 0\\ 1& 1& 0& 0& 1\\ 1& 0& 1& 0& 1\end{array}\right]\,\left[\left[\mathrm{e1}-\mathrm{e5}\,\mathrm{e2}\+\mathrm{e3}-\mathrm{e5}\,\mathrm{e4}\,\mathrm{e2}\,\mathrm{e3}\right]\right]\right]\right]$

$\mathrm{newBasis}≔\mathrm{decomposition}\left[2\right]$

$\mathrm{newBasis}≔\left[\mathrm{e1}-\mathrm{e5}\,\mathrm{e2}\+\mathrm{e3}-\mathrm{e5}\,\mathrm{e4}\,\mathrm{e2}\,\mathrm{e3}\right]$

$\mathrm{L2}≔\mathrm{LieAlgebraData}\left(\mathrm{newBasis}\,\mathrm{Alg2}\right)\:$

$\mathrm{DGsetup}\left(\mathrm{L2}\right)$

$\mathrm{Lie\; algebra:\; Alg2}$

$\mathrm{MultiplicationTable}\left(''LieTable''\right)$

$\left[\begin{array}{ccccccc}& \mathrm{`|`}& \mathrm{e1}& \mathrm{e2}& \mathrm{e3}& \mathrm{e4}& \mathrm{e5}\\ & \mathrm{---}& \mathrm{----}& \mathrm{----}& \mathrm{----}& \mathrm{----}& \mathrm{----}\\ \mathrm{e1}& \mathrm{`|`}& 0& 0& \mathrm{e1}& 0& 0\\ \mathrm{e2}& \mathrm{`|`}& 0& 0& \mathrm{e2}& 0& 0\\ \mathrm{e3}& \mathrm{`|`}& -\mathrm{e1}& -\mathrm{e2}& 0& 0& 0\\ \mathrm{e4}& \mathrm{`|`}& 0& 0& 0& 0& -\mathrm{e4}-\mathrm{e5}\\ \mathrm{e5}& \mathrm{`|`}& 0& 0& 0& \mathrm{e4}\+\mathrm{e5}& 0\end{array}\right]$

$A≔\mathrm{decomposition}\left[1\right]$

$A:=\left[\begin{array}{rrrrr}1& 0& 0& 0& 0\\ -1& 0& 0& 0& -1\\ 0& 0& 0& 1& 0\\ 1& 1& 0& 0& 1\\ 1& 0& 1& 0& 1\end{array}\right]$

$\mathrm{Query}\left(\mathrm{Alg1}\,\mathrm{Alg2}\,A\,''Homomorphism''\right)$

$\mathrm{true}$

$L≔\mathrm{\_DG}\left(\left[\left[''LieAlgebra''\,\mathrm{Alg3}\,\left[4\right]\right]\,\left[\left[\left[1\,3\,2\right]\,-2\right]\,\left[\left[1\,4\,1\right]\,-1\right]\,\left[\left[2\,3\,1\right]\,-1\right]\,\left[\left[2\,4\,2\right]\,-1\right]\right]\right]\right)$

$\mathrm{DGsetup}\left(L\right)\:$

$L≔\left[\left[\mathrm{e1}\,\mathrm{e3}\right]\=-2\mathrm{e2}\,\left[\mathrm{e1}\,\mathrm{e4}\right]\=-\mathrm{e1}\,\left[\mathrm{e2}\,\mathrm{e3}\right]\=-\mathrm{e1}\,\left[\mathrm{e2}\,\mathrm{e4}\right]\=-\mathrm{e2}\right]$

$\mathrm{Decompose}\left(\right)\:$

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.

$\mathrm{DifferentialGeometry}:-\mathrm{Tools}:-\mathrm{CalculationHistory}:-\mathrm{Get}\left(''Decompose''\right)$

$\left[\left[''Alg3''\,\left[-1\+\mathrm{\_z1}\,-\frac{1}{2}\+{\mathrm{\_z1}}^2\right]\right]\right]$

$\mathrm{Decompose}\left(\mathrm{hint}=\left[x\,\left[x^2-\frac{1}{2}\,\left[x-\frac{1}{\mathrm{sqrt}\left(2\right)}\,x+\frac{1}{\mathrm{sqrt}\left(2\right)}\right]\right]\right]\right)\left[2\right]$

$\left[\mathrm{e1}\+\sqrt{2}\mathrm{e2}\,\mathrm{e3}\+\sqrt{2}\mathrm{e4}\,\mathrm{e1}-\sqrt{2}\mathrm{e2}\,\mathrm{e3}-\sqrt{2}\mathrm{e4}\right]$

$\mathrm{L3}≔\mathrm{\_DG}\left(\left[\left[''LieAlgebra''\,\mathrm{Alg3}\,\left[4\right]\right]\,\left[\left[\left[1\,3\,1\right]\,1\right]\,\left[\left[2\,4\,1\right]\,1\right]\,\left[\left[1\,4\,2\right]\,-1\right]\,\left[\left[2\,3\,2\right]\,1\right]\right]\right]\right)$

$\mathrm{L3}≔\left[\left[\mathrm{e1}\,\mathrm{e3}\right]\=\mathrm{e1}\,\left[\mathrm{e1}\,\mathrm{e4}\right]\=-\mathrm{e2}\,\left[\mathrm{e2}\,\mathrm{e3}\right]\=\mathrm{e2}\,\left[\mathrm{e2}\,\mathrm{e4}\right]\=\mathrm{e1}\right]$

$\mathrm{DGsetup}\left(\mathrm{L3}\right)\:$

$\mathrm{Decompose}\left(\mathrm{method}=''AbsolutelyIndecomposable''\,\mathrm{hint}=\left[x\,\left[x^2+1\,\left[x-\mathrm{I}\,x+\mathrm{I}\right]\right]\right]\right)\left[2\right]$

$\left[\mathrm{e1}\+\mathrm{e2}\mathrm{I}\,\mathrm{e3}-\mathrm{I}\mathrm{e4}\,\mathrm{e1}-\mathrm{I}\mathrm{e2}\,\mathrm{e3}\+\mathrm{e4}\mathrm{I}\right]$

$L≔\mathrm{\_DG}\left(\left[\left[''LieAlgebra''\,\mathrm{so4}\,\left[6\right]\right]\,\left[\left[\left[1\,2\,4\right]\,-1\right]\,\left[\left[1\,3\,5\right]\,-1\right]\,\left[\left[1\,4\,2\right]\,1\right]\,\left[\left[1\,5\,3\right]\,1\right]\,\left[\left[2\,3\,6\right]\,-1\right]\,\left[\left[2\,4\,1\right]\,-1\right]\,\left[\left[2\,6\,3\right]\,1\right]\,\left[\left[3\,5\,1\right]\,-1\right]\,\left[\left[3\,6\,2\right]\,-1\right]\,\left[\left[4\,5\,6\right]\,-1\right]\,\left[\left[4\,6\,5\right]\,1\right]\,\left[\left[5\,6\,4\right]\,-1\right]\right]\right]\right)$

$L≔\left[\left[\mathrm{e1}\,\mathrm{e2}\right]\=-\mathrm{e4}\,\left[\mathrm{e1}\,\mathrm{e3}\right]\=-\mathrm{e5}\,\left[\mathrm{e1}\,\mathrm{e4}\right]\=\mathrm{e2}\,\left[\mathrm{e1}\,\mathrm{e5}\right]\=\mathrm{e3}\,\left[\mathrm{e2}\,\mathrm{e3}\right]\=-\mathrm{e6}\,\left[\mathrm{e2}\,\mathrm{e4}\right]\=-\mathrm{e1}\,\left[\mathrm{e2}\,\mathrm{e6}\right]\=\mathrm{e3}\,\left[\mathrm{e3}\,\mathrm{e5}\right]\=-\mathrm{e1}\,\left[\mathrm{e3}\,\mathrm{e6}\right]\=-\mathrm{e2}\,\left[\mathrm{e4}\,\mathrm{e5}\right]\=-\mathrm{e6}\,\left[\mathrm{e4}\,\mathrm{e6}\right]\=\mathrm{e5}\,\left[\mathrm{e5}\,\mathrm{e6}\right]\=-\mathrm{e4}\right]$

$\mathrm{DGsetup}\left(L\right)$

$\mathrm{Lie\; algebra:\; so4}$

$\mathrm{infolevel}\left[\mathrm{Decompose}\right]≔3$

${\mathrm{infolevel}}_{\mathrm{Decompose}}≔3$

$\mathrm{decomposition}≔\mathrm{Decompose}\left(\right)$

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]]

$\mathrm{decomposition}:=\left[\left[\left[\begin{array}{cccccc}\frac{1}{2}& 0& 0& 0& 0& -\frac{1}{2}\\ 0& \frac{1}{2}& 0& 0& \frac{1}{2}& 0\\ 0& 0& \frac{1}{2}& -\frac{1}{2}& 0& 0\\ \frac{1}{2}& 0& 0& 0& 0& \frac{1}{2}\\ 0& \frac{1}{2}& 0& 0& -\frac{1}{2}& 0\\ 0& 0& \frac{1}{2}& \frac{1}{2}& 0& 0\end{array}\right]\,\left[\left[\mathrm{e1}-\mathrm{e6}\,\mathrm{e2}\+\mathrm{e5}\,\mathrm{e3}-\mathrm{e4}\,\mathrm{e1}\+\mathrm{e6}\,\mathrm{e2}-\mathrm{e5}\,\mathrm{e3}\+\mathrm{e4}\right]\right]\right]\right]$