 MatrixAlgebra - Maple Help

LieAlgebraData[MatrixAlgebra] - convert a list of matrices to a Lie algebra data structure

Calling Sequence

Parameters

MatrixAlgebra   - a list of linearly independent square matrices (all of the same dimension)

AlgName         - a name or string, the name to be assigned to the Lie algebra

option          - the keyword argument coefficienttype = "real" Description

 • In the LieAlgebras package, the command DGsetup is used to initialize a Lie algebra -- that is, to define the basis elements for the Lie algebra and its dual and to store the structure constants for the Lie algebra in memory.  The first argument for DGsetup is a Lie algebra data structure which contains the structure constants in a standard format used by the LieAlgebras package.
 • An important class of Lie algebras is given by the matrix Lie algebras, where the Lie algebra bracket is given by the matrix commutator   The program LieAlgebraData(MatrixAlgebra, AlgName) will create the Lie algebra data structure for the matrix algebra whose basis is the given list of matrices MatrixAlgebra.
 • If the given matrices are complex matrices which form a real Lie algebra, then the keyword argument coefficienttype = "real" will ensure that real structure constants are calculated.
 • The command LieAlgebraData is part of the DifferentialGeometry:-LieAlgebras package.  It can be used in the form LieAlgebraData(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:-LieAlgebras:-LieAlgebraData(...). Examples

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

Example 1.

In this example we create a 5-dimensional Lie algebra from a list of 3 x 3 matrices. First we define the matrices.

 > $M≔\mathrm{map}\left(\mathrm{Matrix},\left[\left[\left[-1,2,-1\right],\left[0,1,0\right],\left[2,-2,2\right]\right],\left[\left[1,0,0\right],\left[2,-1,1\right],\left[2,-2,2\right]\right],\left[\left[0,0,0\right],\left[1,-1,1\right],\left[2,-2,2\right]\right],\left[\left[-1,2,-1\right],\left[1,0,0\right],\left[4,-4,2\right]\right],\left[\left[4,-4,2\right],\left[2,-2,1\right],\left[-2,2,-1\right]\right]\right]\right)$ Before continuing we note that it is not difficult to directly check that the span of the matrices M is closed under matrix commutators.  For example, the matrix commutator of ${M}_{1}$ and is $-{M}_{1}+{M}_{2}$. (The matrices P and Q computed below are equal.)

 > $P≔\mathrm{.}\left({M}_{1},{M}_{5}\right)-\mathrm{.}\left({M}_{5},{M}_{1}\right)$ > $\mathrm{GetComponents}\left(P,M\right)$
 $\left[{-}{1}{,}{1}{,}{0}{,}{0}{,}{0}\right]$ (2.1)
 > $Q≔-{M}_{1}+{M}_{2}$ The program LieAlgebraData will calculate all the required matrix commutators of the matrices in M as linear combinations in M and return the result as a Lie algebra data structure.

 > $L≔\mathrm{LieAlgebraData}\left(M,\mathrm{Ex1}\right)$
 ${L}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{\mathrm{e2}}{-}{\mathrm{e1}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e4}}{+}{\mathrm{e3}}{-}{\mathrm{e1}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e2}}{-}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e4}}{+}{\mathrm{e3}}{-}{\mathrm{e2}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{2}{}{\mathrm{e2}}{-}{2}{}{\mathrm{e1}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e4}}{+}{\mathrm{e3}}{-}{\mathrm{e1}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e4}}{+}{\mathrm{e3}}{+}{\mathrm{e2}}{-}{2}{}{\mathrm{e1}}\right]$ (2.2)

Here represent the matrices respectively.  Thus the structure equation  coincides with that explicitly computed above.

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

Finally, we remark that this Lie algebra is decomposable.

 Ex1 > $\mathrm{Query}\left("Indecomposable"\right)$
 ${\mathrm{false}}$ (2.3)

A decomposition of this Lie algebra can be found using the program Decompose.

Example 2.

The set of trace-free complex matrices define a real 6-dimensional Lie algebra. Here is a basis for this matrix algebra.

 Ex1 > $M≔\mathrm{map}\left(\mathrm{Matrix},\left[\left[\left[1,0\right],\left[0,-1\right]\right],\left[\left[0,1\right],\left[0,0\right]\right],\left[\left[0,0\right],\left[1,0\right]\right],\left[\left[I,0\right],\left[0,-I\right]\right],\left[\left[0,I\right],\left[0,0\right]\right],\left[\left[0,0\right],\left[I,0\right]\right]\right]\right)$ Because these matrices are linearly dependent (over the complex numbers), we must use the keyword argument coefficienttype = "real" to calculate the real structure equations.

 > $\mathrm{LieAlgebraData}\left(M,\mathrm{alg2},\mathrm{coefficienttype}="real"\right)$
 $\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{2}{}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{-}{2}{}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{2}{}{\mathrm{e5}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e6}}\right]{=}{-}{2}{}{\mathrm{e6}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{-}{2}{}{\mathrm{e5}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{2}{}{\mathrm{e6}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{-}{2}{}{\mathrm{e2}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e6}}\right]{=}{2}{}{\mathrm{e3}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e1}}\right]$ (2.4)