 StandardRepresentation - Maple Help

LieAlgebras[StandardRepresentation] - find the standard matrix representation or linear vector field representation of a classical matrix algebra

Calling Sequences

StandardRepresentation(alg, option)

StandardRepresentation(alg,M)

Parameters

alg      - a name or string, the name of an initialized classical matrix algebra

M        - a name or string, the name of an initialized manifold

- the keyword argument representationspace = V, where is the name of an initialized space Description

 • The first calling sequence for StandardRepresentation returns a list of matrices defining any one of the classical simple matrix algebras:

$-$

For convenience the following matrix algebras can also be constructed.

For the definitions and examples of all these algebras, see Details for SimpleLieAlgebraData.

 • With the keyword argument representationspace = V, a representation is returned
 • The second calling sequence gives the corresponding list of linear vectors fields for these algebras.
 • The Lie algebra for these classical matrix algebras must first be created using the command SimpleLieAlgebraData.
 • The command Query/[MatrixAlgebra] can be used to verify that a given list of matrices belongs to any one of these matrix algebras. Examples

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

Example 1.

We obtain the explicit matrix representation for $\mathrm{sl}\left(3\right)$, the 8 dimensional Lie algebra of trace-free matrices. First create the data for $\mathrm{sl}\left(3\right)$using the SimpleLieAlgebraData command.

 > $\mathrm{LD1}≔\mathrm{SimpleLieAlgebraData}\left("sl\left(3\right)",\mathrm{sl3},\mathrm{labelformat}="gl",\mathrm{labels}=\left['E','\mathrm{θ}'\right]\right)$
 ${\mathrm{LD1}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{2}{}{\mathrm{e4}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e7}}\right]{=}{-}{2}{}{\mathrm{e7}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e8}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e6}}\right]{=}{2}{}{\mathrm{e6}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e8}}\right]{=}{-}{2}{}{\mathrm{e8}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e1}}{-}{\mathrm{e2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e8}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e2}}\right]{,}\left[{\mathrm{E11}}{,}{\mathrm{E22}}{,}{\mathrm{E12}}{,}{\mathrm{E13}}{,}{\mathrm{E21}}{,}{\mathrm{E23}}{,}{\mathrm{E31}}{,}{\mathrm{E32}}\right]{,}\left[{\mathrm{θ11}}{,}{\mathrm{θ22}}{,}{\mathrm{θ12}}{,}{\mathrm{θ13}}{,}{\mathrm{θ21}}{,}{\mathrm{θ23}}{,}{\mathrm{θ31}}{,}{\mathrm{θ32}}\right]$ (2.1)

Initialize this Lie algebra.

 > $\mathrm{DGsetup}\left(\mathrm{LD1}\right)$
 ${\mathrm{Lie algebra: sl3}}$ (2.2)

Now get the explicit matrices defining $\mathrm{sl}\left(3\right)$.

 sl3 > $\mathrm{A1}≔\mathrm{StandardRepresentation}\left(\mathrm{sl3}\right)$ The notation for the basis for the abstract Lie algebra was constructed to match this list of matrices:

 sl3 > $\mathrm{Tools}:-\mathrm{DGinfo}\left(\mathrm{sl3},"FrameBaseVectors"\right)$
 $\left[{\mathrm{E11}}{,}{\mathrm{E22}}{,}{\mathrm{E12}}{,}{\mathrm{E13}}{,}{\mathrm{E21}}{,}{\mathrm{E23}}{,}{\mathrm{E31}}{,}{\mathrm{E32}}\right]$ (2.3)

The structure equations for the Lie algebra defined by the matrices $\mathrm{A1}$ coincides exactly with the Lie algebra structure equations generated by the SimpleLieAlgebraData command in equation (2.1).

 sl3 > $\mathrm{LieAlgebraData}\left(\mathrm{A1},\mathrm{sl3a}\right)$
 $\left[\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{2}{}{\mathrm{e4}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e7}}\right]{=}{-}{2}{}{\mathrm{e7}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e8}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e6}}\right]{=}{2}{}{\mathrm{e6}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e8}}\right]{=}{-}{2}{}{\mathrm{e8}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e1}}{-}{\mathrm{e2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e8}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e2}}\right]$ (2.4)

One can see by inspection that all the matrices in  are trace-free. This can also be verified using the Query command.

 sl3 > $\mathrm{Query}\left(\mathrm{A1},"sl\left(3\right)","MatrixAlgebra"\right)$
 ${\mathrm{true}}$ (2.5)

To obtain the standard vector field representation for first define a manifold with coordinates $\mathrm{x1},\mathrm{x2},\mathrm{x3}.$

 sl3 > $\mathrm{DGsetup}\left(\left[\mathrm{x1},\mathrm{x2},\mathrm{x3}\right],V\right)$
 ${\mathrm{frame name: V}}$ (2.6)

We get the desired vector fields with the second calling sequence.

 V > $\mathrm{Γ1}≔\mathrm{StandardRepresentation}\left(\mathrm{sl3},V\right)$
 ${\mathrm{Γ1}}{:=}\left[{\mathrm{x1}}{}{\mathrm{D_x1}}{-}{\mathrm{x3}}{}{\mathrm{D_x3}}{,}{\mathrm{x2}}{}{\mathrm{D_x2}}{-}{\mathrm{x3}}{}{\mathrm{D_x3}}{,}{\mathrm{x1}}{}{\mathrm{D_x2}}{,}{\mathrm{x1}}{}{\mathrm{D_x3}}{,}{\mathrm{x2}}{}{\mathrm{D_x1}}{,}{\mathrm{x2}}{}{\mathrm{D_x3}}{,}{\mathrm{x3}}{}{\mathrm{D_x1}}{,}{\mathrm{x3}}{}{\mathrm{D_x2}}\right]$ (2.7)

Again the structure equations for are identical to those in equations (2.1)or (2.4).

 V > $\mathrm{LieAlgebraData}\left(\mathrm{Γ1}\right)$
 $\left[\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{2}{}{\mathrm{e4}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e7}}\right]{=}{-}{2}{}{\mathrm{e7}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e8}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e6}}\right]{=}{2}{}{\mathrm{e6}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e8}}\right]{=}{-}{2}{}{\mathrm{e8}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e1}}{-}{\mathrm{e2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e8}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e2}}\right]$ (2.8)

Here is the standard representation, given as a representation mapping.

 V > $\mathrm{ρ}≔\mathrm{StandardRepresentation}\left(\mathrm{sl3},\mathrm{representationspace}=V\right)$ Example 2.

In this example we construct 2 different matrix representations for the Lorentz Lie algebra . This is the 6-dimensional Lie algebra of 4×4 matrices $A$ which are skew-symmetric with respect to a signature  quadratic form $Q$, that is, There are two standard forms foreither

or

which give rise to two forms for the matrices Either form can be generated. The default is since this form is better for calculating the Cartan subalgebra, the root space decomposition, the Cartan decomposition and so on.

 V > $\mathrm{LD2a}≔\mathrm{SimpleLieAlgebraData}\left("so\left(3, 1\right)",\mathrm{so31a},\mathrm{labelformat}="gl",\mathrm{labels}=\left['B','\mathrm{β}'\right]\right)$
 ${\mathrm{LD2a}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e4}}\right]{,}\left[{\mathrm{B11}}{,}{\mathrm{B13}}{,}{\mathrm{B14}}{,}{\mathrm{B23}}{,}{\mathrm{B24}}{,}{\mathrm{B34}}\right]{,}\left[{\mathrm{β11}}{,}{\mathrm{β13}}{,}{\mathrm{β14}}{,}{\mathrm{β23}}{,}{\mathrm{β24}}{,}{\mathrm{β34}}\right]$ (2.9)
 V > $\mathrm{DGsetup}\left(\mathrm{LD2a}\right)$
 ${\mathrm{Lie algebra: so31a}}$ (2.10)

Here are the defining matrices for  with respect to

 so31a > $\mathrm{A2a}≔\mathrm{StandardRepresentation}\left(\mathrm{so31a}\right)$ To get the alternative form for using add the keyword argument to the arguments for SimpleLieAlgebraData.

 V > $\mathrm{LD2b}≔\mathrm{SimpleLieAlgebraData}\left("so\left(3,1\right)",\mathrm{so31b},\mathrm{version}=2,\mathrm{labelformat}="gl",\mathrm{labels}=\left['C','\mathrm{gamma}'\right]\right)$
 ${\mathrm{LD2b}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e3}}\right]{,}\left[{\mathrm{C12}}{,}{\mathrm{C13}}{,}{\mathrm{C23}}{,}{\mathrm{C14}}{,}{\mathrm{C24}}{,}{\mathrm{C34}}\right]{,}\left[{\mathrm{γ12}}{,}{\mathrm{γ13}}{,}{\mathrm{γ23}}{,}{\mathrm{γ14}}{,}{\mathrm{γ24}}{,}{\mathrm{γ34}}\right]$ (2.11)
 so31a > $\mathrm{DGsetup}\left(\mathrm{LD2b}\right)$
 ${\mathrm{Lie algebra: so31b}}$ (2.12)

Here are the defining matrices for with respect to ${Q}_{2}.$

 so31b > $\mathrm{A2b}≔\mathrm{StandardRepresentation}\left(\mathrm{so31b}\right)$ We check that these matrices satisfy the defining equations and respectively.

 so31b > $\mathrm{Query}\left(\mathrm{A2a},"so\left(3,1\right)","MatrixAlgebra"\right)$
 ${\mathrm{true}}$ (2.13)
 so31b > $\mathrm{Query}\left(\mathrm{A2b},"so\left(3, 1\right)",\mathrm{version}=2,"MatrixAlgebra"\right)$
 ${\mathrm{true}}$ (2.14)

Example 3.

We give the standard representation for the Lie algebra of skew-Hermitian matrices.



 so31b > $\mathrm{LD3}≔\mathrm{SimpleLieAlgebraData}\left("u\left(3\right)",\mathrm{u3},\mathrm{version}=2,\mathrm{labelformat}="gl",\mathrm{labels}=\left['S','\mathrm{σ}'\right]\right)$
 ${\mathrm{LD3}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{-}{2}{}{\mathrm{e4}}{+}{2}{}{\mathrm{e7}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e8}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e8}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e6}}\right]{=}{-}{2}{}{\mathrm{e4}}{+}{2}{}{\mathrm{e9}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e9}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e8}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e8}}\right]{=}{-}{2}{}{\mathrm{e7}}{+}{2}{}{\mathrm{e9}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e9}}\right]{=}{-}{\mathrm{e8}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e3}}\right]{,}\left[{\mathrm{S12}}{,}{\mathrm{S13}}{,}{\mathrm{S23}}{,}{\mathrm{Si11}}{,}{\mathrm{Si12}}{,}{\mathrm{Si13}}{,}{\mathrm{Si22}}{,}{\mathrm{Si23}}{,}{\mathrm{Si33}}\right]{,}\left[{\mathrm{σ12}}{,}{\mathrm{σ13}}{,}{\mathrm{σ23}}{,}{\mathrm{sigmai11}}{,}{\mathrm{sigmai12}}{,}{\mathrm{sigmai13}}{,}{\mathrm{sigmai22}}{,}{\mathrm{sigmai23}}{,}{\mathrm{sigmai33}}\right]$ (2.15)

 so31b > $\mathrm{DGsetup}\left(\mathrm{LD3}\right)$
 ${\mathrm{Lie algebra: u3}}$ (2.16)
 u3 > $\mathrm{A3}≔\mathrm{StandardRepresentation}\left(\mathrm{u3}\right)$ To calculate the structure equations for this list of matrices, as a real Lie algebra, include the keyword argument " in the calling sequence for the LieAlgebraData command.

 u3 > $\mathrm{LieAlgebraData}\left(\mathrm{A3},\mathrm{u3a},\mathrm{method}="real"\right)$
 $\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{-}{2}{}{\mathrm{e4}}{+}{2}{}{\mathrm{e7}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e8}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e8}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e6}}\right]{=}{-}{2}{}{\mathrm{e4}}{+}{2}{}{\mathrm{e9}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e9}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e8}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e8}}\right]{=}{-}{2}{}{\mathrm{e7}}{+}{2}{}{\mathrm{e9}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e9}}\right]{=}{-}{\mathrm{e8}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e3}}\right]$ (2.17)

We obtain the same structure equations as in (2.15).

We check that these matrices are skew-Hermitian but they are not all trace-free.

 u3 > $\mathrm{Query}\left(\mathrm{A3},"u\left(3\right)","MatrixAlgebra"\right)$
 ${\mathrm{true}}$ (2.18)
 u3 > $\mathrm{Query}\left(\mathrm{A3},"sl\left(3\right)","MatrixAlgebra"\right)$
 ${\mathrm{false}}$ (2.19)