 MultiplicationTable - Maple Help

LieAlgebras[MultiplicationTable] - display the multiplication table of a Lie algebra or a general non-commutative algebra

Calling Sequences

MultiplicationTable(LieAlgebraName, keyword)

Parameters

LieAlgebraName  - (optional) name or string, the name assigned to a Lie algebra

keyword         - keyword string, one of "LieBracket", "ExteriorDerivative", "LieDerivative", "AlgebraTable" Description

 • MultiplicationTable(LieAlgebraName, keyword) displays the form of structure equations for the Lie algebra or algebra dictated by the keyword.
 • If the keyword is "LieBracket", then the Lie brackets of the basis elements are displayed in a two-dimensional array.
 • If the keyword is "AlgebraTable", then the non-commutative products of the basis elements are displayed in a two-dimensional array.
 • If the keyword is "ExteriorDerivative", then the exterior derivatives $d\left({\mathrm{θ}}^{i}\right)$of the dual basis elements are printed.
 • If the keyword is "LieDerivative", then the Lie derivatives ${\mathrm{ℒ}}_{{e}_{i}}{\mathrm{θ}}^{j}$ of the dual 1-forms  with respect to the basis vectors are displayed in a two-dimensional array.
 • If LieAlgebraName is omitted, then the appropriate multiplication table of the current algebra is displayed.
 • The command MultiplicationTable is part of the DifferentialGeometry:-LieAlgebras package.  It can be used in the form MultiplicationTable(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:-LieAlgebras:-MultiplicationTable(...). Examples

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

Example 1.

First we initialize a 5 dimensional Lie algebra.

 > $\mathrm{L1}â‰”\mathrm{_DG}\left(\left[\left["LieAlgebra",\mathrm{Alg1},\left[5\right]\right],\left[\left[\left[2,3,1\right],1\right],\left[\left[2,5,3\right],1\right],\left[\left[4,5,4\right],1\right]\right]\right]\right):$
 > $\mathrm{DGsetup}\left(\mathrm{L1}\right):$

Display the Lie bracket multiplication table.

 Alg1 > $\mathrm{MultiplicationTable}\left("LieBracket"\right)$
 $\left[\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e4}}\right]$ (2.1)

Display the exterior derivatives of the dual 1-forms.

 Alg1 > $\mathrm{MultiplicationTable}\left("ExteriorDerivative"\right)$
 ${d}{}\left({\mathrm{θ1}}\right){=}{-}{\mathrm{θ2}}{}{\bigwedge }{}{\mathrm{θ3}}$
 ${d}{}\left({\mathrm{θ2}}\right){=}{0}{}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ2}}$
 ${d}{}\left({\mathrm{θ3}}\right){=}{-}{\mathrm{θ2}}{}{\bigwedge }{}{\mathrm{θ5}}$
 ${d}{}\left({\mathrm{θ4}}\right){=}{-}{\mathrm{θ4}}{}{\bigwedge }{}{\mathrm{θ5}}$
 ${d}{}\left({\mathrm{θ5}}\right){=}{0}{}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ2}}$ (2.2)

Display the Lie derivatives of the dual 1-forms.

 Alg1 > $\mathrm{MultiplicationTable}\left("LieDerivative"\right)$ Example 2.

We initialize a 4 dimensional Lie algebra. Instead of using the standard default labels for the basis vectors we use and for the dual 1-forms we use .

 Alg1 > $\mathrm{L2}â‰”\mathrm{_DG}\left(\left[\left["LieAlgebra",\mathrm{Alg1},\left[4\right]\right],\left[\left[\left[2,3,1\right],1\right],\left[\left[2,4,3\right],1\right],\left[\left[4,2,4\right],1\right]\right]\right]\right):$
 Alg1 > $\mathrm{DGsetup}\left(\mathrm{L2},\left[X,Y,U,V\right],\left[\mathrm{α},\mathrm{β},\mathrm{σ},\mathrm{τ}\right]\right):$

Display the Lie bracket multiplication table.

 Alg1 > $\mathrm{MultiplicationTable}\left("LieBracket"\right)$
 $\left[\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e3}}\right]$ (2.3)

Display the exterior derivatives of the dual 1-forms.

 Alg1 > $\mathrm{MultiplicationTable}\left("ExteriorDerivative"\right)$
 ${d}{}\left({\mathrm{α}}\right){=}{-}{\mathrm{β}}{}{\bigwedge }{}{\mathrm{σ}}$
 ${d}{}\left({\mathrm{β}}\right){=}{0}{}{\mathrm{α}}{}{\bigwedge }{}{\mathrm{β}}$
 ${d}{}\left({\mathrm{σ}}\right){=}{-}{\mathrm{β}}{}{\bigwedge }{}{\mathrm{τ}}$
 ${d}{}\left({\mathrm{τ}}\right){=}{-}{\mathrm{β}}{}{\bigwedge }{}{\mathrm{τ}}$ (2.4)

Display the Lie derivatives of the dual 1-forms.

 Alg1 > $\mathrm{MultiplicationTable}\left("LieDerivative"\right)$ Example 3.

We initialize the quaternions $\mathrm{ℍ}$ and display the multiplication table.

 Alg1 > $\mathrm{L3}â‰”\mathrm{AlgebraLibraryData}\left("Quaternions",H\right)$
 ${\mathrm{L3}}{:=}\left[{{\mathrm{e1}}}^{{2}}{=}{\mathrm{e1}}{,}{\mathrm{e1}}{.}{\mathrm{e2}}{=}{\mathrm{e2}}{,}{\mathrm{e1}}{.}{\mathrm{e3}}{=}{\mathrm{e3}}{,}{\mathrm{e1}}{.}{\mathrm{e4}}{=}{\mathrm{e4}}{,}{\mathrm{e2}}{.}{\mathrm{e1}}{=}{\mathrm{e2}}{,}{{\mathrm{e2}}}^{{2}}{=}{-}{\mathrm{e1}}{,}{\mathrm{e2}}{.}{\mathrm{e3}}{=}{\mathrm{e4}}{,}{\mathrm{e2}}{.}{\mathrm{e4}}{=}{-}{\mathrm{e3}}{,}{\mathrm{e3}}{.}{\mathrm{e1}}{=}{\mathrm{e3}}{,}{\mathrm{e3}}{.}{\mathrm{e2}}{=}{-}{\mathrm{e4}}{,}{{\mathrm{e3}}}^{{2}}{=}{-}{\mathrm{e1}}{,}{\mathrm{e3}}{.}{\mathrm{e4}}{=}{\mathrm{e2}}{,}{\mathrm{e4}}{.}{\mathrm{e1}}{=}{\mathrm{e4}}{,}{\mathrm{e4}}{.}{\mathrm{e2}}{=}{\mathrm{e3}}{,}{\mathrm{e4}}{.}{\mathrm{e3}}{=}{-}{\mathrm{e2}}{,}{{\mathrm{e4}}}^{{2}}{=}{-}{\mathrm{e1}}\right]$ (2.5)
 Alg1 > $\mathrm{DGsetup}\left(\mathrm{L3},\left[e,i,j,k\right],\left[\mathrm{θ}\right]\right)$
 ${\mathrm{algebra name: H}}$ (2.6)
 Alg1 > $\mathrm{MultiplicationTable}\left("AlgebraTable"\right)$ 