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
Examples
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 ei, ej of the basis elements e1, e2, ..., en are displayed in a two-dimensional array.
If the keyword is "AlgebraTable", then the non-commutative products ei⋅ej of the basis elements e1, e2, ..., en are displayed in a two-dimensional array.
If the keyword is "ExteriorDerivative", then the exterior derivatives dθiof the dual basis elements θ1, θ2, ... , θn are printed.
If the keyword is "LieDerivative", then the Lie derivatives ℒeiθj of the dual 1-forms θ1, θ2, ... , θn with respect to the basis vectors e1, e2, ..., en 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(...).
withDifferentialGeometry:withLieAlgebras:
Example 1.
First we initialize a 5 dimensional Lie algebra.
L1≔_DGLieAlgebra,Alg1,5,2,3,1,1,2,5,3,1,4,5,4,1:
DGsetupL1:
Display the Lie bracket multiplication table.
MultiplicationTableLieBracket
e2,e3=e1,e2,e5=e3,e4,e5=e4
Display the exterior derivatives of the dual 1-forms.
MultiplicationTableExteriorDerivative
dθ1=−θ2⋀θ3
dθ2=0θ1⋀θ2
dθ3=−θ2⋀θ5
dθ4=−θ4⋀θ5
dθ5=0θ1⋀θ2
Display the Lie derivatives of the dual 1-forms.
MultiplicationTableLieDerivative
| θ1θ2θ3θ4θ5------------------------e1| 00000e2| −θ30−θ500e3| θ20000e4| 000−θ50e5| 00θ2θ40
Example 2.
We initialize a 4 dimensional Lie algebra. Instead of using the standard default labels for the basis vectors we use X, Y, U,V and for the dual 1-forms we use α, β, σ, τ.
L2≔_DGLieAlgebra,Alg1,4,2,3,1,1,2,4,3,1,4,2,4,1:
DGsetupL2,X,Y,U,V,α,β,σ,τ:
e2,e3=e1,e2,e4=e3
dα=−β⋀σ
dβ=0α⋀β
dσ=−β⋀τ
dτ=−β⋀τ
| αβστ--------------------X| 0000Y| −σ0−ττU| β000V| 00β−β
Example 3.
We initialize the quaternions ℍ and display the multiplication table.
L3≔AlgebraLibraryDataQuaternions,H
L3:=e12=e1,e1.e2=e2,e1.e3=e3,e1.e4=e4,e2.e1=e2,e22=−e1,e2.e3=e4,e2.e4=−e3,e3.e1=e3,e3.e2=−e4,e32=−e1,e3.e4=e2,e4.e1=e4,e4.e2=e3,e4.e3=−e2,e42=−e1
DGsetupL3,e,i,j,k,θ
algebra name: H
MultiplicationTableAlgebraTable
| eijk---- ---- ---- ---- ---- e| eijki| i−ek−jj| j−k−eik| kj−i−e
See Also
DifferentialGeometry
LieAlgebras
ExteriorDerivative
LieBracket
LieDerivative
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