Let $G$ be an $n$-dimensional Lie group with Lie algebra $\mathrm{\𝔤}$$$and let $\left[e_i\, e_j\right] \= C_{\mathrm{ij}}^k{}_{ }{e}_k$ be the structure equations for $\mathrm{\𝔤}$. If $x^i$are coordinates for the dual vector space $\mathrm{\𝔤}$${}^{\*}$, then the infinitesimal generators for the co-adjoint action of $G$ on$\mathit{ }\mathrm{\𝔤}$${}^{\*}$are the vector fields $X_i\= C_{\mathrm{ij}}^k{}_{ }{x}^j\frac{\partial }{\partial x^k}$ .

The command InfinitesimalCoadjointAction(Algebra, Manifold) calculates the vector fields $X_i$for the Lie algebra Algebra using the coordinates for the dual space provide by M.

The command InfinitesimalCoadjointAction is part of the DifferentialGeometry:-LieAlgebras package. It can be used in the form InfinitesimalCoadjointAction(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:- LieAlgebras:- InfinitesimalCoadjointAction(...).

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

$\mathrm{LD1}≔\mathrm{\_DG}\left(\left[\left[''LieAlgebra''\,\mathrm{alg1}\,\left[3\right]\right]\,\left[\left[\left[1\,3\,1\right]\,1\right]\,\left[\left[2\,3\,1\right]\,1\right]\,\left[\left[2\,3\,2\right]\,1\right]\right]\right]\right)$

$\mathrm{LD1}:=\left[\left[\mathrm{e1}\,\mathrm{e3}\right]\=\mathrm{e1}\,\left[\mathrm{e2}\,\mathrm{e3}\right]\=\mathrm{e1}\+\mathrm{e2}\right]$

$\mathrm{DGsetup}\left(\mathrm{LD1}\right)$

$\mathrm{Lie\; algebra:\; alg1}$

$\mathrm{DGsetup}\left(\left[x\,y\,z\right]\,N\right)$

$\mathrm{frame\; name:\; N}$

$\mathrm{Gamma}≔\mathrm{InfinitesimalCoadjointAction}\left(\mathrm{alg1}\,N\right)$

$\mathrm{\Γ}:=\left[x\mathrm{D\_z}\,\left(x\+y\right)\mathrm{D\_z}\,-x\mathrm{D\_x}\+\left(-x-y\right)\mathrm{D\_y}\right]$

$\mathrm{LieAlgebraData}\left(\mathrm{Gamma}\right)$

$\left[\left[\mathrm{e1}\,\mathrm{e3}\right]\=\mathrm{e1}\,\left[\mathrm{e2}\,\mathrm{e3}\right]\=\mathrm{e1}\+\mathrm{e2}\right]$

$A≔\mathrm{Adjoint}\left(\mathrm{e3}\right)$

$\mathrm{convert}\left(\mathrm{LinearAlgebra}:-\mathrm{Transpose}\left(A\right)\,\mathrm{DGvector}\,N\right)$

$-x\mathrm{D\_x}\+\left(-x-y\right)\mathrm{D\_y}$

$\mathrm{LD2}≔\mathrm{\_DG}\left(\left[\left[''LieAlgebra''\,\mathrm{alg2}\,\left[4\right]\right]\,\left[\left[\left[2\,4\,1\right]\,1\right]\,\left[\left[3\,4\,3\right]\,1\right]\right]\right]\right)$

$\mathrm{LD2}:=\left[\left[\mathrm{e2}\,\mathrm{e4}\right]\=\mathrm{e1}\,\left[\mathrm{e3}\,\mathrm{e4}\right]\=\mathrm{e3}\right]$

$\mathrm{DGsetup}\left(\mathrm{LD2}\right)$

$\mathrm{Lie\; algebra:\; alg2}$

$\mathrm{DGsetup}\left(\left[w\,x\,y\,z\right]\,\mathrm{N2}\right)$

$\mathrm{frame\; name:\; N2}$

$\mathrm{\Γ2}≔\mathrm{InfinitesimalCoadjointAction}\left(\mathrm{alg2}\,\mathrm{N2}\right)$

$\mathrm{\Γ2}:=\left[w\mathrm{D\_z}\,y\mathrm{D\_z}\,-w\mathrm{D\_x}-y\mathrm{D\_y}\right]$

$\mathrm{Center}\left(\mathrm{alg2}\right)$

$\left[\mathrm{e1}\right]$

$\mathrm{QuotientAlgebra}\left(\left[\mathrm{e1}\right]\,\left[\mathrm{e2}\,\mathrm{e3}\,\mathrm{e4}\right]\right)$

$\left[\left[\mathrm{e2}\,\mathrm{e3}\right]\=\mathrm{e2}\right]$

$\mathrm{LieAlgebraData}\left(\mathrm{\Γ2}\right)$

$\left[\left[\mathrm{e2}\,\mathrm{e3}\right]\=\mathrm{e2}\right]$

$\mathrm{LD3}≔\mathrm{Library}:-\mathrm{Retrieve}\left(''Winternitz''\,1\,\left[5\,12\right]\,\mathrm{alg3}\right)$

$\mathrm{LD3}:=\left[\left[\mathrm{e1}\,\mathrm{e5}\right]\=\mathrm{e1}\,\left[\mathrm{e2}\,\mathrm{e5}\right]\=\mathrm{e1}\+\mathrm{e2}\,\left[\mathrm{e3}\,\mathrm{e5}\right]\=\mathrm{e2}\+\mathrm{e3}\,\left[\mathrm{e4}\,\mathrm{e5}\right]\=\mathrm{e3}\+\mathrm{e4}\right]$

$\mathrm{DGsetup}\left(\mathrm{LD3}\right)$

$\mathrm{Lie\; algebra:\; alg3}$

$\mathrm{DGsetup}\left(\left[\mathrm{x1}\,\mathrm{x2}\,\mathrm{x3}\,\mathrm{x4}\,\mathrm{x5}\right]\,\mathrm{N3}\right)$

$\mathrm{frame\; name:\; N3}$

$\mathrm{\Γ3}≔\mathrm{InfinitesimalCoadjointAction}\left(\mathrm{alg3}\,\mathrm{N3}\right)$

$\mathrm{\Γ3}:=\left[\mathrm{x1}\mathrm{D\_x5}\,\left(\mathrm{x1}\+\mathrm{x2}\right)\mathrm{D\_x5}\,\left(\mathrm{x2}\+\mathrm{x3}\right)\mathrm{D\_x5}\,\left(\mathrm{x3}\+\mathrm{x4}\right)\mathrm{D\_x5}\,-\mathrm{x1}\mathrm{D\_x1}\+\left(-\mathrm{x1}-\mathrm{x2}\right)\mathrm{D\_x2}\+\left(-\mathrm{x2}-\mathrm{x3}\right)\mathrm{D\_x3}\+\left(-\mathrm{x3}-\mathrm{x4}\right)\mathrm{D\_x4}\right]$

$C≔\mathrm{expand}\left(\mathrm{GroupActions}:-\mathrm{InvariantGeometricObjectFields}\left(\mathrm{\Γ3}\,\left[1\right]\,\mathrm{output}=''list''\right)\right)$

$C:=\left[-\ln \left(\mathrm{x1}\right)\+\frac{\mathrm{x2}}{\mathrm{x1}}\,\frac{1}{2}{\ln \left(\mathrm{x1}\right)}^2-\frac{\ln \left(\mathrm{x1}\right)\mathrm{x2}}{\mathrm{x1}}\+\frac{\mathrm{x3}}{\mathrm{x1}}\,-\frac{1}{6}{\ln \left(\mathrm{x1}\right)}^3\+\frac{1}{2}\frac{{\ln \left(\mathrm{x1}\right)}^2\mathrm{x2}}{\mathrm{x1}}-\frac{\ln \left(\mathrm{x1}\right)\mathrm{x3}}{\mathrm{x1}}\+\frac{\mathrm{x4}}{\mathrm{x1}}\right]$

$\mathrm{expand}\left(\left[\mathrm{expand}\left(\exp \left(-C\left[1\right]\right)\,\mathrm{symbolic}\right)\,2C\left[2\right]-{C\left[1\right]}^2\,3C\left[3\right]+{C\left[1\right]}^3-3C\left[1\right]C\left[2\right]\right]\right)$

$\left[\frac{\mathrm{x1}}{{\ⅇ}^{\frac{\mathrm{x2}}{\mathrm{x1}}}}\,\frac{2\mathrm{x3}}{\mathrm{x1}}-\frac{{\mathrm{x2}}^2}{{\mathrm{x1}}^2}\,\frac{3\mathrm{x4}}{\mathrm{x1}}\+\frac{{\mathrm{x2}}^3}{{\mathrm{x1}}^3}-\frac{3\mathrm{x2}\mathrm{x3}}{{\mathrm{x1}}^2}\right]$