Let $\mathrm{\𝔤}\mathit{ }$be a Lie algebra and $x \in \mathrm{\𝔤}\mathit{\.}$ Then the adjoint transformation defined by $x$ is the linear transformation ad$\left(x\right) \: \mathrm{\𝔤}\mathit{ }\mathit{\to }\mathit{ }\mathrm{\𝔤}\mathit{ }$defined by ad$\left(x\right)\left(y\right) \= \left[x\, y\right]$for all $y \in \mathrm{\𝔤}$. The transformation $\mathrm{ad}\left(x\right)$always defines a derivation on $\mathrm{\𝔤}\mathit{\,}\mathit{ }$that is, ad$\left(x\right)\left(\left[y\, z\right]\right) \= \left[\mathrm{ad}\left(x\right)\left(y\right)\, z\right] \+ \left[y\, \mathrm{ad}\left(x\right)\left(z\right)\right]$. The mapping $x\to$ad$\left(x\right)$defines a representation of $\mathrm{\𝔤}\mathit{\.}$ The exponential of $\mathrm{ad}\left(x\right)\,$usually denoted by Ad($x$), is a Lie algebra isomorphism.

Adjoint(x) returns the matrix representing the linear transformation $\mathrm{ad}\left(x\right)$.

AdjointExp(x) returns the matrix representing the linear transformation Ad(x) = exp($\mathrm{ad}\left(x\right)\)$.

Adjoint() returns the list of adjoint matrices for the basis vectors of the current algebra $\mathrm{\𝔤}$.

Adjoint(alg) returns the list of adjoint matrices for the basis vectors of the algebra alg.

Adjoint(alg , representationspace = V) returns the adjoint representation of $\mathrm{\𝔤}$$$, with representation space V.

Adjoint(x, h) calculates the restriction of ad($x\)$to the subspace h (h must be an ad($x$) invariant subspace).

Adjoint(x, h, k) calculates Adjoint(x) on the vector space quotient g/k with respect to the basis determined by h (k must be an ad($x$) invariant subspace).

The commands Adjoint and AdjointExp are part of the DifferentialGeometry:-LieAlgebras package. They can be used in the form Adjoint(...) and AdjointExp(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:-LieAlgebras:-Adjoint(...) and DifferentialGeometry:-LieAlgebras:-AdjointExp(...).

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

$\mathrm{L1}≔\mathrm{\_DG}\left(\left[\left[''LieAlgebra''\,\mathrm{Alg1}\,\left[4\right]\right]\,\left[\left[\left[1\,3\,1\right]\,1\right]\,\left[\left[2\,4\,1\right]\,1\right]\,\left[\left[1\,4\,2\right]\,-1\right]\,\left[\left[2\,3\,2\right]\,1\right]\right]\right]\right)$

$\mathrm{L1}:=\left[\left[\mathrm{e1}\,\mathrm{e3}\right]\=\mathrm{e1}\,\left[\mathrm{e1}\,\mathrm{e4}\right]\=-\mathrm{e2}\,\left[\mathrm{e2}\,\mathrm{e3}\right]\=\mathrm{e2}\,\left[\mathrm{e2}\,\mathrm{e4}\right]\=\mathrm{e1}\right]$

$\mathrm{DGsetup}\left(\mathrm{L1}\right)\:$

$\mathrm{Adjoint}\left(t\mathrm{e4}\right)$

$\mathrm{AdjointExp}\left(t\mathrm{e4}\right)$

$\mathrm{Adjoint}\left(\mathrm{e1}+2\mathrm{e3}\right)$

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

$\mathrm{Adjoint}\left(\mathrm{e4}+2\mathrm{e3}\right),\mathrm{Adjoint}\left(\mathrm{e4}+2\mathrm{e3}\,\left[\mathrm{e1}\,\mathrm{e2}\right]\,\left[\mathrm{e3}\,\mathrm{e4}\right]\right)$

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

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

$\mathrm{\rho }≔\mathrm{Adjoint}\left(\mathrm{Alg1}\,\mathrm{representationspace}=V\right)$

$\mathrm{Query}\left(\mathrm{\rho }\,''Representation''\right)$

$\mathrm{true}$