The Jacobson radical of a matrix algebra $\mathrm{\𝔸}$is the set of matrices $b \in \mathrm{\𝔸}$ such that trace$\left(\mathrm{ab}\right) \=0$for all $a \in \mathrm{\𝔸}$. The Jacobson radical consists entirely of nilpotent matrices and coincides with the nilradical of $\mathrm{\𝔸}$.

A list of matrices defining a basis for the Jacobson radical is returned. If the Jacobson radical is trivial, then an empty list is returned.

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

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

$M≔\mathrm{map}\left(\mathrm{Matrix}\,\left[\left[\left[1\,0\,0\,0\right]\,\left[0\,0\,0\,0\right]\,\left[0\,0\,1\,0\right]\,\left[0\,0\,0\,0\right]\right]\,\left[\left[0\,1\,0\,0\right]\,\left[0\,0\,0\,0\right]\,\left[0\,0\,0\,0\right]\,\left[0\,0\,0\,0\right]\right]\,\left[\left[0\,0\,0\,0\right]\,\left[0\,1\,0\,0\right]\,\left[0\,0\,-1\,0\right]\,\left[0\,0\,0\,0\right]\right]\,\left[\left[0\,0\,1\,0\right]\,\left[0\,0\,0\,1\right]\,\left[0\,0\,0\,0\right]\,\left[0\,0\,0\,0\right]\right]\,\left[\left[0\,0\,0\,1\right]\,\left[0\,0\,0\,0\right]\,\left[0\,0\,0\,0\right]\,\left[0\,0\,0\,0\right]\right]\right]\right)$

$M≔\left[\left[\left[\begin{array}{rrrr}1& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 0\end{array}\right]\,\left[\begin{array}{rrrr}0& 1& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]\,\left[\begin{array}{rrrr}0& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& 0\end{array}\right]\,\left[\begin{array}{rrrr}0& 0& 1& 0\\ 0& 0& 0& 1\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]\,\left[\begin{array}{rrrr}0& 0& 0& 1\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]\right]\right]$

$J≔\mathrm{JacobsonRadical}\left(M\right)$

$J≔\left[\left[\left[\begin{array}{rrrr}0& 0& 0& 1\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]\,\left[\begin{array}{rrrr}0& 0& 1& 0\\ 0& 0& 0& 1\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]\,\left[\begin{array}{rrrr}0& 1& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]\right]\right]$

$L≔\mathrm{LieAlgebraData}\left(M\,\mathrm{Alg1}\right)\:$

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

$\mathrm{Nilradical}\left(\right)$

$\left[\mathrm{e2}\,\mathrm{e4}\,\mathrm{e5}\right]$