Calling Sequences

Parameters

M        - a list of square matrices which define a basis for a matrix Lie algebra $\mathrm{𝔸}$.

Description

 • The Jacobson radical of a matrix algebra is the set of matrices  such that tracefor all . 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(...).

Examples

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

Example 1.

Find the Jacobson radical of the set of matrices M.

 > $M≔\mathrm{map}\left(\mathrm{Matrix},\left[\left[\left[1,0,0,0\right],\left[0,0,0,0\right],\left[0,0,1,0\right],\left[0,0,0,0\right]\right],\left[\left[0,1,0,0\right],\left[0,0,0,0\right],\left[0,0,0,0\right],\left[0,0,0,0\right]\right],\left[\left[0,0,0,0\right],\left[0,1,0,0\right],\left[0,0,-1,0\right],\left[0,0,0,0\right]\right],\left[\left[0,0,1,0\right],\left[0,0,0,1\right],\left[0,0,0,0\right],\left[0,0,0,0\right]\right],\left[\left[0,0,0,1\right],\left[0,0,0,0\right],\left[0,0,0,0\right],\left[0,0,0,0\right]\right]\right]\right)$
 ${M}{≔}\left[\left[\left[\begin{array}{rrrr}{1}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {1}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrr}{0}& {1}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrr}{0}& {0}& {0}& {0}\\ {0}& {1}& {0}& {0}\\ {0}& {0}& {-}{1}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrr}{0}& {0}& {1}& {0}\\ {0}& {0}& {0}& {1}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrr}{0}& {0}& {0}& {1}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]\right]\right]$ (2.1)

 > $J≔\mathrm{JacobsonRadical}\left(M\right)$
 ${J}{≔}\left[\left[\left[\begin{array}{rrrr}{0}& {0}& {0}& {1}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrr}{0}& {0}& {1}& {0}\\ {0}& {0}& {0}& {1}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrr}{0}& {1}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]\right]\right]$ (2.2)

Clearly each one of these matrices is nilpotent. Note that J = [M[2], M[4], M[5]]. We check that J is also the nilradical of M, when viewed as an abstract Lie algebra.

 > $L≔\mathrm{LieAlgebraData}\left(M,\mathrm{Alg1}\right):$
 > $\mathrm{DGsetup}\left(L\right):$
 Alg1 > $\mathrm{Nilradical}\left(\right)$
 $\left[{\mathrm{e2}}{,}{\mathrm{e4}}{,}{\mathrm{e5}}\right]$ (2.3)