LieAlgebras[JacobsonRadical] - find the Jacobson radical for a matrix Lie algebra
M - a list of square matrices which define a basis for a matrix Lie algebra 𝔸.
The Jacobson radical of a matrix algebra 𝔸 is the set of matrices b ∈𝔸 such that traceab =0 for all a ∈ 𝔸. The Jacobson radical consists entirely of nilpotent matrices and coincides with the nilradical of 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(...).
Find the Jacobson radical of the set of matrices M.
M ≔ map⁡Matrix,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,−1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0
M ≔ 1000000000100000,0100000000000000,0000010000−100000,0010000100000000,0001000000000000
J ≔ JacobsonRadical⁡M
J ≔ 0001000000000000,0010000100000000,0100000000000000
Clearly each one of these matrices is nilpotent. Note that J = [M, M, M]. We check that J is also the nilradical of M, when viewed as an abstract Lie algebra.
L ≔ LieAlgebraData⁡M,Alg1:
DifferentialGeometry, LieAlgebras, LieAlgebraData, Nilradical
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