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LieAlgebras[JacobsonRadical] - find the Jacobson radical for a matrix Lie algebra

Calling Sequences

     JacobsonRadical(M)

Parameters

     M        - a list of square matrices which define a basis for a matrix Lie algebra 𝔸.

 

Description

Examples

See Also

Description

• 

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(...).

Examples

with(DifferentialGeometry): with(LieAlgebras):

 

Example 1.

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:=1000000000100000,0100000000000000,0000010000100000,0010000100000000,0001000000000000

(2.1)

 

J := JacobsonRadical(M);

J:=0001000000000000,0010000100000000,0100000000000000

(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 := LieAlgebraData(M, Alg1):

DGsetup(L):

Alg1 > 

Nilradical();

e2,e4,e5

(2.3)

See Also

DifferentialGeometry, LieAlgebras, LieAlgebraData, Nilradical