JacobsonRadical - Maple Help
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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 .

 

Description

Examples

See Also

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 .

• 

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

 

Example 1.

Find the Jacobson radical of the set of matrices M.

(2.1)

 

(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.

Alg1 > 

(2.3)

See Also

DifferentialGeometry, LieAlgebras, LieAlgebraData, Nilradical


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