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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
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Description
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The Jacobson radical of a matrix algebra A is the set of matrices b in A such that tr(ab) = 0 for all a in A. The Jacobson radical consists entirely of nilpotent matrices and coincides with the nilradical of A.
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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.
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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(...).
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Examples
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Example 1.
Find the Jacobson radical of the set of matrices M.
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![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]]])](/support/helpjp/helpview.aspx?si=6621/file05809/math63.png)
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![M := [Matrix([[1, 0, 0, 0], [0, 0, 0, 0], [0, 0, 1, 0], [0, 0, 0, 0]]), Matrix([[0, 1, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]]), Matrix([[0, 0, 0, 0], [0, 1, 0, 0], [0, 0, -1, 0], [0, 0, 0, 0]]), Matrix([[0, 0, 1, 0], [0, 0, 0, 1], [0, 0, 0, 0], [0, 0, 0, 0]]), Matrix([[0, 0, 0, 1], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]])]](/support/helpjp/helpview.aspx?si=6621/file05809/math66.png)
| (2.1) |
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![J := [Matrix([[0, 0, 0, 1], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]]), Matrix([[0, 0, 1, 0], [0, 0, 0, 1], [0, 0, 0, 0], [0, 0, 0, 0]]), Matrix([[0, 1, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]])]](/support/helpjp/helpview.aspx?si=6621/file05809/math75.png)
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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.
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Alg1 >
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![[e2, e4, e5]](/support/helpjp/helpview.aspx?si=6621/file05809/math102.png)
| (2.3) |
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