LieAlgebras[Radical] - find the radical of a Lie algebra
Calling Sequences
Radical(LieAlgName)
Parameters
LieAlgName - (optional) name or string, the name of a Lie algebra
Description
Examples
The radical of a Lie algebra is the largest solvable ideal contained in . The radical of can be calculated as the orthogonal complement of the derived algebra of with respect to the Killing form , that is, rad = for all . See, for example, Fulton and Harris Representation Theory, Graduate Texts in Mathematics 129, Springer 1991, Proposition C.22 page 484.
Radical(LieAlgName) calculates the radical of the Lie algebra defined by LieAlgName. If no argument is given, then the radical of the current Lie algebra is found.
A list of vectors defining a basis for the rad(is returned. If rad( is trivial, then an empty list is returned.
The command Radical is part of the DifferentialGeometry:-LieAlgebras package. It can be used in the form Radical(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:-LieAlgebras:-Radical(...).
Example 1.
First we initialize a Lie algebra.
We calculate the radical of Alg1 to be the 4-dimensional ideal with basis and check that the result is indeed a solvable ideal.
We remark that the span of the vectors is a 4-dimensional solvable subalgebra but it is not an ideal.
See Also
DifferentialGeometry
LieAlgebras
LeviDecomposition
Nilradical
Query[Ideal]
Query[Solvable]
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