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YCor
YCor
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What is an example of a real solvable simply-connected Lie group G$G$ whose nilradical does not have a complement (that is, G$G$ is not a semidirect product of the nilradical and another subgroup)? Is it possible to produce such a group if G$G$ is supersolvable (that is, G$G$ can be embedded in a group of upper triangular matrices)? Or the nilradical of a triangular real Lie group always split?

I know that such examples cannot be isomorphic to an algebraic group, because by the Jordan-Chevalley decomposition the unipotent radical (and therefore the nilradical) has a complement.

What is an example of a real solvable simply-connected Lie group G whose nilradical does not have a complement (that is, G is not a semidirect product of the nilradical and another subgroup)? Is it possible to produce such a group if G is supersolvable (that is, G can be embedded in a group of upper triangular matrices)? Or the nilradical of a triangular real Lie group always split?

I know that such examples cannot be isomorphic to an algebraic group, because by the Jordan-Chevalley decomposition the unipotent radical (and therefore the nilradical) has a complement.

What is an example of a real solvable simply-connected Lie group $G$ whose nilradical does not have a complement (that is, $G$ is not a semidirect product of the nilradical and another subgroup)? Is it possible to produce such a group if $G$ is supersolvable (that is, $G$ can be embedded in a group of upper triangular matrices)? Or the nilradical of a triangular real Lie group always split?

I know that such examples cannot be isomorphic to an algebraic group, because by the Jordan-Chevalley decomposition the unipotent radical (and therefore the nilradical) has a complement.

added 190 characters in body; edited title
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Guest7819
Guest7819
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Example of a supersolvable Lie algebra/Lie group/algebra whose nilradical does not have a complement

What is an example of a real solvable simply-connected Lie group G whose nilradical does not have a complement (that is, G is not a semidirect product of the nilradical and another subgroup)? Is it possible to produce such a group if G is supersolvable (that is, G can be embedded in a group of upper triangular matrices)? Or the nilradical of a triangular real Lie group always split?

I know that such examples cannot be isomorphic to an algebraic group, because by the Jordan-Chevalley decomposition the unipotent radical (and therefore the nilradical) has a complement.

Example of a supersolvable Lie algebra/Lie group whose nilradical does not have a complement

What is an example of a real solvable simply-connected Lie group G whose nilradical does not have a complement (that is, G is not a semidirect product of the nilradical and another subgroup)? Is it possible to produce such a group if G is supersolvable (that is, G can be embedded in a group of upper triangular matrices)? Or the nilradical of a triangular real Lie group always split?

Example of a supersolvable Lie group/algebra whose nilradical does not have a complement

What is an example of a real solvable simply-connected Lie group G whose nilradical does not have a complement (that is, G is not a semidirect product of the nilradical and another subgroup)? Is it possible to produce such a group if G is supersolvable (that is, G can be embedded in a group of upper triangular matrices)? Or the nilradical of a triangular real Lie group always split?

I know that such examples cannot be isomorphic to an algebraic group, because by the Jordan-Chevalley decomposition the unipotent radical (and therefore the nilradical) has a complement.

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Guest7819
Guest7819
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Example of a supersolvable Lie algebra/Lie group whose nilradical does not have a complement

What is an example of a real solvable simply-connected Lie group G whose nilradical does not have a complement (that is, G is not a semidirect product of the nilradical and another subgroup)? Is it possible to produce such a group if G is supersolvable (that is, G can be embedded in a group of upper triangular matrices)? Or the nilradical of a triangular real Lie group always split?