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The Lie-Kolchin triangularizability theorem states that any connected, solvable subgroup $G$ of $\mathrm{GL}(n, \mathbf{C})$ is upper-triangularizable (there is some basis of $\mathbf{C}^n$ in which every matrix of $G$ is upper-triangular).

Is there any reasonable use case where this theorem really helps showing that $G$ is triangularizable?

Any connected and solvable matrix group I can come up with is trivially triangularizable (and most of the interesting groups are either not connected or not solvable). More generally, I would like to know if there is any useful application or corollary of this theorem.

Thanks!

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One immediate consequence of Lie-Kolchin is, that every irreducible finite-dimensional representation of a connected and solvable linear algebraic group G has dimension one. The same is true for solvable Lie algebras, by Lie's theorem, which is the analogue of Lie-Kolchin for Lie algebras. This has several more applications in matrix theory, representation theory and for geometric structures on solvable Lie groups.

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  • $\begingroup$ Thanks ! So this theorem is not as much about triangularizable linear groups than it is about irreducible representations... $\endgroup$
    – Kovomaka
    Commented Jun 1, 2023 at 12:56

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