Talk:Lie–Kolchin theorem

Perron-Frobenius theorem

I added a link to above, but maybe this was not a good idea, as maybe it confuses the issue. (Its confusing me right now). Revert if you want to. linas 30 June 2005 14:54 (UTC)

Are there any theorems about the eigenvectors of Borel subgroups? Seems to me that as long as all the diagonal elements are 1, then the eigenspace is one dimensional; or more generally, if there are p 1's on the diagonal of an n by n matrix, then the eigenspace is n-p+1 dimensional, or so it seems, for real/complex matrices; but is it true? linas 30 June 2005 15:13 (UTC)

Dear Linas, good question about "multiplicity one". To give first a positive result: if the representation of the Borel subgroup B(in a semi-simple of more generally reductive linear algebraic group G over an algebraically closed field) comes from an irreducible representation of the bigger algebraic group G then the common eigenspace of B is in fact one-dimensional. This plays an important role in the classification of irreducible representations of semi-simple or reductive algebraic groups (like GL(n,C), Sp(2n,C), SO(n,C) and the exceptional series), in fact the definition of the highest weight vector of a finite-dimensional rep is based upon this fact. In general, for a general solvable and connected algebraic group G, a necessary condition (for uniqueness of common eigenvector) is that the representation is indecomposable (i.e. does not decompose into a direct sum of two sub-representations). Beyond that not much can be said about the one-dimensional subrepresentations. For example, take in the three-dimensional Heisenberg group over the complex numbers C (upper-triangular 3 &times 3 matrices with one's on the diagonal ) the subgroup G of matrices where the upper 2 &times 2 block is the identity matrix. This is a three-dimensional rep of a two-dimensional algebraic group isomorphic to C^2 (the two elements in the last columns add componentwise upon multiplication). The space of fixed vectors has dimension 2. On the other hand the representation is in fact indecomposable. The general rule maybe is (?): the more "degenerate" (i.e. closer to abelian) the solvable group G is the "bigger" is the space of common eigenvectors resp. one-dimensional subrepresentations in an indecomposable repr of G (please dont't take that as a precise mathematical statement...).

--212.18.24.11 12:32, 13 July 2005 (UTC)

Kolchin's theorem

Kolchin's theorem redirects to this article. Kaplansky,Fields and Rings (1972) states it in the form:

Let S be a multiplicative semigroup of unipotent matrices. The elements of S can be simultaneously triangularised.

The same formulation is used by Herstein (1986) [1], where it is deduced from a theorem of Wedderburn, that a finite dimensional algebra over a field having a basis of nilpotent elements is nilpotent.

In addition, this form of the theorem does not require the fielkd to be algebraically closed, which is an assumption throughout this article. So, does this form merit a separate section, or its own article? Deltahedron (talk) 20:06, 20 June 2014 (UTC)