Lie–Kolchin theorem

In mathematics, the Lie–Kolchin theorem is a theorem in the representation theory of linear algebraic groups; Lie's theorem is the analog for linear Lie algebras.

It states that if G is a connected and solvable linear algebraic group defined over an algebraically closed field and

\rho \colon G\to GL(V)

a representation on a nonzero finite-dimensional vector space V, then there is a one-dimensional linear subspace L of V such that

\rho (G)(L)=L.

That is, ρ(G) has an invariant line L, on which G therefore acts through a one-dimensional representation. This is equivalent to the statement that V contains a nonzero vector v that is a common (simultaneous) eigenvector for all $\rho (g),\,\,g\in G$ .

It follows directly that every irreducible finite-dimensional representation of a connected and solvable linear algebraic group G has dimension one. In fact, this is another way to state the Lie–Kolchin theorem.

Lie's theorem states that any nonzero representation of a solvable Lie algebra on a finite dimensional vector space over an algebraically closed field of characteristic 0 has a one-dimensional invariant subspace.

The result for Lie algebras was proved by Sophus Lie (1876) and for algebraic groups was proved by Ellis Kolchin (1948, p.19).

The Borel fixed point theorem generalizes the Lie–Kolchin theorem.

Triangularization

Sometimes the theorem is also referred to as the Lie–Kolchin triangularization theorem because by induction it implies that with respect to a suitable basis of V the image $\rho (G)$ has a triangular shape; in other words, the image group $\rho (G)$ is conjugate in GL(n,K) (where n = dim V) to a subgroup of the group T of upper triangular matrices, the standard Borel subgroup of GL(n,K): the image is simultaneously triangularizable.

The theorem applies in particular to a Borel subgroup of a semisimple linear algebraic group G.

Lie's theorem

Lie's theorem states that if V is a finite dimensional vector space over an algebraically closed field of characteristic 0, then for any solvable Lie algebra of endomorphisms of V there is a vector that is an eigenvector for every element of the Lie algebra.

Applying this result repeatedly shows that there is a basis for V such that all elements of the Lie algebra are represented by upper triangular matrices. This is a generalization of the result of Frobenius that commuting matrices are simultaneously upper triangularizable, as commuting matrices form an abelian Lie algebra, which is a fortiori solvable.

A consequence of Lie's theorem is that any finite dimensional solvable Lie algebra over a field of characteristic 0 has a nilpotent derived algebra.

Counter-examples

If the field K is not algebraically closed, the theorem can fail. The standard unit circle, viewed as the set of complex numbers $\{x+iy\in {\mathbb {C}}\,|\,x^{2}+y^{2}=1\}$ of absolute value one is a one-dimensional commutative (and therefore solvable) linear algebraic group over the real numbers which has a two-dimensional representation into the special orthogonal group SO(2) without an invariant (real) line. Here the image $\rho (z)$ of $z=x+iy$ is the orthogonal matrix

{\begin{pmatrix}x&y\\-y&x\end{pmatrix}}.

For algebraically closed fields of characteristic p>0 Lie's theorem holds provided the dimension of the representation is less than p, but can fail for representations of dimension p. An example is given by the 3-dimensional nilpotent Lie algebra spanned by 1, x, and d/dx acting on the p-dimensional vector space k[x]/(x^p), which has no eigenvectors. Taking the semidirect product of this 3-dimensional Lie algebra by the p-dimensional representation (considered as an abelian Lie algebra) gives a solvable Lie algebra whose derived algebra is not nilpotent.

References

Gorbatsevich, V.V. (2001) [1994], "l/l058710", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
Kolchin, E. R. (1948), "Algebraic matric groups and the Picard-Vessiot theory of homogeneous linear ordinary differential equations", Annals of Mathematics. Second Series, 49: 1–42, ISSN 0003-486X, JSTOR 1969111, MR 0024884, Zbl 0037.18701
Lie, Sophus (1876), "Theorie der Transformationsgruppen. Abhandlung II", Archiv for Mathematik og Naturvidenskab, 1: 152–193
William C. Waterhouse, Introduction to Affine Group Schemes, Graduate Texts in Mathematics vol. 66, Springer Verlag New York, 1979 (chapter 10, in particular section 10.2).

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