Lie–Kolchin theorem

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 {displaystyle rho colon Gto 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 {displaystyle 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 {displaystyle rho (g),,,gin 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.

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 {displaystyle rho (G)} has a triangular shape; in other words, the image group {displaystyle 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.

Counter-example If the field K is not algebraically closed, the theorem can fail. The standard unit circle, viewed as the set of complex numbers {displaystyle {x+iyin mathbb {C} mid 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 {displaystyle rho (z)} of {displaystyle z=x+iy} is the orthogonal matrix {displaystyle {begin{pmatrix}x&y\-y&xend{pmatrix}}.} References Gorbatsevich, V.V. (2001) [1994], "Lie-Kolchin theorem", Encyclopedia of Mathematics, EMS Press 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): 1–42, doi:10.2307/1969111, 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 Waterhouse, William C. (2012) [1979], "10. Nilpotent and Solvable Groups §10.2 The Lie-Kolchin Triangularization Theorem", Introduction to Affine Group Schemes, Graduate texts in mathematics, vol. 66, Springer, pp. 74–75, ISBN 978-1-4612-6217-6 Categories: Lie algebrasRepresentation theory of algebraic groupsTheorems in representation theory