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 {stile di visualizzazione rho (G)(l)=L.} Questo è, r(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 {stile di visualizzazione 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. Infatti, 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 {stile di visualizzazione rho (G)} has a triangular shape; in altre parole, the image group {stile di visualizzazione 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 {stile di visualizzazione {x+iyin mathbb {C} mid x^{2}+si^{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 (vero) line. Here the image {stile di visualizzazione rho (z)} di {displaystyle z=x+iy} is the orthogonal matrix {stile di visualizzazione {inizio{pmatrice}x&y\-y&xend{pmatrice}}.} References Gorbatsevich, V.V. (2001) [1994], "Lie-Kolchin theorem", Enciclopedia della matematica, EMS Press Kolchin, e. R. (1948), "Algebraic matric groups and the Picard-Vessiot theory of homogeneous linear ordinary differential equations", Annali di matematica, Seconda serie, 49 (1): 1–42, doi:10.2307/1969111, ISSN 0003-486X, JSTOR 1969111, SIG 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, Testi di laurea in matematica, vol. 66, Springer, pp. 74–75, ISBN 978-1-4612-6217-6 Categorie: Lie algebrasRepresentation theory of algebraic groupsTheorems in representation theory