Lie–Kolchin theorem

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In mathematics, the Lie–Kolchin theorem is a theorem in the representation theory of linear algebraic groups.

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

$\rho: G \rightarrow GL(V)$

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

ρ(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 there exists a non-zero eigenvector v which is a common (simultaneous) eigenvector for all $\rho(g), \,\, g \in G$ . Sometimes the theorem is also referred to as the Lie–Kolchin triangularization theorem because it implies that with respect to a suitable basis of V the image $ρ(G)$ has a triangular shape or in other words, the image group $ρ(G)$ is conjugate to a subgroup of the group T of upper triangular matrices (in GL(n,K) where n = dim V), the standard Borel subgroup of GL(n, K). Because every (finite-dimensional) representation of G has a one-dimensional invariant subspace according to the Lie–Kolchin theorem every irreducible finite-dimensional representation of a connected and solvable linear algebraic group G has dimension one (which is yet another way to state the Lie–Kolchin theorem).

The theorem applies in particular to a Borel subgroup of a semi-simple linear algebraic group G (which is defined as a maximal connected solvable subgroup of G).

This result is named for Sophus Lie and Ellis Kolchin (1916-1991).

Remark: If the field K is not algebraically closed the theorem does not hold in general. 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 abelian (and therefore solvable) algebraic group over the real numbers which has a two-dimensional representation into the special orthogonal group SO(2) without invariant (real) line. Here the image $ρ(z)$ of $z = x + i y$ is the orthogonal matrix

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

[edit] Lie's theorem

Let $\mathfrak{g}$ be a finite-dimensional complex solvable Lie algebra, and $V$ a representation of $\mathfrak{g}.$ Then there exists an element of $V$ which is a simultaneous eigenvector for all elements of $\mathfrak{g}.$

Applying this result inductively, we find that there is a basis of $V$ with respect to which all elements of $\mathfrak{g}$ are upper triangular.

[edit] References

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).

This article incorporates material from Lie's theorem on PlanetMath, which is licensed under the GFDL.