Lie–Kolchin theorem

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

:$ho:\; G\; ightarrow\; GL(V)$

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

: $ho(G)(L)\; =\; L.$

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 $ho(g),\; ,,\; g\; in\; G$. 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 $ho(G)$ has a "triangular shape" or in other words, the image group $ho(G)$ 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 $ho(z)$ of $z=x+iy$ is the orthogonal matrix

:

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.

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).

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