Lie–Kolchin theorem

Last updated April 17, 2019

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.

Mathematics Field of study concerning quantity, patterns and change

Mathematics includes the study of such topics as quantity, structure, space, and change.

Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and the algebraic operations in terms of matrix addition and matrix multiplication. The algebraic objects amenable to such a description include groups, associative algebras and Lie algebras. The most prominent of these is the representation theory of groups, in which elements of a group are represented by invertible matrices in such a way that the group operation is matrix multiplication.

In mathematics, a linear algebraic group is a subgroup of the group of invertible n×n matrices that is defined by polynomial equations. An example is the orthogonal group, defined by the relation M^TM = 1 where M^T is the transpose of M.

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure, which is widely used in algebra, number theory and many other areas of mathematics.

\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

In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication. Representations of groups are important because they allow many group-theoretic problems to be reduced to problems in linear algebra, which is well understood. They are also important in physics because, for example, they describe how the symmetry group of a physical system affects the solutions of equations describing that system.

A vector space is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars. Scalars are often taken to be real numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers, or generally any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called axioms, listed below.

\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.

In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation $or irrep of an algebraic structure is a nonzero representation that has no proper subrepresentation closed under the action of .$

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 SophusLie ( 1876 ) and for algebraic groups was proved by EllisKolchin ( 1948 , p.19).

Marius Sophus Lie was a Norwegian mathematician. He largely created the theory of continuous symmetry and applied it to the study of geometry and differential equations.

Ellis Robert Kolchin was an American mathematician at Columbia University. Kolchin earned a doctorate in mathematics from Columbia University in 1941 under supervision of Joseph Ritt. He was awarded a Guggenheim Fellowship in 1954 and 1961.

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 $\{x+iy\in \mathbb {C} \mid x^{2}+y^{2}=1\}$ 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 $\rho (z)$ 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.

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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, 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
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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