In mathematics, the Lie–Kolchin theorem is a theorem in the

representation theory 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 essen ...

linear algebraic group In mathematics, a linear algebraic group is a subgroup of the group of invertible n\times n matrices (under matrix multiplication) that is defined by polynomial equations. An example is the orthogonal group, defined by the relation M^TM = I_n ...

Lie's theorem In mathematics, specifically the theory of Lie algebras, Lie's theorem states that, over an algebraically closed field of characteristic zero, if \pi: \mathfrak \to \mathfrak(V) is a finite-dimensional representation of a solvable Lie algebra, then ...

is the analog for linear Lie algebras. It states that if ''G'' is a connected and solvable

defined over an

algebraically closed In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . Examples As an example, the field of real numbers is not algebraically closed, becaus ...

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

and :

\rho\colon G \to GL(V)

representation Representation may refer to: Law and politics *Representation (politics), political activities undertaken by elected representatives, as well as other theories ** Representative democracy, type of democracy in which elected officials represent a ...

on a nonzero finite-dimensional

vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...

''V'', then there is a one-dimensional linear subspace ''L'' of ''V'' such that :

\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. The result for Lie algebras was proved by and for algebraic groups was proved by . 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 In mathematics, a triangular matrix is a special kind of square matrix. A square matrix is called if all the entries ''above'' the main diagonal are zero. Similarly, a square matrix is called if all the entries ''below'' the main diagonal are ...

matrices, the standard

Borel subgroup In the theory of algebraic groups, a Borel subgroup of an algebraic group ''G'' is a maximal Zariski closed and connected solvable algebraic subgroup. For example, in the general linear group ''GLn'' (''n x n'' invertible matrices), the subgrou ...

of GL(''n'',''K''): the image is

simultaneously triangularizable In mathematics, a triangular matrix is a special kind of square matrix. A square matrix is called if all the entries ''above'' the main diagonal are zero. Similarly, a square matrix is called if all the entries ''below'' the main diagonal are ...

. The theorem applies in particular to a

of a

semisimple In mathematics, semi-simplicity is a widespread concept in disciplines such as linear algebra, abstract algebra, representation theory, category theory, and algebraic geometry. A semi-simple object is one that can be decomposed into a sum of ''sim ...

''G''.

Counter-example

If the field ''K'' is not algebraically closed, the theorem can fail. The standard

unit circle In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...

, viewed as the set of

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...

\

of absolute value one is a one-dimensional commutative (and therefore solvable)

over the real numbers which has a two-dimensional representation into the

special orthogonal group In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. ...

SO(2) without an invariant (real) line. Here the image

\rho(z)

z=x+iy

is the

orthogonal matrix In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors. One way to express this is Q^\mathrm Q = Q Q^\mathrm = I, where is the transpose of and is the identity ...

\begin x & y \\ -y & x \end.

References

* * * * {{DEFAULTSORT:Lie-Kolchin theorem Lie algebras Representation theory of algebraic groups Theorems in representation theory