In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 ...
of
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 wh ...
s;
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 algebra In algebra, a linear Lie algebra is a subalgebra \mathfrak of the Lie algebra \mathfrak(V) consisting of endomorphisms of a vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often ca ...
s.
It states that if ''G'' is a
connected
Connected may refer to:
Film and television
* ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular''
* '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film
* ''Connected'' (2015 TV ...
and
solvable 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 wh ...
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, because ...
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
:
a
representation 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 can ...
''V'', then there is a one-dimensional linear subspace ''L'' of ''V'' such that
:
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
.
It follows directly that every
irreducible
In philosophy, systems theory, science, and art, emergence occurs when an entity is observed to have properties its parts do not have on their own, properties or behaviors that emerge only when the parts interact in a wider whole.
Emergence ...
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
In mathematics, the Borel fixed-point theorem is a fixed-point theorem in algebraic geometry generalizing the Lie–Kolchin theorem. The result was proved by .
Statement
If ''G'' is a connected, solvable, linear algebraic group acting regularly ...
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
has a ''triangular shape''; in other words, the image group
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 subgroup ...
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
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 subgroup ...
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 ...
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 wh ...
''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 Eucl ...
, 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 form ...
s
of absolute value one is a one-dimensional commutative (and therefore solvable)
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 wh ...
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. T ...
SO(2) without an invariant (real) line. Here the image
of
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 ma ...
:
References
*
*
*
*
{{DEFAULTSORT:Lie-Kolchin theorem
Lie algebras
Representation theory of algebraic groups
Theorems in representation theory