In mathematics, specifically the theory of
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
s, Lie's theorem states that,
over an algebraically closed field of characteristic zero, if
is a finite-dimensional
representation of a
solvable Lie algebra
In mathematics, a Lie algebra \mathfrak is solvable if its derived series terminates in the zero subalgebra. The ''derived Lie algebra'' of the Lie algebra \mathfrak is the subalgebra of \mathfrak, denoted
: mathfrak,\mathfrak/math>
that consist ...
, then there is a
flag
A flag is a piece of textile, fabric (most often rectangular) with distinctive colours and design. It is used as a symbol, a signalling device, or for decoration. The term ''flag'' is also used to refer to the graphic design employed, and fla ...
of
invariant subspaces
Invariant and invariance may refer to:
Computer science
* Invariant (computer science), an expression whose value doesn't change during program execution
** Loop invariant, a property of a program loop that is true before (and after) each iterati ...
of
with
, meaning that
for each
and ''i''.
Put in another way, the theorem says there is a basis for ''V'' such that all linear transformations in
are represented by upper triangular matrices. This is a generalization of the result of Frobenius that
commuting matrices
In linear algebra, two matrices A and B are said to commute if AB=BA, or equivalently if their commutator ,B AB-BA is zero. Matrices A that commute with matrix B are called the commutant of matrix B (and vice versa).
A set of matrices A_1, \ldot ...
are simultaneously
upper triangularizable, as commuting matrices generate 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 (see
#Consequences). Also, to each flag in a finite-dimensional vector space ''V'', there correspond a
Borel subalgebra (that consist of linear transformations stabilizing the flag); thus, the theorem says that
is contained in some Borel subalgebra of
.
Counter-example
For algebraically closed fields of characteristic ''p''>0 Lie's theorem holds provided the dimension of the representation is less than ''p'' (see the proof below), 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.
Proof
The proof is by induction on the dimension of
and consists of several steps. (Note: the structure of the proof is very similar to that for
Engel's theorem.) The basic case is trivial and we assume the dimension of
is positive. We also assume ''V'' is not zero. For simplicity, we write
.
Step 1: Observe that the theorem is equivalent to the statement:
*There exists a vector in ''V'' that is an eigenvector for each linear transformation in
.
Indeed, the theorem says in particular that a nonzero vector spanning
is a common eigenvector for all the linear transformations in
. Conversely, if ''v'' is a common eigenvector, take
to be its span and then
admits a common eigenvector in the quotient
; repeat the argument.
Step 2: Find an ideal
of codimension one in
.
Let