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 ...

, a branch of mathematics, Engel's theorem states that a finite-dimensional Lie algebra

\mathfrak g

is a

nilpotent Lie algebra In mathematics, a Lie algebra \mathfrak is nilpotent if its lower central series terminates in the zero subalgebra. The ''lower central series'' is the sequence of subalgebras : \mathfrak \geq mathfrak,\mathfrak\geq mathfrak,

adjoint map In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is GL(n ...

\operatorname(X)\colon \mathfrak \to \mathfrak,

given by

\operatorname(X)(Y) = [X, Y]

, is a nilpotent endomorphism on

\mathfrak

; i.e.,

\operatorname(X)^k = 0

for some ''k''. It is a consequence of the theorem, also called Engel's theorem, which says that if a Lie algebra of matrices consists of nilpotent matrices, then the matrices can all be simultaneously brought to a

strictly 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 ...

form. Note that if we merely have a Lie algebra of matrices which is nilpotent ''as a Lie algebra'', then this conclusion does ''not'' follow (i.e. the naïve replacement in

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 ...

of "solvable" with "nilpotent", and "upper triangular" with "strictly upper triangular", is false; this already fails for the one-dimensional Lie subalgebra of scalar matrices). The theorem is named after the mathematician Friedrich Engel, who sketched a proof of it in a letter to

Wilhelm Killing Wilhelm Karl Joseph Killing (10 May 1847 – 11 February 1923) was a German mathematician who made important contributions to the theories of Lie algebras, Lie groups, and non-Euclidean geometry. Life Killing studied at the University of Mü ...

dated 20 July 1890 . Engel's student K.A. Umlauf gave a complete proof in his 1891 dissertation, reprinted as .

Statements

Let

\mathfrak(V)

be the Lie algebra of the endomorphisms of a finite-dimensional vector space ''V'' and

\mathfrak g \subset \mathfrak(V)

a subalgebra. Then Engel's theorem states the following are equivalent: # Each

X \in \mathfrak

is a nilpotent endomorphism on ''V''. # There exists a flag

V = V_0 \supset V_1 \supset \cdots \supset V_n =
0, \, \operatorname V_i = i

such that

\mathfrak g \cdot V_i \subset V_

; i.e., the elements of

\mathfrak g

are simultaneously strictly upper-triangulizable. Note that no assumption on the underlying base field is required. We note that Statement 2. for various

\mathfrak g

and ''V'' is equivalent to the statement *For each nonzero finite-dimensional vector space ''V'' and a subalgebra

\mathfrak g \subset \mathfrak(V)

, there exists a nonzero vector ''v'' in ''V'' such that

X(v) = 0

for every

X \in \mathfrak g.

This is the form of the theorem proven in #Proof. (This statement is trivially equivalent to Statement 2 since it allows one to inductively construct a flag with the required property.) In general, a Lie algebra

\mathfrak g

is said to be

nilpotent In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0. The term was introduced by Benjamin Peirce in the context of his work on the cla ...

if the

lower central series In mathematics, especially in the fields of group theory and Lie theory, a central series is a kind of normal series of subgroups or Lie subalgebras, expressing the idea that the commutator is nearly trivial. For groups, the existence of a centra ...

of it vanishes in a finite step; i.e., for

C^0 \mathfrak g = \mathfrak g, C^i \mathfrak g = mathfrak g, C^ \mathfrak g /math> = (''i''+1)-th power of \mathfrak g, there is some ''k'' such that C^k \mathfrak g = 0 . Then Engel's theorem implies the following theorem (also called Engel's theorem): when \mathfrak g has finite dimension,
* \mathfrak g is nilpotent if and only if \operatorname(X) is nilpotent for each X \in \mathfrak g .
Indeed, if \operatorname(\mathfrak g) consists of nilpotent operators, then by 1. \Leftrightarrow 2. applied to the algebra \operatorname(\mathfrak g) \subset \mathfrak(\mathfrak g), there exists a flag \mathfrak g = \mathfrak_0 \supset \mathfrak_1 \supset \cdots \supset \mathfrak_n = 0 such that mathfrak g, \mathfrak g_i \subset \mathfrak g_. Since C^i \mathfrak g\subset \mathfrak g_i, this implies \mathfrak g is nilpotent. (The converse follows straightforwardly from the definition.)

Proof

We prove the following form of the theorem: ''if

\mathfrak \subset \mathfrak(V)

is a Lie subalgebra such that every

X \in \mathfrak

is a nilpotent endomorphism and if ''V'' has positive dimension, then there exists a nonzero vector ''v'' in ''V'' such that

X(v) = 0

for each ''X'' in

\mathfrak

.'' The proof is by induction on the dimension of

\mathfrak

and consists of a few steps. (Note the structure of the proof is very similar to that for

, which concerns a solvable algebra.) The basic case is trivial and we assume the dimension of

\mathfrak

is positive. Step 1: Find an ideal

\mathfrak

of codimension one in

\mathfrak

. :This is the most difficult step. Let

\mathfrak

be a maximal (proper) subalgebra of

\mathfrak

, which exists by finite-dimensionality. We claim it is an ideal of codimension one. For each

X \in \mathfrak h

, it is easy to check that (1)

\operatorname(X)

induces a linear endomorphism

\mathfrak/\mathfrak \to \mathfrak/\mathfrak

and (2) this induced map is nilpotent (in fact,

\operatorname(X)

is nilpotent as

X

is nilpotent; see Jordan–Chevalley decomposition#Lie algebras). Thus, by inductive hypothesis applied to the Lie subalgebra of

\mathfrak(\mathfrak/\mathfrak)

generated by

\operatorname(\mathfrak)

, there exists a nonzero vector ''v'' in

\mathfrak/\mathfrak

such that

\operatorname(X)(v) = 0

for each

X \in \mathfrak

. That is to say, if

v = /math> for some ''Y'' in \mathfrak but not in \mathfrak h, then, Y = \operatorname(X)(Y) \in \mathfrak for every X \in \mathfrak . But then the subspace \mathfrak' \subset \mathfrak spanned by \mathfrak and ''Y'' is a Lie subalgebra in which \mathfrak is an ideal of codimension one. Hence, by maximality, \mathfrak' = \mathfrak g . This proves the claim.

Step 2: Let W = \ . Then \mathfrak stabilizes ''W''; i.e., X (v) \in W for each X \in \mathfrak, v \in W .

:Indeed, for Y in \mathfrak and X in \mathfrak, we have: X(Y(v)) = Y(X(v)) +, Y v) = 0 since \mathfrak is an ideal and so, Y \in \mathfrak . Thus, Y(v) is in ''W''.

Step 3: Finish up the proof by finding a nonzero vector that gets killed by \mathfrak .

:Write \mathfrak = \mathfrak + L where ''L'' is a one-dimensional vector subspace. Let ''Y'' be a nonzero vector in ''L'' and ''v'' a nonzero vector in ''W''. Now, Y is a nilpotent endomorphism (by hypothesis) and so Y^k(v) \ne 0, Y^(v) = 0 for some ''k''. Then Y^k(v) is a required vector as the vector lies in ''W'' by Step 2. \square

Notes

Citations

Works cited

* * * * * * {{refend Representation theory of Lie algebras Theorems in representation theory

Statements

Proof

See also

Notes

Citations

Works cited