Engel’s theorem

Engel’s theorem

Before proceeding, it will be useful to recall the

definition

of a

nilpotent Lie algebra

. Let

𝔤

be a

Lie algebra

. The

lower central series

of

𝔤

is defined to be the

filtration

of ideals

𝒟

0

⁢

𝔤

⊃

𝒟

1

⁢

𝔤

⊃

𝒟

2

⁢

𝔤

⊃

…

,

where

𝒟

0

⁢

𝔤

=

𝔤

,

𝒟

k

+

1

⁢

𝔤

=

[

𝔤

,

𝒟

k

⁢

𝔤

]

,

k

∈

ℕ

.

To say that

𝔤

is

nilpotent

is to say that the lower
central series has a

trivial

termination, i.e. that there exists a

k

such that

𝒟

k

⁢

𝔤

=

0

,

or equivalently, that

k

nested bracket

operations

always vanish.

Theorem 1

(Engel)

Let

g

⊂

End

V

be a Lie algebra of

endomorphisms

of a

finite-dimensional

vector space

V

. Suppose that all elements of

g

are nilpotent transformations. Then,

g

is a nilpotent Lie
algebra.

Lemma 1

Let

X

:

V

→

V

be a nilpotent endomorphism of a vector
space

V

. Then, the

adjoint action

ad

(

X

)

:

End

V

→

End

V

is also a nilpotent endomorphism.

Proof.

Suppose that

X

k

=

0

for some

k

∈

ℕ

. We will show that

ad

(

X

)

2

⁢

k

-

1

=

0

.

Note that

ad

(

X

)

=

l

⁢

(

X

)

-

r

⁢

(

X

)

,

where

l

⁢

(

X

)

,

r

⁢

(

X

)

:

End

V

→

End

V

,

are the endomorphisms
corresponding, respectively, to left and right

multiplication

by

X

.
These two endomorphisms commute, and hence we can use the binomial
formula to write

ad

(

X

)

2

⁢

k

-

1

=

∑

i

=

0

2

⁢

k

-

1

(

-

1

)

i

⁢

l

⁢

(

X

)

2

⁢

k

-

1

-

i

⁢

r

⁢

(

X

)

i

.

Each of terms in the above sum vanishes because

l

⁢

(

X

)

k

=

r

⁢

(

X

)

k

=

0

.

QED

Lemma 2

Let

g

be as in the

theorem

, and suppose, in

addition

, that

g

is a nilpotent Lie algebra. Then the joint kernel,

ker

⁡

𝔤

=

⋂

a

∈

𝔤

ker

⁡

a

,

is non-trivial.

Proof.

We proceed by

induction

on the

dimension

of

𝔤

.
The claim is true for dimension 1, because then

𝔤

is generated by
a single nilpotent transformation, and all nilpotent transformations
are

singular

.

Suppose then that the claim is true for all Lie algebras of dimension
less than

n

=

dim

⁡

𝔤

. We note that

𝒟

1

⁢

𝔤

fits the hypotheses
of the lemma, and has dimension less than

n

, because

𝔤

is
nilpotent. Hence, by the

induction hypothesis

V

0

=

ker

⁡

𝒟

1

⁢

𝔤

is non-trivial. Now, if we restrict all
actions to

V

0

, we obtain a

representation

of

𝔤

by

abelian

transformations

. This is because for all

a

,

b

∈

𝔤

and

v

∈

V

0

we have

a

⁢

b

⁢

v

-

b

⁢

a

⁢

v

=

[

a

,

b

]

⁢

v

=

0

.

Now a finite number of mutually
commuting linear endomorphisms admits a mutual

eigenspace

decomposition

. In particular, if all of the commuting endomorphisms
are singular, their joint kernel will be non-trivial. We apply
this result to a basis of

𝔤

/

𝒟

1

⁢

𝔤

acting on

V

0

,
and the desired

conclusion

follows. QED

Proof of the theorem.

We proceed by induction on the dimension of

𝔤

. The theorem is
true in dimension 1, because in that circumstance

𝒟

1

⁢

𝔤

is
trivial.

Next, suppose that the theorem holds for all Lie algebras of dimension
less than

n

=

dim

⁡

𝔤

. Let

𝔥

⊂

𝔤

be a properly

contained

subalgebra

of minimum codimension. We claim that there exists an

a

∈

𝔤

but not in

𝔥

such that

[

a

,

𝔥

]

⊂

𝔥

.

By the induction hypothesis,

𝔥

is nilpotent. To prove the claim
consider the

isotropy representation

of

𝔥

on

𝔤

/

𝔥

. By
Lemma 1, the action of each

a

∈

𝔥

on

𝔤

/

𝔥

is a nilpotent
endomorphism. Hence, we can apply Lemma 2 to deduce that the joint
kernel of all these actions is non-trivial, i.e. there exists a

a

∈

𝔤

but not in

𝔥

such that

[

b

,

a

]

≡

0

mod

𝔥

,

for all

b

∈

𝔥

. Equivalently,

[

𝔥

,

a

]

⊂

𝔥

and the claim
is proved.

Evidently then, the span of

a

and

𝔥

is a subalgebra of

𝔤

.
Since

𝔥

has minimum codimension, we infer that

𝔥

and

a

span all of

𝔤

, and that

𝒟

1

⁢

𝔤

⊂

𝔥

.

(1)

Next, we claim that all the

𝒟

k

⁢

𝔥

are ideals of

𝔤

. It is
enough to show that

[

a

,

𝒟

k

⁢

𝔥

]

⊂

𝒟

k

⁢

𝔥

.

We argue by induction on

k

. Suppose the
claim is true for some

k

. Let

b

∈

𝔥

,

c

∈

𝒟

k

⁢

𝔥

be given. By
the

Jacobi identity

[

a

,

[

b

,

c

]

]

=

[

[

a

,

b

]

,

c

]

+

[

b

,

[

a

,

c

]

]

.

The first term on the right hand-side in

𝒟

k

+

1

⁢

𝔥

because

[

a

,

b

]

∈

𝔥

. The second term is in

𝒟

k

+

1

⁢

𝔥

by the induction
hypothesis. In this way the claim is established.

Now

a

is nilpotent, and hence by Lemma 1,

ad

(

a

)

n

=

0

(2)

for some

n

∈

ℕ

. We now claim that

𝒟

n

+

1

⁢

𝔤

⊂

𝒟

1

⁢

𝔥

.

By (

1

) it suffices to show that

[

𝔤

,

[

…

[

𝔤

⏞

n

⁢

times

,

𝔥

]

…

]

]

⊂

𝒟

1

𝔥

.

Putting

𝔤

1

=

𝔤

/

𝒟

1

⁢

𝔥

,

𝔥

1

=

𝔥

/

𝒟

1

⁢

𝔥

,

this is

equivalent

to

[

𝔤

1

,

[

…

[

𝔤

1

⏞

n

⁢

times

,

𝔥

1

]

…

]

]

=

0

.

However,

𝔥

1

is abelian, and hence, the above follows directly
from (

2

).

Adapting this

argument

in the

obvious

fashion we can show that

𝒟

k

⁢

n

+

1

⁢

𝔤

⊂

𝒟

k

⁢

𝔥

.

Since

𝔥

is nilpotent,

𝔤

must be nilpotent as well. QED

Historical remark.

In the traditional formulation of Engel’s theorem, the hypotheses are
the same, but the conclusion is that there exists a basis

B

of

V

,
such that all elements of

𝔤

are represented by

nilpotent matrices

relative to

B

.

Let us put this another way. The vector space of nilpotent matrices

Nil

, is a nilpotent Lie algebra, and indeed all subalgebras of

Nil

are
nilpotent Lie algebras. Engel’s theorem asserts that the

converse

holds, i.e. if all elements of a Lie algebra

𝔤

are nilpotent
transformations, then

𝔤

is

isomorphic

to a subalgebra of

Nil

.

The classical result follows straightforwardly from our version of the
Theorem and from Lemma 2. Indeed, let

V

1

be the joint kernel

𝔤

. We then let

U

2

be the joint kernel of

𝔤

acting on

V

/

V

0

, and let

V

2

⊂

V

be the

subspace

obtained by pulling

U

2

⁢

x

back to

V

. We do this a finite number of times and obtain a
flag of subspaces

0

=

V

0

⊂

V

1

⊂

V

2

⊂

…

⊂

V

n

=

V

,

such that

𝔤

⁢

V

k

+

1

=

V

k

for all

k

. The choose an

adapted basis

relative to this flag, and we’re done.

Title

Engel’s theorem

Canonical

name

EngelsTheorem

Date of creation

2013-03-22 12:42:25

Last modified on

2013-03-22 12:42:25

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

5

Author

rmilson (146)

Entry type

Theorem

Classification

msc 17B30

Classification

msc 15A57

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