In
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
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, if and only if for each , the adjoint representation of a Lie algebra">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 G ...
:
given by
, is a nilpotent endomorphism on
; i.e.,
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 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, the ...
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
be the Lie algebra of the endomorphisms of a finite-dimensional vector space ''V'' and
a subalgebra. Then Engel's theorem states the following are equivalent:
# Each
is a nilpotent endomorphism on ''V''.
# There exists a flag
such that
; i.e., the elements of
are simultaneously strictly upper-triangulizable.
Note that no assumption on the underlying base field is required.
We note that Statement 2. for various
and ''V'' is equivalent to the statement
*For each nonzero finite-dimensional vector space ''V'' and a subalgebra
, there exists a nonzero vector ''v'' in ''V'' such that
for every
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
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