Lie's theorem

In mathematics, specifically the theory of Lie algebras, Lie's theorem states that,^[1] over an algebraically closed field of characteristic zero, if $\pi :{\mathfrak {g}}\to {\mathfrak {gl}}(V)$ is a finite-dimensional representation of a solvable Lie algebra, then there's a flag $V=V_{0}\supset V_{1}\supset \cdots \supset V_{n}=0$ of invariant subspaces of $\pi ({\mathfrak {g}})$ with $\operatorname {codim} V_{i}=i$ , meaning that $\pi (X)(V_{i})\subseteq V_{i}$ for each $X\in {\mathfrak {g}}$ and i.

Put in another way, the theorem says there is a basis for V such that all linear transformations in $\pi ({\mathfrak {g}})$ are represented by upper triangular matrices.^[2] This is a generalization of the result of Frobenius that commuting matrices 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 $\pi ({\mathfrak {g}})$ is contained in some Borel subalgebra of ${\mathfrak {gl}}(V)$ .^[1]

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 ${\mathfrak {g}}$ 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 ${\mathfrak {g}}$ is positive. We also assume V is not zero. For simplicity, we write $X\cdot v=\pi (X)(v)$ .

Step 1: Observe that the theorem is equivalent to the statement:^[3]

There exists a vector in V that is an eigenvector for each linear transformation in $\pi ({\mathfrak {g}})$ .

Indeed, the theorem says in particular that a nonzero vector spanning $V_{n-1}$ is a common eigenvector for all the linear transformations in $\pi ({\mathfrak {g}})$ . Conversely, if v is a common eigenvector, take $V_{n-1}$ to be its span and then $\pi ({\mathfrak {g}})$ admits a common eigenvector in the quotient $V/V_{n-1}$ ; repeat the argument.

Step 2: Find an ideal ${\mathfrak {h}}$ of codimension one in ${\mathfrak {g}}$ .

Let $D{\mathfrak {g}}=[{\mathfrak {g}},{\mathfrak {g}}]$ be the derived algebra. Since ${\mathfrak {g}}$ is solvable and has positive dimension, $D{\mathfrak {g}}\neq {\mathfrak {g}}$ and so the quotient ${\mathfrak {g}}/D{\mathfrak {g}}$ is a nonzero abelian Lie algebra, which certainly contains an ideal of codimension one and by the ideal correspondence, it corresponds to an ideal of codimension one in ${\mathfrak {g}}$ .

Step 3: There exists some linear functional $\lambda$ in ${\mathfrak {h}}^{*}$ such that

V_{\lambda }=\{v\in V|X\cdot v=\lambda (X)v,X\in {\mathfrak {h}}\}

is nonzero. This follows from the inductive hypothesis (it is easy to check that the eigenvalues determine a linear functional).

Step 4: $V_{\lambda }$ is a ${\mathfrak {g}}$ -invariant subspace. (Note this step proves a general fact and does not involve solvability.)

Let $Y\in {\mathfrak {g}}$ , $v\in V_{\lambda }$ , then we need to prove $Y\cdot v\in V_{\lambda }$ . If $v=0$ then it's obvious, so assume $v\neq 0$ and set recursively $v_{0}=v,\,v_{i+1}=Y\cdot v_{i}$ . Let $U=\operatorname {span} \{v_{i}|i\geq 0\}$ and $\ell \in \mathbb {N} _{0}$ be the largest such that $v_{0},\ldots ,v_{\ell }$ are linearly independent. Then we'll prove that they generate U and thus $\alpha =(v_{0},\ldots ,v_{\ell })$ is a basis of U. Indeed, assume by contradiction that it's not the case and let $m\in \mathbb {N} _{0}$ be the smallest such that $v_{m}\notin \langle v_{0},\ldots ,v_{\ell }\rangle$ , then obviously $m\geq \ell +1$ . Since $v_{0},\ldots ,v_{\ell +1}$ are linearly dependent, $v_{\ell +1}$ is a linear combination of $v_{0},\ldots ,v_{\ell }$ . Applying the map $Y^{m-\ell -1}$ it follows that $v_{m}$ is a linear combination of $v_{m-\ell -1},\ldots ,v_{m-1}$ . Since by the minimality of m each of these vectors is a linear combination of $v_{0},\ldots ,v_{\ell }$ , so is $v_{m}$ , and we get the desired contradiction. We'll prove by induction that for every $n\in \mathbb {N} _{0}$ and $X\in {\mathfrak {h}}$ there exist elements $a_{0,n,X},\ldots ,a_{n,n,X}$ of the base field such that $a_{n,n,X}=\lambda (X)$ and

X\cdot v_{n}=\sum _{i=0}^{n}a_{i,n,X}v_{i}.

The $n=0$ case is straightforward since $X\cdot v_{0}=\lambda (X)v_{0}$ . Now assume that we have proved the claim for some $n\in \mathbb {N} _{0}$ and all elements of ${\mathfrak {h}}$ and let $X\in {\mathfrak {h}}$ . Since ${\mathfrak {h}}$ is an ideal, it's $[X,Y]\in {\mathfrak {h}}$ , and thus

X\cdot v_{n+1}=Y\cdot (X\cdot v_{n})+[X,Y]\cdot v_{n}=Y\cdot \sum _{i=0}^{n}a_{i,n,X}v_{i}+\sum _{i=0}^{n}a_{i,n,[X,Y]}v_{i}=a_{0,n,[X,Y]}v_{0}+\sum _{i=1}^{n}(a_{i-1,n,X}+a_{i,n,[X,Y]})v_{i}+\lambda (X)v_{n+1},

and the induction step follows. This implies that for every $X\in {\mathfrak {h}}$ the subspace U is an invariant subspace of X and the matrix of the restricted map $\pi (X)|_{U}$ in the basis $\alpha$ is upper triangular with diagonal elements equal to $\lambda (X)$ , hence $\operatorname {tr} (\pi (X)|_{U})=\dim(U)\lambda (X)$ . Applying this with $[X,Y]\in {\mathfrak {h}}$ instead of X gives $\operatorname {tr} (\pi ([X,Y])|_{U})=\dim(U)\lambda ([X,Y])$ . On the other hand, U is also obviously an invariant subspace of Y, and so

\operatorname {tr} (\pi ([X,Y])|_{U})=\operatorname {tr} ([\pi (X),\pi (Y)]|_{U}])=\operatorname {tr} ([\pi (X)|_{U},\pi (Y)|_{U}])=0

since commutators have zero trace, and thus $\dim(U)\lambda ([X,Y])=0$ . Since $\dim(U)>0$ is invertible (because of the assumption on the characteristic of the base field), $\lambda ([X,Y])=0$ and

X\cdot (Y\cdot v)=Y\cdot (X\cdot v)+[X,Y]\cdot v=Y\cdot (\lambda (X)v)+\lambda ([X,Y])v=\lambda (X)(Y\cdot v),

and so $Y\cdot v\in V_{\lambda }$ .

Step 5: Finish up the proof by finding a common eigenvector.

Write ${\mathfrak {g}}={\mathfrak {h}}+L$ where L is a one-dimensional vector subspace. Since the base field is algebraically closed, there exists an eigenvector in $V_{\lambda }$ for some (thus every) nonzero element of L. Since that vector is also eigenvector for each element of ${\mathfrak {h}}$ , the proof is complete. $\square$

Consequences

The theorem applies in particular to the adjoint representation $\operatorname {ad} :{\mathfrak {g}}\to {\mathfrak {gl}}({\mathfrak {g}})$ of a (finite-dimensional) solvable Lie algebra ${\mathfrak {g}}$ over an algebraically closed field of characteristic zero; thus, one can choose a basis on ${\mathfrak {g}}$ with respect to which $\operatorname {ad} ({\mathfrak {g}})$ consists of upper triangular matrices. It follows easily that for each $x,y\in {\mathfrak {g}}$ , $\operatorname {ad} ([x,y])=[\operatorname {ad} (x),\operatorname {ad} (y)]$ has diagonal consisting of zeros; i.e., $\operatorname {ad} ([x,y])$ is a strictly upper triangular matrix. This implies that $[{\mathfrak {g}},{\mathfrak {g}}]$ is a nilpotent Lie algebra. Moreover, if the base field is not algebraically closed then solvability and nilpotency of a Lie algebra is unaffected by extending the base field to its algebraic closure. Hence, one concludes the statement (the other implication is obvious):^[4]

A finite-dimensional Lie algebra ${\mathfrak {g}}$ over a field of characteristic zero is solvable if and only if the derived algebra $D{\mathfrak {g}}=[{\mathfrak {g}},{\mathfrak {g}}]$ is nilpotent.

Lie's theorem also establishes one direction in Cartan's criterion for solvability:

If V is a finite-dimensional vector space over a field of characteristic zero and ${\mathfrak {g}}\subseteq {\mathfrak {gl}}(V)$ a Lie subalgebra, then ${\mathfrak {g}}$ is solvable if and only if $\operatorname {tr} (XY)=0$ for every $X\in {\mathfrak {g}}$ and $Y\in [{\mathfrak {g}},{\mathfrak {g}}]$ .^[5]

Indeed, as above, after extending the base field, the implication $\Rightarrow$ is seen easily. (The converse is more difficult to prove.)

Lie's theorem (for various V) is equivalent to the statement:^[6]

For a solvable Lie algebra ${\mathfrak {g}}$ over an algebraically closed field of characteristic zero, each finite-dimensional simple ${\mathfrak {g}}$ -module (i.e., irreducible as a representation) has dimension one.

Indeed, Lie's theorem clearly implies this statement. Conversely, assume the statement is true. Given a finite-dimensional ${\mathfrak {g}}$ -module V, let $V_{1}$ be a maximal ${\mathfrak {g}}$ -submodule (which exists by finiteness of the dimension). Then, by maximality, $V/V_{1}$ is simple; thus, is one-dimensional. The induction now finishes the proof.

The statement says in particular that a finite-dimensional simple module over an abelian Lie algebra is one-dimensional; this fact remains true over any base field since in this case every vector subspace is a Lie subalgebra.^[7]

Here is another quite useful application:^[8]

Let ${\mathfrak {g}}$ be a finite-dimensional Lie algebra over an algebraically closed field of characteristic zero with radical $\operatorname {rad} ({\mathfrak {g}})$ . Then each finite-dimensional simple representation $\pi :{\mathfrak {g}}\to {\mathfrak {gl}}(V)$ is the tensor product of a simple representation of ${\mathfrak {g}}/\operatorname {rad} ({\mathfrak {g}})$ with a one-dimensional representation of ${\mathfrak {g}}$ (i.e., a linear functional vanishing on Lie brackets).

By Lie's theorem, we can find a linear functional $\lambda$ of $\operatorname {rad} ({\mathfrak {g}})$ so that there is the weight space $V_{\lambda }$ of $\operatorname {rad} ({\mathfrak {g}})$ . By Step 4 of the proof of Lie's theorem, $V_{\lambda }$ is also a ${\mathfrak {g}}$ -module; so $V=V_{\lambda }$ . In particular, for each $X\in \operatorname {rad} ({\mathfrak {g}})$ , $\operatorname {tr} (\pi (X))=\dim(V)\lambda (X)$ . Extend $\lambda$ to a linear functional on ${\mathfrak {g}}$ that vanishes on $[{\mathfrak {g}},{\mathfrak {g}}]$ ; $\lambda$ is then a one-dimensional representation of ${\mathfrak {g}}$ . Now, $(\pi ,V)\simeq (\pi ,V)\otimes (-\lambda )\otimes \lambda$ . Since $\pi$ coincides with $\lambda$ on $\operatorname {rad} ({\mathfrak {g}})$ , we have that $V\otimes (-\lambda )$ is trivial on $\operatorname {rad} ({\mathfrak {g}})$ and thus is the restriction of a (simple) representation of ${\mathfrak {g}}/\operatorname {rad} ({\mathfrak {g}})$ . $\square$

References

^ ^a ^b Serre 2001, Theorem 3
^ Humphreys 1972, Ch. II, § 4.1., Corollary A.
^ Serre 2001, Theorem 3″
^ Humphreys 1972, Ch. II, § 4.1., Corollary C.
^ Serre 2001, Theorem 4
^ Serre 2001, Theorem 3'
^ Jacobson 1979, Ch. II, § 6, Lemma 5.
^ Fulton & Harris 1991, Proposition 9.17.

Sources

Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. Vol. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. MR 1153249. OCLC 246650103.
Humphreys, James E. (1972), Introduction to Lie Algebras and Representation Theory, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90053-7.
Jacobson, Nathan (1979), Lie algebras (Republication of the 1962 original ed.), New York: Dover Publications, Inc., ISBN 0-486-63832-4, MR 0559927
Serre, Jean-Pierre (2001), Complex Semisimple Lie Algebras, Berlin: Springer, doi:10.1007/978-3-642-56884-8, ISBN 3-5406-7827-1, MR 1808366

[Serre_theorem-1] Serre 2001, Theorem 3

[2] Humphreys 1972, Ch. II, § 4.1., Corollary A.

[3] Serre 2001, Theorem 3″

[4] Humphreys 1972, Ch. II, § 4.1., Corollary C.

[5] Serre 2001, Theorem 4

[6] Serre 2001, Theorem 3'

[7] Jacobson 1979, Ch. II, § 6, Lemma 5.

[8] Fulton & Harris 1991, Proposition 9.17.

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