CHAPTER VI

The Theorem of Ado-Iwasawa

In this chapter we shall prove that every finite-dimensional Lie algebra has a faithful finite-dimensional representation. We shall treat the two cases: characteristic 0 and characteristic p separately. The result in the first case is known as Ado’s theorem. For this we shall give a proof which is essentially a simplification of one due to Harish-Chandra. For the characteristic p case the result is due to Iwasawa. The proof we shall give is simpler than his and leads to several other results on representations in the characteristic p case.

1. Preliminary results

If R is a homomorphism of a Lie algebra into _L, where is a finite-dimensional algebra (associative with 1), then we know that R has a unique extension to a homomorphism R of into . If is the kernel, is finite-dimensional. In general, if is a subspace of a vector space , then the dimension of / will be called the co-dimension of in . Thus R determines an ideal in of finite co-dimension. The homomorphism R is an isomorphism of if and only if . Conversely, let be an ideal in such that and has finite codimension in . Then the restriction to of the canonical homomorphism of into is an isomorphism of into the finitedimensional algebra _L. Since any finite-dimensional algebra has a faithful finite-dimensional representation it is clear that a Lie algebra will have a faithful finite-dimensional representation if and only if the universal enveloping algebra of contains an ideal of finite co-dimension satisfying .

We recall that an element a of an algebra is called algebraic if there exists a non-zero polynomial φ(λ) such that φ(a) = 0. This is equivalent to the assumption that the subalgebra generated by a is finite-dimensional. Consequently, every element of a finite-dimensional algebra is algebraic. If is an ideal in we shall say that a ∈ is algebraic modulo if there exists a non-zero polynomial φ(λ) such that φ(a) ∈ . This is equivalent to saying that the coset is algebraic in . It follows that if is of finite co-dimension, then every element of is algebraic modulo . We now state the following criterion for the universal enveloping algebra of a Lie algebra.

LEMMA 1. Let be a finite-dimensional Lie algebra, (u₁, u₂, …, u_n) a basis for , the universal enveloping algebra of , an ideal in . Then is of finite co-dimension in if and only if every u_i is algebraic modulo .

Proof: The necessity of the condition has been established above. Now let φ_i(λ) be a non-zero polynomial such that φ_i(u_i) ∈ and let n_i = deg φ_i(λ). Then every u_i^k is congruent modulo to a linear combination of the elements 1, u_i, u_i², …, u_i^n_i-1. The set of standard monomials , is a basis for and the remark just made implies that these monomials are congruent modulo to a linear combination of the monomials with 0 λ_i < n_i. Since this is a finite set, / is finite-dimensional.

LEMMA 2. Same assumptions as Lemma 1, and ideals in . If and are of finite co-dimension, then is of finite co-dimension in .

Proof: Let be non-zero polynomials such that , . Then has the property that and the result follows from Lemma 1.

LEEMMA 3. Let be an algebra, B a set of generators for and D a derivation in . Suppose that for every u ∈ B there exists a positive integer n(u) such that uD^n(u) = 0. Then for every a ∈ there exists a positive integer n(a) such that aD^n(a) = 0. If is finite-dimensional then D is nilpotent.

Proof: Let be the subset of elements b such that bD^n(b) = 0 for some integer n(b) 0. If b₁, b₂ ∈ and b₁D^n₁ = 0 = b₂D^n₂ then (b₁ + b₂)Dⁿ = 0 for n = max (n₁, n₂). (αb₁)D^n₁ = 0 α ∈ Φ and

if N = n₁ + n₂ – 1. Hence is a subalgebra and since = . This proves the first statement. The second is an immediate consequence.

2. The characteristic zero case

The key lemma for our proof of Ado’s theorem is the following

LEMMA 4. Let be a finite-dimensional solvable Lie algebra over a field of characteristic zero, the nil radical of , the universal enveloping algebra of . Suppose is an ideal of of finite codimension such that every element of is nilpotent modulo . Then there exists an ideal in such that: is of finite co-dimension, for every derivation D of (extended to ), (4) every element of is nilpotent modulo .

Proof. Let be the ideal in generated by and . Then and / is the ideal in / generated by . Since is an ideal in the Lie algebra and the elements of this ideal are nilpotent in the finite-dimensional enveloping associative algebra it follows from Theorem 2.2 that is contained in the radical of /. Hence / is in the radical. This implies that there exists an integer r such that and D is a derivation in then we know that (Theorem 3.7). It follows that . Hence , which implies that Hence (3) holds for and we have already noted that (1) holds. Since contains , is of finite co-dimension and is of finite co-dimension in . This proves (2). If for some positive integer n. Hence This proves (4).

THEOREM 1. Let where is a solvable ideal and ₁ is a subalgebra of the finite dimensional Lie algebra of characteristic 0. Suppose we have a finite dimensional representation S of such that z^s is nilpotent for every z in the nil radical of . Then there exists a finite-dimensional representation R of such that: (1) if x^R = 0 for x in then x^s = 0, (2) y^R is nilpotent for every y of the form y = z + u where and u ∈ ₁ is such that is nilpotent.

Proof: S defines a homomorphism of the universal enveloping algebra of whose kernel is of finite co-dimension. Also if then (z^s)ⁿ = 0 so zⁿ ∈ and satisfies the hypothesis of Lemma 4. Let be the ideal in the conclusion of this lemma. We shall define the required representation R in . We first define a representation R′ of acting in the space . If s ∈ we set s^R′ = s_R, the right multiplication in determined by s. If I ∈ ₁ we define l^R′ to be the derivation in which extends the derivation . The R′’s defined on and on ₁ define a unique linear transformation R′ on which is a representation of and ₁ separately. To prove that R′ is a representation for it suffices to show that . Now . On the other hand, if D is a derivation in and a ∈ , then the derivation condition gives [a_RD] = (aD)_R. Hence we have , as required. Since is an ideal in such that for any derivation D of , is a subspace of which is invariant relative to the representation R′ of acting in . Hence we have an induced representation R in the finite-dimensional factor space /. Let x ∈ satisfy x^R = 0. This means that x^R maps into Hence and so x^s = 0. Let . Then, by Lemma 4, z is nilpotent modulo Hence z^R is nilpotent. Since is an ideal in (Theorem 3.7) it follows that z^R is in the radical of the algebra of linear transformations generated by ^R. Now let y = z + u where and is nilpotent. Since , in order to prove that y^R is nilpotent, it suffices to prove that u^R is nilpotent. By definition u^R′ is the derivation in which coincides with on and nilpotent. Since generates it follows from Lemma 3 that for every a ∈ there is an integer n(a) such that a(u^R′)^{n (a)} = 0. Hence for every we have n() such that (u^R)^n(a) = 0. Since / is finite-dimensional this implies that u^R is nilpotent. Thus R satisfies the conditions (1) and (2).

We can now prove

Ado’s theorem. Every finite-dimensional Lie algebra of characteristic zero has a faithful finite-dimensional representation.

Proof: We recall that the kernel of the adjoint representation A is the center of . It will therefore suffice to prove the existence of a finite-dimensional representation R of which is faithful on the center . For then we can form the direct sum representation of R and A. The kernel of this is the intersection of the kernels of R and of A. Hence this representation is faithful as well as finite-dimensional. We proceed to construct R. Let be the radical, the nil radical. Let where each _i is an ideal in the next and dim _i+1 = dim _i + 1. Such a sequence exists since is solvable and contains . If dim = c then in a c + 1-dimensional vector space there exists a nilpotent linear transformation z such that z^c ≠ 0. Then is isomorphic to the Lie algebra with basis (z, z², …, z^c), so has a faithful representation by nilpotent linear transformations in a finitedimensional space. Since each _i is nilpotent and where Φu_i+1 is a subalgebra, the preceding theorem can be applied successively to obtain a finite-dimensional representation T of by nilpotent linear transformations such that T is faithful on . Next we obtain a sequence of subspaces, such that _{i + 1} is an ideal in _i and dim _{i + 1} = dim _i + 1. Then . Also is the nil radical of every _i (Theorem 3.7). Hence the theorem can be applied again beginning with T to obtain a representation S of which is finite-dimensional, faithful on and represents the elements of by nilpotent linear transformations. Finally we write a subalgebra (Levi’s theorem). Then we can apply Theorem 1 again to obtain the required representation R of .

Remark: The R constructed has the property that z^R is nilpotent for every z ∈ . The same holds for the adjoint representation. Hence the direct sum has the property too. We therefore have a faithful finite-dimensional representation such that the transformations corresponding to the elements of are nilpotent—and hence are in the radical of the enveloping associative algebra.

3 The characteristic p case

We recall that if Φ is of characteristic p then a polynomial of the form is called a p-polynomial. If μ(λ) is a polynomial of degree m, then we can write

where the r_i(λ) are of degree < m. Since the space of polynomials of degree < m is m-dimensional there exist α_i, i = 0, …, m, not all 0 such that . Then (1) implies that We have therefore proved that every polynomial is a factor of a suitable non-zero p-polynomial.

Now let be a finite-dimensional Lie algebra over Φ, the universal enveloping algebra. Let a ∈ and let μ(λ) be a nonzero polynomial such that μ(ad a) = 0. Such a polynomial exists since the algebra of transformations in is finite-dimensional. Let be a p-polynomial divisible by μ(λ).

Then we have

In other words, for every b ∈ we have

On the other hand, we know that . Iteratio of this gives

Hence (2′) implies

b ∈ which implies that the element

is in the center of . We have therefore proved the following

LEMMA 5. Let be a finite-dimensional Lie algebra over a field of characteristic p ≠ 0 and let be the universal enveloping algebra. Then for every a ∈ there exists a polynomial m_a(λ) such that m_a(a) is in the center of .

The result just proved and Lemma 5.4 are the main steps in our proof of

Iwasawa’s theorem. Every finite-dimensional Lie algebra of characteristic p ≠ 0 has a faithful finite-dimensional representation.

Proof : Let (u₁, u₂, …, u_n) be a basis for and let m_i(λ) be a p-polynomial such that , the center of the universal enveloping algebra. If deg m_i(λ) = p^m_i then z_i = u_i^pm_i + v_i where . Hence, by Lemma 5.4, the elements , form a basis for . Let be the ideal in generated by the Z_i. As in the proof of Theorem 5.11, the cosets of the elements form a basis for /. Hence this algebra is finite-dimensional and the canonical mapping a → = a + , a ∈ , is an isomorphism of into . It follows that there exists a faithful finite-dimensional representation of .

We shall show next that in the characteristic p ≠ 0 case there is no connection between structure of Lie algebras and complete reducibility of modules. In the following theorem we shall need a result proved in Chapter II (Theorem 2.10) that an algebra of linear transformations in a finite-dimensional vector space which has a non-zero radical cannot be completely reducible. We shall need also a result which is somewhat more difficult to prove, namely, that if z is an element of a finite-dimensional algebra and z does not belong to the radical of the algebra, then there exists an irreducible representation R of the algebra such that z^R ≠ 0 (See, for example, Jacobson [3], Theorem 3.1 and Definition 1.1.)

THEOREM 2. Every finite-dimensional Lie algebra over a field of characteristic p ≠ 0 has a 1:1 finite-dimensional representation which is not completely reducible and a 1:1 finite-dimensional completely reducible representation.

Proof: Let the u_i and z_i = m_i(u_i) be as in the proof of Iwasawa’s theorem. Let ₁ be the ideal in generated by (z₁², z₂, …, z_n). Then the argument shows that but . Hence Z₁ + ₁ is a non-zero center nilpotent element in the finitedimensional algebra /₁. The ideal generated by such an element is nilpotent. Hence /₁ is not semi-simple. Hence any 1:1 representation of this algebra is not completely reducible. Since generates /₁ this representation provides a representation for which is not completely reducible. The argument used before shows that the canonical mapping of into /₁ is an isomorphism. Hence the representation we have indicated is 1: 1 for and this proves our first assertion. Next let a be any nonzero element of and take u₁ = a in the basis (u₁, u₂, …, u_n) for . Let α ≠ 0 be in Φ. Then m₁(λ) − α is not divisible by λ and this is the minimum polynomial of a + ₂ in /₂ where ₂ is the ideal generated by m₁(u₁) − α, m₂(u₂), … m_n(u_n). Thus a + ₂ is not nilpotent and so it does not belong to the radical. It follows that there exists a finite-dimensional irreducible representation of /₂ such that a + ₂ is not represented by 0. This gives a finitedimensional irreducible representation R_a of such that a^R_a ≠ 0. Let _a denote the kernel of R_a (in ). Then . Since is finite-dimensional we can find a finite number a₁, a₂, …, a_m of the a’s in such that . We now form the module which is a direct sum of the m irreducible modules _j corresponding to the representations R_aj. Then evidently is completely reducible and the kernel of the associated representation is . Hence this gives a faithful finite-dimensional completely reducible representation for .

Exercises

1. Show that any finite-dimensional Lie algebra of characteristic p has indecomposable modules of arbitrarily high finite-dimensionalities.

Exercises 2-4 are designed to prove the following theorem: Let be an algebra over an algebraically closed field of characteristic 0, a finitedimensional simple subalgebra of _L which contains a non-zero algebraic element. Then the subalgebra of generated by is finite dimensional. We may as well assume that this subalgebra is itself and it suffices to show that has a basis consisting of algebraic elements.

2. Show that contains a non-zero nilpotent element e. (Hint: use Exercise 3.11.)

3. Show that contains a non-zero algebraic element h which is contained in some Cartan subalgebra of . (Hint: use Theorem 3.17, and Exercise 3.13.)

4. If e_α have the usual significance relative to show that there exists a root α ≠ 0 such that h_α, e_α, e – α are algebraic. Then show that this holds for every root α and hence that has a basis of algebraic elements. use this to prove the theorem stated.

5. Extend the theorem stated above to semi-simple under the stronger hypothesis that contains a set of algebraic elements such that the ideal in generated by this set is all of .

6. Extend the result in 5 to the case in which the base field is any field of characteristic 0.

7. (Harish-Chandra). Let be a finite-dimensional Lie algebra over a field of characteristic 0 and let R be a faithful finite-dimensional representation of by linear transformations of trace 0 in . Let R_i, i = 1, 2, … denote the representation in , i times and let _i denote the kernel in of R_i. Prove that .

8. Show that every finite-dimensional Lie algebra has a faithful finitedimensional representation by linear transformations of trace 0.