CHAPTER VII

Classification of Irreducible Modules

The principal objective of this chapter is the classification of the finite-dimensional irreducible modules for a finite-dimensional split semi-simple Lie algebra over a field of characteristic 0. The main result—due to Cartan—gives a 1:1 correspondence between the modules of the type specified and the “dominant integral” linear functions on a splitting Cartan subalgebra of . The existence of a finite-dimensional irreducible module corresponding to any dominant integral function was established by Cartan by separate case investigations of the simple Lie algebras and so it depended on the classification of these algebras. A more elegant method for handling this question was devised by Chevalley and by Harish-Chandra (independently). This does not require case considerations. Moreover, it yields a uniform proof of the existence of a split semi-simple Lie algebra corresponding to every Cartan matrix or Dynkin diagram and another proof of the uniqueness (in the sense of isomorphism) of this algebra.

Harish-Chandra’s proof of these results is quite complicated.* The version we shall give is a comparatively simple one which is based on an explicit definition of a certain infinite-dimensional Lie algebra defined by an integral matrix (A_ij) satisfying certain conditions which are satisfied by the Cartan matrices, and the study of certain cyclic modules, “e-extreme modules” for . The principal tools which are needed in our discussion are the Poincaré-Birkhoff-Witt theorem and the representation theory for split three-dimensional simple Lie algebras.

1. Definition of certain Lie algebras

Let (A_ij), i,j = 1,2, …,l, be a matrix of integers A_ij having the following properties (which are known to hold for the Cartan matrix of any finite-dimensional split semi-simple Lie algebra over a field of characteristic 0)

* Chevalley’s proof has not been published.

(α)  A_ij = 2,       A_ij 0 if i ≠ j,       A_ij = 0 implies A_ji = 0.

(β)  det (A_ij) ≠ 0.

(γ)  If (α₁, α₂, …, α_l) is a basis for an l-dimensional vector space over the rationals, then the group W generated by the l linear transformations Sα_i defined by

is a finite group.

Let Φ be an arbitrary field of characteristic 0. We shall define a Lie algebra over Φ which is determined by the matrix (A_ij). We begin with the free Lie algebra (§ 5.4) generated by the free generators e_i, f_i, h_i, i = 1,2, …, l, and let be the (Lie) ideal in generated by the elements

Let . Let be the subspace of spanned by the h_i and let α_i be the linear function on such that

The condition (β) implies that the l α_i form a basis for the conjugate space * of .

Since is freely generated by the e_i, f_i, h_i, i = 1,2, …, l, any mapping e_i → E_i, f_i → F_i, h_i → H_i of the generators into linear transformations of a vector space defines a (unique) representation of .In other words, if is any vector space with basis {u_j}, then can be made into an -module by defining the module products u_je_i, u_jf_i, u_jh_i in a completely arbitrary manner as elements of . We now let be the free (associative) algebra generated by l free generators x₁, x₂, …, x_l. Then has the basis 1, x_i1 … x_ir, i_j = 1,2,…,l, r = 1, 2, …. Let Λ ≡ Λ(h) be a linear function on . Then the foregoing remark implies that we can turn into an -module by defining

In these equations and in those which will appear subsequently we abbreviate Λ(h) by Λ, etc., but write in full Λ(h_i), etc. Let ' be the kernel of the representation of in . We proceed to show that the ideal defining . This will imply that can be considered as defining a representation, hence a module, for . The linear transformation in corresponding to h has a diagonal matrix relative to the chosen basis. Hence any two of these transformations commute and so [h_ih_j] ε '. We note next that the linear transformation corresponding to f_i is the right multiplication x_iR in determined by x_i. The last equation and fourth equation in (4) imply that

This implies that [e_if_j]—δ_ijh_i is in the kernal '. We have

Hence [f_ih] + α_if_i ε '. This implies that if x is an element of such that xh = M(h)x ≡ Mx, then (xf_i)h = (M-α_i)(xf_i), or (xx_i)h = (M-α_i)(xx_i). We now assert that (xi₁ … xi_re_i)h = (Λ-αi₁- … - αi_r + α_i)xi₁ … xi₁e_i. This is clear for r = 0 if we adopt the convention that the corresponding base element is 1. We now assume the result for r - 1. Then

This proves our assertion and it implies that

Hence . We have therefore proved that all the generators of are contained in '. Consequently and so can be regarded as a module for . We can now prove

THEOREM 1. Let denote the free Lie algebra generated by 3l elements e_i, f_i, h_i, i = 1,2, …, l, let be the ideal in generated by the elements (2), and let . Then: (i) The canonical homomorphism of onto maps isomorphic ally into , so we can identify the corresponding subspaces. (ii) The subspace is a subalgebra which is a split three-dimensional simple algebra. (iii) The subalgebra _- of generated by the f_i is the free Lie algebra generated by these elements and a similar statement holds for the subalgebra ₊ generated by the e_i. is an abelian subalgebra and we have the vector space decomposition

(iv) If is the universal enveloping algebra of , then where is the subalgebra generated by the subalgebra generated by ₊ and _- the subalgebra generated by _-.

Proof. For the moment let . Consider the representation R of determined in by the linear function Λ = 0 on . If which implies that (αi₁ + … + αi_r)(h) = 0 for all choices of the αi_j, Since α₁, α₂, …, α_l form a basis for the conjugate space this implies that h = 0. Hence is 1:1. We have , hence is a subalgebra of which is a homomorphic image of the split three-dimensional simple Lie algebra. Because of the simplicity of the latter the image is either 0 or is isomorphic to the three-dimensional split simple algebra. Since by our first result, we have an isomorphism. This proves (ii). We have for any and , . Since the linear functions 0, ± α_i are all different the usual weight argument implies that a relation of the form implies ξ_i = 0, η_i = 0 a for all i and . Then h′ = 0 by our first result. Thus we see that is an isomorphism on . We make the identification of this space with its image and from now on we write e_i, f_i, h_i etc. for etc. We write and we have seen that this is an l-dimensional abelian subalgebra of . We have noted before that in the representation R of acting in the right multiplication in determined by x_i. Let _- denote the universal enveloping algebra of _-, the subalgebra of generated by the f_i. Then we have a homomorphism of _- into the algebra of linear transformations in mapping f_i into x_iR. If we combine this with the inverse of the isomorphism a → a_R of (the regular representation) we obtain a homomorphism of _- into sending f_i into x_i. On the other hand, since is freely generated by the x_i we have a homomorphism of into mapping x_i into f_i. It follows that both our homomorphisms are surjective isomorphisms. Since the free Lie algebra is obtained by taking the Lie algebra generated by the generators of a free associative algebra it is now clear that _- is the free Lie algebra generated by the f_i and _- is the free associative algebra generated by the f_i. Also the Poincaré-Birkhoff-Witt theorem permits the identification of _- with the subalgebra of generated by _-, hence by the f_i. The basic property of free generators implies that we have an automorphism of sending e_i → f_i, f_i → e_i, h_i → — h_i. This maps the generators (2) of into and so it induces an automorphism in which maps f_i into e_i. Since the subalgebra generated by the f_i is free it follows that the subalgebra ₊ generated by the e_i is free and its universal enveloping algebra ₊ is the free associative algebra generated by the e_i. This algebra can be identified with the subalgebra of generated by the e_i. It remains to prove (5); for once this is done then the relation follows from the Poincaré-Birkhoff-Witt theorem by choosing an ordered basis for to consist of an ordered basis for followed by one for ₊ followed by one for _-. To prove (5) we show first that is a subalgebra of . The argument is similar to one we have used before (§ 4.3). We observe first that every element of ₊ is a linear combination of the elements [e_i1e_i2 … e_ir] ≡ [… [e_i1e_i2] … e_ir] and every element of _- is a linear combination of the elements [f_i1 … f_ir]. The Jacobi identity and induction on r implies that

Hence . We have and by induction on r 2 we can show that [[e_i1 … e_ir]f_j] ε ₊. It follows that Iteration of this and the Jacobi identity implies that . Similarly, These and imply that ₁ is a subalgebra. Since e_i, f_i, h_i ε ₁ it follows that , that is, . Equations (6) imply that is a direct sum of root spaces relative to and the non-zero roots are the functions ±(α_i1 + … + α_ir). It is clear that ₊ is the sum of the root spaces corresponding to the roots α_i1 + … + α_ir and _- is the sum of those corresponding to the roots – (α_i1 + … + α_ir). It follows that . This completes the proof.

2. On certain cyclic modules for

A Lie module is cyclic with generator x if is the smallest submodule of containing x. If is the universal enveloping algebra of the Lie algebra, then is the smallest submodule containing x. Hence is cyclic with x as generator if and only if . The module for which we constructed in § 1 is cyclic with 1 as generator since 1f_i1 … f_ir = x_i1 … x_i2 and these elements and 1 form a basis for .

We shall call a module for e-extreme if it is cyclic and the generator x can be chosen so that xh = Λ(h)x and xe_i = 0, i = 1,2, …, l. It is apparent from (4) that is e-extreme with 1 as generator of the required type. Thus we see that for every linear function Λ(h) on there exists an e-extreme module for which the generator x satisfies xh = Λ(h)x, xe_i = 0. We shall now consider the theory of e-extreme modules for . A similar theory can be developed for f-extreme modules which are defined to be cyclic with generator y such that yh = Λ(h)y, yf_i = 0, i = 1, 2, …, l. We shall stick to the e-extreme modules but shall make use of the corresponding results for f-extreme modules when needed.

Let be e-extreme with generator x satisfying xh = Λ(h)x, xe_i = 0. We know that the universal enveloping algebra of can be factored as where are the subalgebras generated by respectively. Then . Since and since . Hence . Moreover, ₋ is generated by the elements f_i. Hence every element of is a linear combination of the elements

where we now adopt the convention that f_i1 … f_ir = 1 if r = 0. We have [f_i1 … f_ir, h] = — (α_i1 + … + α_ir)f_i1 … f_ir (induction on r) and xh = Λx and these relations imply that (xf_i1 … f_ir)h = x[f_i1 … f_ir, h] + (xh)f_i1 … f_ir = (Λ – α_i1 – … – α_ir)xf_i1 … f_ir. Hence

Thus is a direct sum of weight spaces relative to and the weights are of the form

where the k_i are non-negative integers. Also it is clear that the restriction to a weight space of the linear transformation corresponding to any h is a scalar multiplication by a field element. It is clear also that the weight space _Λ corresponding to Λ has x as basis and so is one-dimensional. The weight space _M corresponding to the weight M = Λ — Σ k_iα_i is spanned by the vectors (7) such that α_i1 + … + α_ir = Σ k_iα_i. Clearly there are only a finite number of sequences (i₁, i₂, …, i_r) such that α_i1 + … + α_ir = Σ k_iα_i where the k_i are fixed non-negative integers. Hence _M is finite-dimensional.

The weight Λ can be characterized as the only weight of in such that every weight has the form Λ - Σ k_iα_i, k_i non-negative integers. We shall call Λ the highest weight of in or of . If is isomorphic to , then is also e-extreme and has the highest weight Λ. It follows that two e-extreme -modules having distinct highest weights cannot be isomorphic.

Let be a submodule of = Σ ⊕ _M where _M is the weight module corresponding to M. If y ε , y ε _M1 + … + _Mk where {M₁, … M_k} is a finite subset of the set of weights. Hence y ε (_M1 + … + _Mk) and this is a submodule of the finite dimensional -module . Such an -module is split and is a direct sum of weight modules whose weights are in the set {M_j} (cf. § 2.4). This means that . Since y is any element of we have also that = Σ ⊕ _M where _M = _M. If _M ≠ 0, M is a weight for in . At any rate, it is clear that is a direct sum of weight modules and the weights of in are among the weights of in .

Now let ' be the subspace of spanned by the ^M with M ≠ Λ and assume that the submodule ≠ . In this case, we must have _Λ = 0 since, otherwise, _Λ = _Λ which is one-dimensional. Then contrary to assumption. Thus we see that any proper submodule . It folfows from this that the sum of all the proper submodules of is contained in ' ⊂ and so this is a proper submodule. This proves the existence of a maximal (proper) submodule of . Moreover, is unique.

We consider again the module constructed in § 1 which we shall now show is a “universal” e-extreme module with highest weight Λ in the sense that every module of this type is a homomorphic image of . For this purpose we define θ to be the linear mapping of onto such that (x_i1 … x_ir) θ = xf_i1 … f_ir (x_i1 … x_ir = 1 if r = 0). Then

If we use induction on r we can use this to establish the formula

Since the elements e_i, f_i, h_i , equations (10), (11) and (12) imply that θ is a module homomorphism of onto .

The result just established shows that any e-extreme -module with highest weight Λ is isomorphic to a module of the form /, a submodule of the module . If is irreducible we have where is a maximal submodule of . We have seen that there is only one such submodule. Hence it is clear that any two irreducible e-extreme modules with the same highest weight are isomorphic. The existence of an irreducible e-extreme module with highest weight Λ is clear also, for, the module satisfies these requirements.

We summarize our main results in the following

THEOREM 2. Let the notations be as in Theorem1 and let Λ(h) be a linear function on . Then there exists an irreducible e-extreme -module with highest weight Λ. The weights for such a module are of the form Λ — Σ k_iα_i,k_i a non-negative integer. The weight space corresponding to Λ is one-dimensional and all the weight spaces are finite-dimensional. Every h ε acts as a scalar multiplication in every weight space. Two irreducible e-extreme -modules are isomorphic if and only if they have the same highest weight.

3. Finite-dimensional irreducible modules

We shall call a linear function Λ on integral if Λ(h_i) is an integer for every i = 1, 2, …, l, and we shall call an integral linear function dominant if Λ(h_i) 0 for all i. In this section we establish a 1:1 correspondence between these linear functions and the isomorphism classes of finite-dimensional irreducible modules for . In view of the correspondence between the isomorphism classes of irreducible e-extreme -modules and the highest weights which we established in Theorem 2 it suffices to prove two things: (1) Every finite-dimensional irreducible module is e-extreme with highest weight dominant integral, (2) Any irreducible e-extreme module with highest weight a dominant integral linear function is finite-dimensional. We prove first

THEOREM 3. Let be as before and let be a finite-dimensional irreducible -module. Then is e-extreme and its highest weight is a dominant integral linear function on .

Proof: is a finite-dimensional module for the subalgebra _i = Φe_i + Φf_i + Φh_i which is a split three-dimensional simple Lie algebra. Hence is completely reducible as _i-module. The form of the irreducible modules for _i (§ 3.8) shows that there exists a basis for such that where the m_ik are integers. Since [h_ih_j] = 0 the linear transformations associated with the different h_i commute; hence, we can find a basis (x_l, x₂, …, x_N) such that x_kh_i = m_ikx_k, i = 1, …, l, k = 1, …, N. If Λ_k denotes the linear function on such that Λ_k(h_i) = m_ik, then Λ_k is integral and

Then Λ_k are the weights of in . Let be the rational vector space spanned by the linear functions α_i. It is easy to see that a linear function if and only if the values α(h_i) are rational for i = 1, 2, …, l (cf. the proof of XIII in § 4.2). Hence the weights and so we may pick out among these weights the highest weight Λ in the ordering of which is specified by saying that if the first λ_i ≠ 0 is positive. In this case it is clear that Λ + α_i is not a weight for any α_i. Let x be a non-zero vector such that xh = Λx. Then (xe_i)h = (Λ + α_i)(xe_i) so xe_i = 0 by the maximality of Λ. Since is irreducible and is a submodule of it is clear that . Hence is e-extreme with Λ as its highest weight also in the sense of the last section. The results of § 2 show that every weight is of the form Λ — Σ k_iα_i, k_i a non-negative integer. On the other hand, the proof of the representation theorem, Theorem 4.1, (applied to + Φe_i + Φf_i) shows that if M is a weight for in , then M — M(h_i)α_i is also a weight. Hence for each i, Λ — Λ(h_i)α_i is a weight and so has the form Λ — Σ k_iα_i. It follows that Λ(h_i) = k_i 0. Thus we see that Λ is a dominant integral function. This completes the proof.

Next let Λ be any dominant integral linear function on and let be the irreducible module furnished by Theorem 2 with maximum weight Λ. The weights of in have the form Λ — Σ k_iα_i, k_i integral and non-negative. Hence these are integral and so they can be ordered by the ordering in . We shall prove that is finite-dimensional. The proof will be based on several lemmas, as follows.

LEMMA 1. Let θ_ij = f_j(adf_i)^-A_ij+1, i ≠ j = 1,2, …, l. Then [θ_ije_k] = 0, k = 1,2, …, l, and θ_ij = 0 for any e-extreme irreducible -module .

Proof: If k ≠ i, [f_ie_k] = 0; hence [ad f_i ade_k] = 0. Then

If k ≠ j this is 0. If k = j we obtain [θ_ije_k] = — A_jif_i(adf_i)^-A_ij. If A_ij = 0, A_ji = 0 by the hypothesis (α) of § 1, so the result is 0 in this case. Otherwise, —A_ij > 0 and f_i(adf_i)^-A_ij = 0. Next let k = i. Then [θ_ije_k] = f_j(adf_i)^-A_ij+1 ad e_i. We recall the commutation formula: , where x' = [xa], x'' = [x'a], …. (eq. 2.6). If we make use of this and the table: [e_if_i] = h_i, [[e_if_i]f_i] = 2f_i, [[[_ef_i]f_i]f_i] = 0, we obtain

The first term in both of these formulas can be dropped since f_j ad e_i = [f_je_i] = 0. If A_ij = 0 we have f_j ad h_i = [f_jh_i] = 0. If — A_ij > 0 we use the second formula and f_j ad h_i = [f_jh_i] = — A_ijf_j to obtain

This completes the proof of [θ_ije_k] = 0. Now let be an irreducible e-extreme -module and, as before, let ' be the subspace spanned by the weight spaces corresponding to the weights other than the highest weight Λ. Consider the submodule where are as in Theorem 1. If we use the fact that ad h is a derivation and that [f_ih] = — α_if_i we see that

which implies that θ_ij = θ_ij. Also θ_ije_k = e_kθ_ij and this implies that . Hence . If x is a canonical generator of such that xh = Λx, xe_i = 0, i = 1, 2, …, l, then every element of is a linear combination of the elements of the form xf_i1 … f_ir and every element of ' is a linear combination of these elements for which r 1. It follows from the definition of θ_ij that . Also it is clear that . Hence and so is a proper submodule of . Since is irreducible we must have θ_ij = 0 and the proof is complete.

LEMMA 2. Let be an irreducible e-extreme module for whose highest weight Λ is a dominant integral linear function. Then for any y ε there exist positive integers r_i, s_i such that ye^ri_i = 0 = yf⁸ⁱ_i, i = 1,2, …, l.

Proof: Let x be a generator of such that xh = Λx, xe_i = 0, i = 1, 2, …,l. Then every element of is a linear combination of elements of the form xf_i1f_i2 … f_ir. It suffices to prove the result for every y = xf_i1f_i2 … f_ir. We have yh = My, M = Λ — Σ k_iα_i, k_i an integer 0. Then (ye_i^k_i+l)h = (M + (k_i + l)α_i)ye_i^k_i+1. Since M + (k_i + 1)α_i is not a weight, ye_i^k_i+1 = 0 which proves the assertion for e_i. (This argument is valid for arbitrary e-extreme -modules.) Let Λ(h_i) = m_i 0. We show next that xf_i^m_i+l = 0. We have xh_i = m_ix, xe_i = 0. Hence if we apply the theory of e-extreme modules to the algebra _i = θe_i + θf_i + θh_i (in place of ) we see that the _i-submodule _i generated by x is the space spanned by x, xf_i, xf²_i, …. Suppose this has a proper submodule . Since (xf^k_i)h_i = (m_i — 2k)xf^k_i, the spaces θxf^k_i are the weight spaces relative to θh_i. Hence _i is spanned by certain of the subspaces θxf^k_i and k 1 since _i ⊂ _i. Moreover, if xf^k_i ε _i then xf^q_i ε _i for all q k. It follows that where k is the least positive integer such that xf^k_i ε _i. Evidently _i ⊆ '. We now observe that _i ⊆ _i since and _ie_i ⊆ _i since this is clear for k = i and it holds for k ≠ i since . It follows now that _i = _i and . Then . Thus _i is a proper -submodule of ≠ 0. This contradicts the irreducibility of and so proves that _i is _i-irreducible. Now in § 3.8 we constructed a finite-dimensional irreducible _i-module with a generator x′ such that x′h_i = m_ix′, x′e_i = 0, and x'f_i^m_i+l = 0. It follows from the isomorphism result on irreducible e-extreme modules (Theorem 2) that this module is isomorphic to _i. Hence we have xf_i^m_i+l = 0. Now suppose we have an integer m 0 such that (xf_i1 … f_ir-1)f_i^m. If i_r = i this implies that (xf_i1 … f_ri)f_i^m = 0. If i_r = j ≠ i we use the relation

where θ_ij is as in Lemma 1 and the congruence is used in the sense of the ideal in generated by θ_ij. It follows from Lemma 1 that

This proves the assertion on f_i by induction on r.

LEMMA 3. Let be as in Lemma 2. Then if M is a weight of in , M — M(h_i)α_i is a weight for i = 1,2, …,l.

Proof: Let y be a non-zero vector such that yh = M(h)y. If M(h_i) 0, then we choose q so that z = ye^q_i ≠ 0, ye^q+1_i = 0 and m so that zf^m_i ≠ 0, zf^m+1_i = 0. This can be done by Lemma 2. Then the determination of the finite-dimensional irreducible modules for (§ 3.8) shows that is such a module and zh_i = mz. On the other hand, yh = M(h)y implies zh = (ye^q_i)h = (M+qα_i)(h)z and . Hence (M + qα_i)(h_i) = M(h_i) + 2q = m and

are weights (corresponding to z, zf_i …, zf^m_i). We have M — M(h_i)α_i = M + (2q - m)α_i and q — m 2q — m q since M(h_i) = m — 2q 0 and q 0. Hence M — M(h_i)α_i is in the displayed sequence. If M(h_i) 0 we reverse the roles of e_i and f_i and argue in a similar fashion.

We can now prove

THEOREM 4. Let be an irreducible e-extreme module for such that the highest weight Λ is a dominant integral linear function. Then is finite-dimensional.

Proof: Let S_αi denote the linear mapping in the space of rational linear combinations of the linear functions α_i such that α_i(h_j) = A_ji. We have α_jS_αi = α_j — A_ijα_i so that S_αi is one of the linear transformations specified in our axiom (γ) of § 1. This axiom states that the group W generated by the S_αi is finite. On the other hand, Lemma 3 implies that the set of weights of in is invariant under W. We now consider the set Σ of images under W of the maximal weight Λ. This is a finite set, so it has a least element M in the lexicographic ordering we have defined in (at the beginning of this section).* Let y be a non-zero vector such that yh = My. Since M — M(h_i)α_i ε Σ, M M — M(h_i)α_i so that we have M(h_i) 0, i = 1, 2, …, l. One of the linear functions M — α_i, M + α_i is not a weight. Thus M = ΛS, S ε W and if M — α_i, M + α_i are weights then Λ – α_iS^-1 = (M — α_i)S^-1 and Λ + α_iS^-1 are weights. However, one of these is greater than Λ in the ordering in which contradicts the maximality of Λ. If M + α_i is not a weight we have ye_i = 0 as well as yh_i = M(h)y. As in the proof of Lemma 3, y generates an irreducible _i-module of dimension M(h_i) + 1. Hence M(h_i) 0. Since M(h_i) 0 we have M(h_i) = 0. If M — α_i is a weight we would have also that M — α_i — (M – α_i)(h_i)α_i = M – α_i + 2α_i = M + α_i is a weight contrary to assumption. Thus M — α_i is not a weight if M + α_i is not a weight. Hence in any case, M — α_i is not a weight. Consequently, yf_i = 0, i = 1,2, ···, l. Since is irreducible . Thus we see that is an f-extreme module. It follows that the weights of in have the form M + Σj_iα_i, j_i integer 0. If M = Λ— l_iα_i it is now clear that the weights have the form Λ — Σ k_iα_i where 0 k_i l_i. Thus there are only a finite number of different weights and since every weight space is finite-dimensional we see that is finite-dimensional.

4. Existence theorem and isomorphism theorem for semi-simple Lie algebras

Let F denote the collection of finite-dimensional irreducible representations of and let ₀ be the kernel of the collection F, that is, the set of elements b ε such that b^R = 0 for every R ε F. Then ₀ is an ideal in . If R ε F we can define a representation R for by setting . Since the set of representing transformations in the corresponding module is the same as that furnished by the representation of it is clear that R is a finite-dimensional irreducible representation for . We shall now show that is a split semi-simple Lie algebra and the given matrix (A_ij) is a Cartan matrix for . Thus we have the following

THEOREM 5. Let where is the free Lie algebra generated by e_i, f_i, h_i, i =1,2, …, l and is the ideal in generated by the elements (2). Let ₀ be the kernel of all thefinite-dimensional irreducible representations of and set . Then: (1) is a finite-dimensional split semi-simple Lie algebra. (2) The canonical mapping of into is an isomorphism so we can identify this subspace with its image. (3) is a splitting Cart an subalgebra, the α_i form a simple system of roots and the h_i, e_i, f_i, form a set of canonical generators whose associated Cartan matrix is the given matrix (A_ij). (4) Let be any finite-dimensional split semi-simple Lie algebra with l-dimensional splitting Cartan subalgebra and canonical generators whose associated Cartan matrix is(A_ij). Then there exists an isomorphism of onto sending .

* This is the only place in our discussion in which axion (γ) is used.

Proof. Let λ_i denote the linear function on such that λ_i(h_j) = δ_ij. Then λ_i is a dominant integral linear function and so it corresponds to a finite-dimensional irreducible representation R_i in a module _i for (and for ). The λ_i form a basis for * and every dominant integral linear function Λ has the form Λ = Σ m_iλ_i, m_i integral 0. We shall show first that ₀ can be characterized as the kernel of the finite set of representations R_i. Thus let be any finite-dimensional irreducible module and let Λ = Σ m_iλ_i be its highest weight. Let be the module which is the tensor product of m_l copies of ₁m₂ copies of ₂, …, m_l copies of _l. Let x_j be a generator of _j such that x_jh = λ_jx_j, h ε , x_je_i = 0, i = 1, 2, …, l, and set

In this definition and formula we adopt the convention that if all the m_i = 0, which is the case, if and only if Λ = 0, then is the one-dimensional module with basis x and xl = 0, l ε . Then, in any case, we have xh = Λx, xe_i = 0, so x is an e-extreme module with highest weight Λ. Consequently, the irreducible module corresponding to Λ is a homomorphic image of x. Suppose l ε and l^Ri = 0 for i = 1,2, …, l. Then _il = 0; hence l= 0 and xl = 0. This implies that l = 0 or l^R = 0. We have therefore proved our assertion that ₀ is the kernel of the finite set of representations R_i, or equally well, the kernel of the single representation S, which is the direct sum of the R_i. Evidently S is finite-dimensional completely reducible and a finite-dimensional completely reducible Lie algebra of linear transformations. Since x_ih = λ_i((h)x_i h ε , h^s = 0 implies that λ_i((h) = 0 for all i.

Since the λ_i form a basis for * this implies that h = 0. Hence is mapped faithfully by S and consequently by the canonical mapping of onto . It follows from this that the simple Lie algebras _i are mapped faithfully. Then the argument used to prove the corresponding assertion in Theorem 1 shows that (2) holds. Since is a direct sum of and the spaces _M corresponding to the roots M = ± (Σ k_i & α_i), k_i integral 0, the same holds for . It follows that is a splitting Cartan subalgebra for Consequently, the center of is contained in .

If h₀ ε we must have [e_ih₀] = α_i(h₀)e_i = 0. Then α_i(h₀)= 0 for all i and h₀ = 0. Thus = 0. Since is isomorphic to a finite-dimensional completely reducible Lie algebra of linear transformations and has 0 center it follows that is semi-simple. Hence we have established (1). We have [e_ih] = α_ie_i and every root of relative to has the form ± (Σk_iα_i), k_i integral and non-negative. Hence the form a simple system. We have [e_if_j] = δ_ijh_i, [e_ih_j] = A_jie_i, [f_ih_j] = — A_jif_i. This means that e_i ε _αi, f_i ε _–αi the generators e_i, f_i, h_i are canonical and the associated Cartan matrix is (A_ij) (cf. §4.3.) This completes the proof of (3). Now let be as in (4). Then it is clear that there exists a homomorphism of onto such that . Since is semi-simple it can be identified with a completely reducible Lie algebra of linear transformations in a finite-dimensional vector space and the homomorphism of onto can be considered as a representation. Since ₀ is mapped into 0 in any finite-dimensional irreducible representation, the homomorphism of maps ₀ into 0 and so we have an induced homomorphism of onto such that e_i→e_i, . Since is l-dimensional the homomorphism maps isomorphically. If ₁ is the kernel of the homomorphism of onto , since₁ is an ideal, it is invariant under ad . It follows that if ₁ ≠ 0 then it contains a non-zero element of or it contains one of the (one-dimensional) root spaces _α, α ≠ 0. The first is ruled out since ₁ = 0. If ₁ ⊇ _α, ₁ ⊇ [_α–α] ≠ 0. Since [_α–α] this is ruled out too. Hence₁= 0 and the homomorphism of is an isomorphism. This proves (4).

Theorems 3 and 4 establish a 1:1 correspondence between the isomorphism classes of finite-dimensional irreducible modules for the (infinite-dimensional) Lie algebra and the collection of dominant integral linear functions on . Also it is clear from the definition of that any finite-dimensional irreducible -module is an -module. The converse is also clear since is a homomorphic image of . Hence we see that Theorems 3 and 4 establish a 1:1 correspondence between the isomorphism classes of finite-dimensional irreducible modules for the finite-dimensional split semi-simple Lie algebra and the collection of dominant integral linear functions on a splitting Cartan subalgebra of .

We recall that a set Σ of linear transformations in a vector space over Φ is called absolutely irreducible if the corresponding set (the set of extensions) of linear transformations is irreducible in ₁, for any extension P of the base field. This condition implies irreducibility since one may take P = Φ. The term “absolutely irreducible” will be applied to modules and representations in the obvious way. It is clear from the definition that a module for a Lie algebra is absolutely irreducible if and only if _P is irreducible for _P for any extension P of Φ. Now let be a split semi-simple Lie algebra over Φ as in Theorem 5 and let be a finite-dimensional irreducible -module. We assert that is absolutely irreducible. Thus we know that is an e-extreme module with generator x such that xe_i = 0, i = 1,2, …, l, and xh = Λ(h)x, h ε where Λ is a dominant integral linear function on . We know also that the weight space _Λ corresponding to Λ coincides with Φx. Consider _P, as module for _P. Since this is finite-dimensional and _P is semi-simple, _P is completely reducible. Every irreducible submodule of _P decomposes into weight modules relative to _P. Hence every weight module for _P in _P can be decomposed into submodules contained in the irreducible components of a decomposition of _P into irreducible _P-modules. In particular, this holds for (_Λ)_P = Px. Since this is one dimensional it follows that x is contained in an irreducible submodule of _P. Since x generates _P we have _P and is irreducible. This proves our assertion on the absolute irreducibility of .

5. Existence of E₇ and E₈

In § 4.6 we established the existence of the split simple Lie algebras of the types corresponding to every Dynkin diagram except E₇ and E₈. (Our method—an explicit construction for each type— admittedly was somewhat sketchy for the exceptional types G₂, F₄ and E₆.) An alternative procedure based on Theorem 5 is now available to us. This requires the verification that the Cartan matrices (A_ij) obtained from the Dynkin diagrams satisfy conditions (α), (β), (γ) of § 1. We shall now carry this out for the diagrams E₇ and E₈ and thereby obtain the existence of these Lie algebras. We remark first that in any such verification (α) is immediate once the matrix is written down and (β) is generally easy. To prove (γ) one displays a finite set of vectors in which span and is invariant under the Weyl reflections S_αi. This will prove the finiteness of the group W generated by the S_αi.

E₈. The Cartan matrix is

One can see that det (A_ij) = 1, so (β) is clear. Also (α) is apparent. We introduce the vectors

where (α₁, α₂, …, α₈) is a basis for an 8-dimensional vector space over the rationals. It is immediate that the (λ_i) form a basis. We write S_i for the Weyl reflection S_αi defined by (1). We can verify that S_i, 1 i 7, permutes λ_i and λ_i+1 and leaves the other λ_j fixed while

Let Σ be the following set of vectors:

where the subscripts are all different and are in the set (1,2, …, 8). It is easy to see that these vectors generate the space. Since the S_i, i 7, are permutation transformations of the λ’s it is clear that these S_i map Σ into itself. One checks also directly that S₈ leaves Σ invariant. This implies that the group W generated by the S_i is finite.

E₇. Let, α₁, α₂, …, α₈ as be a simple system of roots of type E₈ in the split Lie algebra E₈. The matrix (A_jk) = (2(α_j, α_k)), j, k = 2, …, 8 is the Cartan matrix E₇. This satisfies (α) and (β). Moreover, (γ) holds since the group generated by S_α2, …, S_α8 is finite; hence the group generated by the restrictions of these mappings to the subspace of ₀* spanned by α₂, …, α₈ is finite. This proves the existence of E₇.

A similar method applies to E₆. We remark also that it is easy to see that if e_i, f_i, h_i, i =1, …, 8, is a set of canonical generators for E₈, then the subalgebra generated by e_j, f_j, h_j, j = 2, ···, 8 is E₇. (Exercise 1, below.)

6. Basic irreducible modules

Let be a split finite-dimensional semi-simple Lie algebra over a field of characteristic 0, and, as in the proof of Theorem 5, let _i be the finite-dimensional irreducible module corresponding to the dominant integral linear function λ_i such that λ_i(h_j) = δ_ij, i, j = 1, 2, …, l. If x_i is defined as in the proof of Theorem 5 and x is as in (14), then the argument used in §4 to prove absolute irreducibility_ml shows that the submodule of generated by x is the irreducible module corresponding to Λ = Σ m_iλ_i. The problem of explicitly determining the irreducible modules of finite-dimensionality for is therefore reduced to that of identifying the modules _i which we shall call the basic irreducible modules for . We shall now carry this out for some of the simple Lie algebras. Others will be indicated in exercises.

A_l. As we have seen in §4.6, A_i is the Lie algebra of (l+1) х (l+1) matrices of trace 0 over Φ. If (e_ij) is the usual matrix basis for the matrix algebra Φ_i+1 and the Cartan subalgebra and simple system of roots are chosen as in § 4.6, then a set of canonical generators corresponding to these is

The Cartan matrix (A_ij) is given by (38) of Chapter IV and we have A_ii = 2, A_{i+1, i} = — 1 = A_i,i+1, A_ij = 0 otherwise. This can be checked by calculating [e_ih_j] = A_jie_i using (16). The simple root α_i is specified by: α_i(h_j) = A_ji. The Weyl reflection S_αi is .

We consider first a representation of A_l by the Lie algebra of linear transformations of trace 0 in an (l + 1)-dimensional vector space . We can choose the basis (u₁, u₂, …, u_l+1) so that u_ih_i = u_i, u_i+1h_i = —u_i+1, u_jh_i = 0, j ≠ i, i +1. Then we have u_ih = Λ_i(h)u_i where the weights Λ_i are given by the table:

It follows from this and the table for the α_i that

Thus S_αi interchanges Λi and Λ_i+1 and leaves fixed the remaining weights Λ_j. The group of transformations of the weights Λ_i generated by these mappings is the symmetric group on the l + 1 weights.

Now let r be an integer, 1 r l, and form the r-fold tensor product ⊗ ⊗ … ⊗ . Let _r denote the subspace of skew symmetric tensors or r-vectors in ⊗ … ⊗ . By definition, this is the space spanned by all vectors of the form

where the summation is taken over all permutations P = (i₁,i₂ … i_r) of (1, 2, …, r) and the sign is + or — according as P is even or odd. It is easy to see that if (u₁, u₂, …, u_l+1) is a basis for = _i then the vectors

form a basis for _r. If a ε = A_l then

which implies that _r is a submodule of ⊗ … ⊗ . We may suppose that u_jh = Λj(h)u_j. Then (20) implies

Hence the basis given in (19) consists of weight vectors. We have tr h = 0, which implies that Λ_l+1 = — (Λ₁ + … + Λ_l). On the other hand, (17) implies that Λ₁ + … + Λ_k = λ_k where λ_k(h_i) = δ_ik, if 1 k l. Hence Λ₁, …, Λ_l is a basis for . If j_r = l + 1, then Λ_j1 + … + Λ_jr = — (Λ_k1 + … + Λ_kl–(r–1) where k₁, …, k_l(r–1) is the complement of (j₁, …, j_r–1) in (1,2, …, l). It follows from this that if where j ₁ < … < j_r and . We now see that the weight space corresponding to the weight Λ_j1 + Λ_j2 + … + Λ_jr is spanned by [u_j1, u_j2, …, u_jr] and so is one-dimensional. Let be a non-zero irreducible submodule of _r. Then contains one of the weight spaces corresponding to, say, Λ′ = Λ′_j1 + … + Λ′_jr. On the other hand, if is any other weight, then there exists an element S of the group generated by the S_αi such that Λ' = ΛS. Then, by Theorem 4.1, Λ′ is a weight of in also and . Thus = _r and so _r is irreducible. We have seen that Λ₁ + … Λ_r = λ_r and this is a dominant integral linear function. It is clear from the definition of [y₁, y₂, …, y_r] that [y₁, y₂, …, y_r] = 0 if two of the y_i are equal. This can be used to check that [u₁, u₂, …, u_r]e_i = 0, l =1, 2, …, l. Since [u₁, …, u_r]h = (Λ₁ + Λ₂ + … + Λ_r) [u₁, …, u_r], , is e-extreme with maximal weight λ_r. We therefore have the following

THEOREM 6. Let A_l be the Lie algebra of (l + 1) x (l + 1) matrices over Φ of trace 0 and let be the Cartan subalgebra of diagonal matrices in A_l, h_i = e_ii — e_{i+1, i+1}, l = 1, 2, …, l. Let _r 1 rl, be the space of r-vectors. Then _r irreducible module for A_l and it is the irreducible module corresponding to the dominant integral linear function λ_r on such that λ_r(h_r) = 1, λ_r(h_j) = 0 if j ≠ r.

B_l, l 2. If we use the simple system of roots of §4.6 for B_l we obtain the canonical generators

For the Cartan matrix (A_ij) = (α_j(h_i)) we have the values A_ii = 2, A_j,_j+1 = –1 = A_j+1, j = 1, …, l – 2, A_{l–1, l} = –1, A_{l, l–1} = –2, all other A_ij = 0. In the representation of B_l by the Lie algebra of skew symmetric linear transformations in the n = 2l + 1 dimensional space we have a basis (u₁, u₂, u_n) such that u₁h = 0, u₂h = Λ₁(h)u₂, …, u_l+1h = Λ_l(h)u_l+1; u_l+2h = – Λ₁(h)u_l+2, u_2l+1 h = – Λ_l(h)u_2l+1, where the linear functions Λ_j(h) satisfy

We have Λ₁ + … + Λ_r = λ_r, r l — 1, Λ₁ + … + Λ_l = 2λ_l where λ_i(h_j) = δ_ij, i, j = 1, …, l. Hence the Λ_i form a basis for ₀*. The Weyl reflection S_αj j = 1, …, l — 1, interchanges Λ_j and Λ_j+1 and leaves the remaining Λ_k invariant. The reflection S_αl leaves fixed every Λ_j, j = 1, …, l — 1 and maps Λ_l into — Λ_l. Hence the group generated by these includes all permutations of the Λ_i and all the mappings which replace any subset of the Λ_i by their negatives leaving the remaining ones fixed. It is easy to see that the group generated by the S_αi is of order 2^ll! and we shall show in the next chapter that this is the complete Weyl group.

Let r be an integer, 1 r l — 1, and let _r be the space of r-vectors. It is easy to see that [u₂ …, u_r+1]h = λ_r(h)[u₂, …, u_r+1] and [u₂, …, u_r+1]e_i = 0, l = 1, 2, …, l. We assert that the cyclic module generated by [u₁, …, u_r+1] is irreducible. Hence this is the required irreducible module corresponding to the dominant integral linear function λ_r. In general, we note that if is a finite-dimensional e-extreme module for a finite-dimensional split semi-simple Lie algebra , then is irreducible. Thus is completely reducible and the generator x corresponding to the highest weight Λ spans the weight space of corresponding to Λ. Hence x is contained in an irreducible submodule ' of . Since x generates we see that = ' is irreducible.

The result we have just indicated on the determination of the irreducible module corresponding to λ_r, r = 1, …, l— 1, can be improved. As a matter of fact, this module is the complete _r module of r-vectors. To prove this one has to show that _r is irreducible. We shall not give a direct proof here but shall obtain this result later as a consequence of Weyl’s formula for the dimensionality of the irreducible module with highest weight Λ, which we shall derive in Chapter VIII. For the representation with Λ = λ_r we shall obtain the dimensionality: . Since this is the dimensionality of _r it will follow that _r is irreducible.

If we consider the space of l-vectors a similar argument will show that this is the irreducible module for B_l corresponding to the highest weight Λ₁ + … + Λ_l = 2λ_l. Hence this does not provide the missing module corresponding to the linear function λ_l. In order to find this one, one has to consider the so-called spin representation of B_l. To give full details of this would be too lengthy. Hence we shall be content to sketch the results without giving complete proofs.

We recall first the definition of the Clifford algebra C(, (x,y)) of a finite-dimensional vector space over a field Φ of characteristic not two relative to a symmetric bilinear form (x, y) on . One forms the tensor algebra = Φ1 ⊕ ⊕ ₂ … ⊕ _i ⊕ …, _i = ⊗ … ⊗ , i times, and one lets be the ideal in generated by all the elements of the form

Then C(, (x, y)) = /. (For details on Clifford algebra the reader should consult Chevalley [3] pp. 36-69, or Artin [1], pp. 186-193.) It is clear that contains all the elements

It is known that the canonical mapping of into C = C(, (x, y)) is an isomorphism, so that one can identify with its image in C. We do this and we write the associative product in C as ab. If (u₁, u₂, …, u_n) is a basis for , then it is known that dim C = 2ⁿ and the set of elements

is a basis for C. Since the elements (25) are in it follows that we have the fundamental “Jordan relation”

for any x, y ε .

The algebra C is not a graded algebra. However, it is a graded vector space in a natural fashion which we shall now indicate. Let x₁,x ₂, … x_r ε . Then we define an r-fold product [x₁, x₂, …, x_r] ε C inductively by the following formulas

where [ab] = ab —ba and a.b = (ab + ba) for a, b ε C.

LEMMA 4. The product [x₁ …, x_r] = 0 if any two of the x_i are equal. Hence for any permutation P = (i₁, i₂, …, i_r) of (1, 2, …, r), [x_i1, …, x_ir] = ±[x₁, …, x_r] where the sign is + or — according to whether P is even or odd.

Proof: Since [x₁, …, x_r] defines a multilinear function of its arguments the second statement is a consequence of the first. We prove the first by induction on r. We have [xx] = 0. Let r > 2 and suppose x_k = x = x_i where 1 k < l r. If l < r then [x₁, …, x_l] = 0 by the induction hypothesis and [x₁, …, x_r] = 0 follows from (28). Hence we may assume that l = r. Also the induction implies that [x₁, …, x_r-1] = ± [… x] so we have to show that [x₁, …, x_r-2xx] = 0. There are two cases: even r and odd r and these will follow by proving the following relations: [a.x, x] = 0 and [ax].x = 0 for a ε C, x ε . For the first of these we have and the second follows from

and the second follows from

This completes the proof.

LEMMA 5. Let denote the subspace of C spanned by all the elements [x₁ …, x_r] where 0 r n = dim and ₀= Φ1. Then dim _r = and C = Φ1 ⊕ ₁⊕ ₂ ⊕ … ⊕ _n.

Proof: If (u₁, u₂, …, u_n) is a basis for , then the skew symmetry and multilinearity of [x₁, …, x_r] imply that every element of _r, r 1, is a linear combination of the elements [u_i1, u_i2, …, u_ir] where i₁ < i₂ < … < i_r = 1, …, n. Now assume that the basis is orthogonal: (u_i, u_j) = 0 if i ≠ j. (It is well known that such bases exist.) The condition (u_i, u_j) = 0 and (27) implies that u_iu_j = —u_ju_i. Hence (u_i1 … u_ir-1)u_ir = ±u_ir(u_i1 … u_ir-1) if i₁ < i₂ < … < i_r and the sign is + or — according as r— 1 is even or odd. The relation just noted and the definition of [x₁ …, x_r] imply that [u_i1, …, u_ir] = 2^ku_i1 … u_ir for (u_i) an orthogonal basis. Since the 2ⁿ elements of the form u_i1 … u_ir, i1 < i2 < … < i_r, form a basis for C it is clear that for a fixed r the elements [u_i1, …, u_ir], i₁, < … < i_r, form a basis for _r. Hence dim _r = and it is clear that C = Φ1 ⊕ ₁⊕ ₂ ⊕ … ⊕ _n.

Let x, y, z ε . Then

On the other hand,

Hence we have the relation

We can now prove

THEOREM 7. The space ₂ is a subalgebra of the Lie algebra C_L. If (x, y) is non-degenerate, then this Lie algebra is isomorphic to the orthogonal Lie algebra determined by (x, y) in .

Proof: The space ₂ is the set of sums of Lie products [xy], x, y ε . By (29), we have

Hence ₂ is a subalgebra of C_L. This relation and (29) imply also that + ₂ is a subalgebra of C_L and the restriction of the adjoint representation of + ₂ to the subalgebra ₂ has as a submodule. If R denotes this representation, then (29) shows that [xz]^R is the mapping

If (x, y) is non-degenerate, then (31) is in the orthogonal Lie algebra. Moreover, every element of the latter Lie algebra is a sum of mappings (31). Hence the image under R of ₂ is the orthogonal Lie algebra. Since both of these algebras have the same dimensionality , R is an isomorphism.

The enveloping algebra of ₂ in C (, (x, y)) will be denoted as C⁺ = C⁺ (, (x, y)). If (u₁, …, u_n) is an orthogonal basis for , then the elements u_iu_j, i < j constitute a basis for ₂. Since u²_i = (u_i, u_i)1 and u_iu_j = – u_ju_i if i ≠ j it is easy to see that the space spanned by the elements u_i1u_i2 … u_i2r, i₁ < i₂ < … < i_2r, r = 0,1,2, …, [n/2] is a subalgebra of C. Since u_i1 u_i2 … u_i2r = (u_i1 u_i2) … (u_i2r–1u_i2r), this subalgebra is contained in C⁺ and since it contains ₂ it coincides with C⁺. It is now clear that C⁺ is the subalgebra of even elements of C, or the so-called second Clifford algebra of (x, y). Its dimensionality is 2^n–1. The structures of C and C⁺ are known. We shall state only what is needed for the representation theory of B_l and D_l. For this the symmetric bilinear form (x, y) is non-degenerate and of maximal Witt index. If n = 2l + 1, then C⁺ is isomorphic to the complete algebra of linear transformations in a 2^l-dimensional vector space over Φ. The isomorphism can be defined explicitly in the following way.

Let (u₁, u₂, …, u_n) be a basis for of the type used to obtain the matrices for B_l (§4.6). Thus we have (u₁, u₁) = 1, (u_i, u_l+1) = 1 = (u_l+1, u_i) if l = 2, l + 1 and all other products are 0. Set v_i = u₁u_i+1, w_i = u₁u_i+l+1, l = 2, …, l. Then it is easily seen that the v_i and w_i generate C⁺, that v_iv v_j = — v_jv v_i, v²_i = 0, w_iw w_j = - w_jw_i, w²_i = 0 if i ≠ j, and that the subalgebras generated by the v’s and the w’s separately are isomorphic to the exterior algebra based on an l-dimensional space. A basis for C⁺ is the set of elements v_i1… v_irw_j1… w_js where i₁ < i₂ < … < i_r, j₁ < j₂ < … < j₈. We have the relations

The subspace spanned by the vectors

is a 2^l-dimensional right ideal in C⁺. The right multiplications by a ε C⁺ in give all the linear transformations of and the correspondence between a and the restriction to of a_R is an isomorphism of C⁺ onto the algebra of linear transformations of.

The representation a → a'_R, a'_R the restriction to of a_R, induces a representation of the subalgebra ₂ of C⁺_L. Since ₂ is isomorphic to B_l this gives a representation of B_l acting in. The B_l-module is irreducible and this is the irreducible module with highest weight λ_l which we require. We shall sketch the argument which can be used to establish this.

We note first that the matrix e_i, l = 1, 2, …, l — 1, in (22) can be identified with the linear transformation x →(x, u_i+_l+2)u_i+1 – (x, u_i+1)u_i+l+2 relative to the basis (u₁, u₂, …, u_n) which we have chosen. Similarly, e_l can be identified with the transformation x →(x,u₁)u_l+1 – (x, u_l+1)u₁, h_i, l = set 1, 2, …, l – 1, with x →(x,u_l+i+1)u_i+1 – (x,u_i+1)u_l+i+1 – (x,u_l+i+2)u_{i+2 +} (x,u_i+2)u_l+i+2 with x → 2(x,u_2l+1)u_l+1 – 2(x,u_l+1)u_2l+1. In this isomorphism with ₂ (cf. (29)) we have

Let z = v₁ … v₁ ε . Then ze_j = 0, j = 1, …, l, zh_i = 0, l = 1, …, l — 1 and zh_l = z. It follows that the cyclic B_l-module generated by z is the irreducible module corresponding to λ_l. It can be shown that this is all of . We shall call the spin module for B_l. We can summarize the results on B_l in the following

THEOREM 8. Let B_l, l 2, be the orthogonal Lie algebra in a (2l + 1)- dimensional space defined by a non-degenerate symmetricbilinear form of maximum Witt index. Let the basis for a Cartan subalgebra of B_l consist of the h_i, i = 1, 2, …, l of (22) and let λ_j be the linear function on such that λ_j(h_i) = δ_ij. Then the irreducible module for B_l with highest weight λ_j, = 1, j = 1, …, l – 1 is the space _j of j-vectors. The irreducible module with highest weight λ_i is the spin module defined above.

G₂. If (α₁, α₂) constitutes a simple system of roots for G₂ then the Cartan matrix is

so that we have

We have seen in §4.3 that the positive roots are α₁, α₂, α₁ + α₂, α₁ + 2α₂, α₁ + 3α₂. The highest of these is 2α₁ + 3α₂. We have (2α₁ + 3α₂)(h₁) = 1, (2α₁ + 3α₂)(h₂) = 0 so that 2α₁ + 3α₂ is the linear function λ₁ such that λ₁(h₁)= 1, λ₁(h₂) = 0. Since G₂ is simple the adjoint representation is irreducible. Thus itself is the irreducible module whose highest weight λ_i. We shall show next that if G₂ is represented by the Lie algebra of derivations in the split Cayley algebra then the representation induced in the seven-dimensional space ₀ of elements of trace 0 is the irreducible representation corresponding to the linear function λ₂ such that h₂(h₁) = 0, λ₂(h₂)= 1. We shall show this by proving that the dimensionality of any module for G₂ satisfying G₂ ± 0 is 7, and if the dimensionality is 7, then the module is irreducible and corresponds to λ₂.

We first express the roots in terms of the basis λ₁, λ ₂. Thus we have α₁ = 2λ₁ - 3λ₂, α₂= –λ₁ + 2λ₂, by (36). Hence the positive roots are

If Λ= m₁λ₁ + m₂λ₂ ε ^*₀ and S₁ = S_α1, ≡ S₂ ≡ Sα₂ are the Weyl reflections determined by α₁ and α₂ respectively, then ΛS₁ = Λ - Λ(h₁)α ₁= Λ - m₁(2λ₁ - 3λ₂) = – m₁λ₁ + (m₂ + 3m₁)λ₂ and ΛS₂ = (m₁ + m₂)λ₁ – m₂λ₂. If Λ is the highest weight of in the module , then the m_i are non-negative and not all 0 and the following linear functions are weights of in :

If m₁ > 0 and m₂ > 0, then this list gives twelve distinct weights which means that dim 12. If m₂ = 0 the list gives six distinct weights: m₁λ₁, - m₁λ₁, 2m₁λ₂ – 3m₁λ₂, - m₁λ₁ + 3m₁λ₂, - 2m₁λ₁ + 3m₁λ₂, m₁λ₁ - 3m₁λ₂. If m₁ = 1, then contains a submodule isomorphic to and so dim 14. If m₁ > 1, then since λ₁ is a root and the λ-string containing m₁λ₁ contains also —m ₁λ₁ there are at least 2m₁ + 1 5 weights in this string. This adds three new weights to the list and shows that dim 9. Now let m₁ = 0. Then the following six weights are all different: m₂λ₂, m₂λ₁ – m₂λ₂, -m₂λ ₁+ 2m₂λ₂, m₂λ₁ – 2m₂λ₂, - 2m₂λ₂, m₂λ ₁+ m₂ λ₂, - m₂λ₁. If m₂ = 1 the λ₂-string containing m₂λ₂ contains 0 as weight so that dim 7. If m₂ > 1 then the λ₂-string containing m₂λ ₂contains at least five weights and this implies that dim 9. We have therefore shown that dim 7 and dim = 7 can hold only if the highest weight Λ= λ₂. Since there exists a module ₀ for G₂ such that dim ₀ = 7 it follows that ₀ is irreducible and its highest weight is λ₂.

THEOREM 9. The irreducible module for = G₂ with highest weight λ₁ is itself. The irreducible module for G₂ with highest weight λ₂ is the seven-dimensional module ₀ of elements of trace 0 in the split Cayley algebra .

Exercises

In these exercises we follow the notations of this chapter: is a finite-dimensional split semi-simple Lie algebra over a field of characteristic 0, splitting Cartan subalgebra, e_i, f_i, h_i canonical generators for , the universal enveloping algebra, etc.

1. Let e_i, f_i, h_i, i = 1,2, …, l, be canonical generators for . Show that the subalgebra ₁ generated by the elements e_j, f_j, h_j, j = 1, …, k, k l is a split semi-simple Lie algebra. Show that E₆ is a subalgebra of E₇, and E₇ is a subalgebra of E₈.

2. Show that has only a finite number of inequivalent irreducible modules of dimension N for N any positive integer.

3. Let be the kernel in of the representation determined by a module and let ^(s) denote the kernel of ⊗ … ⊗ , s-times. Show that a finite-dimensional can be chosen so that if is any ideal in of finite co-dimension, then there exists an s such that .

4. Let be the orthogonal Lie algebra defined by a non-degenerate symmetric bilinear form in an n 3 dimensional space . Prove that the space _r of r-vectors is -irreducible for 1 r l if n = 2l + 1 and 1 r l — 1 if n = 2l.

5. Determine a set of basic irreducible modules for C_l, l 4.

6. Show that the minimum dimensionality for an irreducible module for E₆ such that E₆ ≠ 0 is 27. Hence prove that E₆ and B₆, and E₆ and C₆ are not isomorphic.

7. Prove that the spin representation of D_l in the second Clifford algebra (of even elements) decomposes as a direct sum of two irreducible modules. Show that these two together with the spaces of r-vectors 1 r l — 2 are the basic irreducible modules for D_l, l 4.

8. Show that a basis (g₁, g₂, …, g_l) can be chosen for such that [e_ig_j] = δ_ije_i, [f_ig_j] = −δ_ijf_i. Let h = 2 Σ₁g_i = Σμ_lh_i where the μ_i ε _Φ. Let e = Σγ_ie_i where every γ_i ≠ 0 and let . Prove that (e, f, h) is a canonical basis for a split three-dimensional simple Lie algebra . Prove that ad is a direct sum of l odd-dimensional irreducible representations for . (The subalgebra is called a “principal three-dimensional subalgebra” of . Such subalgebras play an important role in the cohomology theory of . See Kostant [3].)

9. Determine a as in 8, for A_l. Find the characteristic roots of adh and use this to obtain the dimensionalities of the irreducible components of ad.

10. Let be a finite-dimensional module for ₁ let ₊ be the (nilpotent) subalgebra of generated by the e_i and let be the subspace of of elements z such that z_l = 0, l ε ₊. Show that dim is the number of irreducible submodules in a direct decomposition of into irreducible submodules. (This number is independent of the particular decomposition by the Krull-Schmidt theorem. See also § 8.5.)

11. Let and be two finite-dimensional irreducible modules for and let * be the contragredient module of . Let R and S, respectively, be the representations in and , R* the representation in *. Show that the number of submodules in a decomposition of * ⊗ as direct sum of irreducible submodules is dim where is the subspace of the space (, ) ( * ⊗ by §1) of linear mappings of into defined by

12. (Dynkin). If ₁ and ₂ are finite-dimensional irreducible -modules with highest weights Λ₁ and Λ₂ and canonical generators x₁ and x₂, respectively, then ₂ is said to be subordinate to ₁ if x₁u = 0 implies x₂u = 0 for every u in the universal enveloping algebra _- of the subalgebra _- of generated by the f_i. Prove that ₂ is subordinate to ₁ if and only if Λ₁ = Λ₂ + M where M is a dominant integral linear function on *. (Hint: Note that if is the finite-dimensional irreducible module with highest weight M and y is a canonical generator then ₁ can be taken to be the submodule of ₂ ⊗ generated by x₂ ⊗ y.)

13. Note that the definition of subordinate in 12 is equivalent to the following: there exists a -homomorphism of ₁ onto ₂ mapping x₁ onto x₂. Use this to prove that if a finite-dimensional irreducible module with ≠ 0 has minimal dimensionality (for such modules), then is basic.