REPRESENTATION TYPE OF REDUCED ENVELOPING ALGEBRAS OF CLASSICAL LIE ALGEBRAS

JOERG FELDVOSS AND DANIEL K. NAKANO

September 1995

Abstract. In this paper we determine the representation type of reduced enveloping algebras for classical Lie algebras for non-zero character. These results extend Pollack’s earlier work for restricted enveloping algebras.

1 Introduction

1.1 One of the fundamental questions in studying the representation theory of finitedimensional algebras involves determining the structure of the indecomposable modules. Let A be a finite-dimensional algebra over a field k and let mod(A) be the set of finite-dimensional A-modules. The algebra A has finite representation type if mod(A) contains finitely many indecomposable modules up to isomorphism. Otherwise, A has infinite representation type. If A has infinite representation type then one might still be able to classify the indecomposable modules. In this case A has tame representation type. Otherwise, A is of wild representation type. The definitions of tame and wild will be provided in a subsequent section.

In [KW1] [KW2], Kac and Weisfeiler studied non-restricted representations of a restricted Lie algebra by constructing certain finite-dimensional associative algebras. The representation theory of these algebras was developed more extensively by work of Friedlander and Parshall [FP1] [FP2]. Let 𝔤 be a restricted Lie algebra with pth-power map [p] defined over an algebraically closed field of characteristic p > 0. The subalgebra ′ = ⟨x^{[p]} − x^{p}⟩ of the universal enveloping algebra U(𝔤) is a central subalgebra. If χ : O → k is an algebra homomorphism, or equivalently an element in the linear dual 𝔤^{∗} of 𝔤, then one can construct a finite-dimensional associative algebra u(𝔤, χ) = U(𝔤)/⟨x^{[p]} − x^{p} − χ(x)^{p}⟩. We remark that any irreducible representation of 𝔤 will be an irreducible representation of u(𝔤, χ) for a unique choice of χ.

Research of the second author partially supported by NSF grant DMS-9500715

1.2 The main purpose of this paper will be to determine the representation type of u(𝔤, χ) for 𝔤 a classical Lie algebra. For χ = 0, u(𝔤, 0) is the restricted enveloping algebra of 𝔤. These representations of u(𝔤, χ) correspond to looking at representations of the first Frobenius kernel of a reductive algebraic group. Pollack [Po] showed that u(𝔤, 0) has infinite representation type for all classical Lie algebras 𝔤. For 𝔤 = 𝔰l(2), it turns out that u(𝔤, 0) has tame representation type, and by results of Rickard [?], the restricted enveloping algebras of other classical algebras have wild representation type.

One of the major results in [FP1] involves proving a Morita equivalence between algebras to reduce to looking at nilpotent characters. Let G be a reductive algebraic group. If the prime is good, then 𝔤 can be identified with 𝔤^{∗} via a G-isomorphism. The Jordan-Chevalley decomposition in 𝔤 can be used to produce a decomposition of characters χ = χ_{s} + χ_{n} where χ_{s} correspond with the semisimple part and χ_{n} corresponds with the nilpotent part. Given a classical Lie algebra 𝔤 with an arbitrary character χ, one can find a reductive subalgebra 𝔩 of 𝔤 such that u(𝔤, χ) is Morita equivalent to u(𝔩, χ_{n}). Therefore, in order to study the representation type of u(𝔤, χ) one can assume without a loss of generality that χ is nilpotent. The following two theorems summarize our results.

Theorem (A). Let p ≠ 2 and 𝔤 be a classical Lie algebra of type A_{n}, B_{n}, C_{n} or D_{n}. Moreover, let χ ∈ 𝔤^{∗} be a nilpotent character. The algebra u(𝔤, χ) has

(a) finite representation type, for 𝔤 of type A_{1}, χ regular nilpotent;

(b) tame representation type, for 𝔤 of type A_{1}, χ = 0 or of type B_{2}, χ regular nilpotent.

(c) wild representation type, otherwise.

Theorem (B). Let p ≥ 3h−3 and 𝔤 be a classical exceptional Lie algebra. Using the Bala-Carter labels, assume that χ does not correspond to a label with any component of the form E_{i}(σ) where i = 6, 7, 8 and σ is of the form a_{j} or b_{j}. Then u(𝔤, χ) is of wild representation type.

2 General results

In this section we will prove general results which allow us to conclude that an algebra has wild representation type by showing that certain subalgebras have wild representation type.

2.1 We begin by presenting the formal definitions of tame and wild representation type [Erd1]. A finite-dimensional algebra B has tame representation type if and only if for each integer d > 0 there exists finitely many B − k[T] bimodules {M_{i} : i = 1, 2, … t} which are free as k[T]-modules, and all but a finite number of indecomposable B-modules of dimension equal to d are isomorphic to M_{i} ⊗_{k[T]} N for some i and some simple k[T]-module N.

An finite-dimensional k-algebra B has wild representation type if and only if there exists a finitely generated B − k⟨x, y⟩-bimodule Q which is free as right k⟨x, y⟩ - module such that the functor F defined by F (N) = Q ⊗_{k⟨x,y⟩} N from mod(k⟨x, y⟩) to mod(B) preserves indecomposability and isomorphism classes.

2.2 Since k is an algebraically closed field, the simple k[T]-modules are one-dimensional and are of the form k_{a} = k[T]/(T − a) for a ∈ k. Given a subalgebra B of an algebra A let coind_{B}^{A}N = A ⊗_{B} N. The results below relate the representation type of the algebras B and A given suitable conditions on the coinduction functor.

Lemma. Let M be an indecomposable B−k[T]-bimodule which is free as k[T]-module. Then M ⊗_{k[T]} k_{a} is an indecomposable B-module.

Proof. Let M be an indecomposable as B − k[T]-bimodule and let M = m_{1}k[T] ⊕ m_{2}k[T] ⊕ … ⊕ m_{t}k[T]. Suppose that M ⊗_{k[T]} k_{a} ≅ Q_{1} ⊕ Q_{2} as B-modules. A basis for M ⊗_{k[T]} k_{a} is given by ⟨m_{j} ⊗ k_{a} : j = 1, 2, …, t⟩. Let ⟨q_{1}, q_{2}, …, q_{l}⟩ be a basis for Q_{1} and ⟨q_{l+1}, q_{l+2}, …, q_{t}⟩ be a basis for Q_{2}. Furthermore, let

q_{j} = ∑_{i=1}^{t} c_{ij}(m_{j} ⊗ k_{a}) = (∑_{i=1}^{t} c_{ij}m_{j}) ⊗ k_{a} = z_{j} ⊗ k_{a}

for j = 1, 2, …, t. If X_{1} = ⟨z_{1}, z_{2}, …, z_{l}⟩k[T] and X_{2} = ⟨z_{l+1}, z_{l+2}, …, z_{t}⟩k[T] then X_{1} and X_{2} are sub B−k[T] bimodules of M. The invertibility of the matrix C = (c_{ij}) shows immediately that M = X_{1} + X_{2}. Moreover, if z_{1}r_{1} + z_{2}r_{2} + … + z_{t}r_{t} = 0 where r_{j} ∈ k[T] for j = 1, 2, … t then (r_{1}, r_{2}, …, r_{t}) · C = 0 by using the fact that z_{j} = ∑_{i=1}^{t} c_{ij}m_{j}. The invertibility of C implies that r_{j} = 0 for all j. Hence, M = X_{1}⊕X_{2} = z_{1}k[T]⊕z_{2}k[T]⊕…⊕z_{t}k[T] which contradicts the indecomposability of M as B − k[T]-bimodule. _{□}

Theorem. Let B be a subalgebra of a finite-dimensional k-algebra A. Assume that every indecomposable B-module N is a direct summand of coind_{B}^{A} N upon restriction to B. If B has wild representation type then A has wild representation type.

Proof. Observe that by our assumption every indecomposable B-module appears as a summand of some indecomposable A-module (upon restriction to B). It is clear that if A has finite representation type then B has finite representation type. Therefore, it suffices to show that if A has tame representation type then B has tame representation type. Assume that A has tame representation type. Fix an integer d > 0 and let

d′ = {max(dim_{k} coind_{B}^{A}N) : N is indecomposable and dim_{k} N = d}.

We remark that d′ < ∞ because dim_{k} coind_{B}^{A}N < (dim_{k} A) · d. For each σ such that 0 < σ ≤ d′ there exists a finite set of A−k[T]-bimodules B_{σ} = {M_{i}^{σ} : i = 1, 2, …, t_{σ}} such that all but a finitely many indecomposable A-modules of dimension σ are isomorphic to a module in C_{σ} = {M_{i}^{σ} ⊗_{k[T]} k_{a} : i = 1, 2, …, t_{σ}, a ∈ k}. Let

E = {M ∈ mod(A) : M is indecomposable, M ∉ ∪_{0<σ≤d′}C_{σ} and dim_{k} M ≤ d′}.

The set E is finite since A has tame representation type.

For each σ, i such that 0 < σ ≤ d′ and 0 ≤ i ≤ t_{σ} let M_{i}^{σ} = X_{iσ}^{1} ⊕ X_{iσ}^{2} ⊕ … ⊕ X_{iσ}^{s_{iσ}} where X_{iσ}^{j} are indecomposable B − k[T]-bimodules for each j = 1, 2, …, s_{iσ}. According to the preceding lemma any indecomposable B-summand of M_{i}^{σ} ⊗_{k[T]} k_{a} must have the form X_{iσ}^{j} ⊗_{k[T]} k_{a} for some j. Let B̂_{d} = {X_{iσ}^{j} : dim_{k}(X_{iσ}^{j} ⊗_{k[T]} k_{a}) = d} and Ĉ_{d} = {X_{iσ}^{j} ⊗_{k[T]} k_{a} : dim_{k}(X_{iσ}^{j} ⊗_{k[T]} k_{a}) = d}. Furthermore, let Ê_{d} = {N : N is an indecomposable B-module, N is a summand of M ∈ E, and dim_{k} N = d}.

The set B̂_{d} is a finite set of B − k[T]-bimodules which are free as k[T]-modules. Let N be an indecomposable B-module and N′ be an indecomposable A-module of dimension less than or equal to d′ which has N appearing as a direct summand. If N′ ∈ ∪_{0<σ≤d′}C_{σ} then N ∈ Ĉ_{d}. On the other hand if N′ ∉ ∪_{0<σ≤d′}C_{σ} for any such choice of N′ then N is contained in the finite set Ê_{d}. Hence, B has tame representation type. _{□}

2.3

Theorem. Let 𝔤 be a finite-dimensional restricted Lie algebra χ ∈ 𝔤^{∗}. Assume that 𝔤 = 𝔩 ⊕ 𝔞 where 𝔩 is a restricted Lie subalgebra of 𝔤 and 𝔞 has a basis whose p-th powers are in 𝔞. Furthermore, assume that χ(a) = 0 for all a ∈ 𝔞. If u(𝔩, χ) has wild representation type then u(𝔤, χ) has wild representation type.

Proof. Let N ∈ mod(u(𝔩, χ)) and consider the module M = ind_{u(𝔩,χ)}^{u(𝔤,χ)}N as a u(𝔩, χ)- module. Let {a_{1}, a_{2}, …, a_{r}} be a basis for 𝔞 such that a_{i}^{[p]} ∈ 𝔞 for i = 1, 2, …, r. Then

M ≅ ind_{u(𝔩,χ)}^{u(𝔤,χ)}N ≅ ⟨1 ⊗ N⟩ ⊕ [ ⨁_{s_{j}≠0forallj=1,2,...r}⟨a_{1}^{s_{1}}a_{2}^{s_{2}} … a_{r}^{s_{r}} ⊗ N⟩] (2.3.1)

is a direct sum decomposition of M as u(𝔩, χ)-module. Observe that the assumptions are necessary because χ(a_{j})^{p} = a_{j}^{p} −a_{j}^{[p]} so a_{j}^{p} = a_{j}^{[p]} ∈ 𝔞_{j} for j = 1, 2, …, r. Hence, N is a direct summand of M as u(𝔩, χ)-module and the result follows by Theorem—–. _{□}

3 Classical algebras with regular nilpotent characters 3.1 Let χ ∈ 𝔤^{∗} be a nilpotent character. We will assume that χ is regular nilpotent and in standard Levi form. That is, χ(e_{α}) = 1 for α ∈ S, χ(h_{j}) = 0 for j = 1, 2, … n and χ(e_{α}) = 0 for α ∈ R − S. (give facts about simple modules in FP)

3.2

Theorem. Let 𝔤 be a classical Lie algebra and χ be a regular nilpotent character. If the rank of 𝔤 is greater than three then u(𝔤, χ) has wild representation type.

Proof. For every simple root α, let 𝔩_{α} be the corresponding Levi subalgebra in 𝔤 and let 𝔟_{α} = 𝔩_{α} ∩ 𝔟. Furthermore, let Z_{α}(μ) = ind_{u(𝔟_{α},χ)}^{u(𝔩_{α},χ)}μ for each μ ∈ X_{1}(T). Set λ = 0. There exists a non-split exact sequence of u(𝔩_{α}, χ)-modules

0 → Z_{α}(λ) → K → Z_{α}(λ) → 0. (3.2.1)

where K is a projective indecomposable u(𝔩_{α})-module. Extend the action of u(𝔩_{α}, χ) on Z_{α}(λ) to u(p_{α}, χ) by letting u_{α} act trivially. It follows that

Z(λ) ≅ ind_{u(𝔟,χ)}^{u(𝔤,χ)}λ ≅ ind_{u(𝔭_{α},χ)}^{u(𝔤,χ)}Z_{α}(λ)

By applying the functor ind_{u(𝔭_{α},χ)}^{u(𝔤,χ)} to the preceding exact sequence, we have

0 → Z(λ) → K → Z(λ) → 0. (3.2.2)

We will first show that the sequence (–2–) is non-split over u(𝔤, χ). Upon restriction to u(𝔭_{α}^{+}, χ), the sequence (–2–) can be rewritten as

(3.2.3) 0 → ind_{u(𝔩_{α},χ)}^{u(𝔭_{α}^{+},χ)}Z_{α}(λ) → ind_{u(𝔩_{α},χ)}^{u(𝔭_{α}^{+},χ)}K → ind_{u(𝔩_{α},χ)}^{u(𝔭_{α}^{+},χ)}Z_{α}(λ) → 0.

Observe that

Hom_{u(𝔭_{α}^{+},χ)}(ind_{u(𝔩_{α},χ)}^{u(𝔭_{α}^{+},χ)}K, Z_{α}(μ)) ≅ Hom_{u(𝔩_{α},χ)} ≅ \begin{cases}k & if μ = λ \\ 0 & otherwise. \end{cases}

Therefore, ind_{u(𝔩_{α},χ)}^{u(𝔭_{α},χ)}K is an indecomposable u(p_{α}, χ)-module and (–2–) must be nonsplit.over u(𝔭_{α}, χ), thus over u(𝔤, χ).

Now consider (–3–) as a short exact sequence of u(𝔩_{α}, χ)-modules. The module K is a projective u(𝔩_{α}, χ)-module, so ind_{u(𝔩_{α},χ)}^{u(𝔭_{α},χ)}K is a projective u(𝔭_{α}, χ)-module, and also projective over u(𝔩_{α}, χ). If (–3–) splits as an exact sequence of u(𝔩_{α}, χ) -modules, then ind_{u(𝔩_{α},χ)}^{u(𝔭_{α},χ)}Z_{α}(λ) is a projective u(𝔩_{α}, χ)-module. This is a contradiction because

Ext_{u(𝔭_{α},χ)}^{1}(ind_{u(𝔩_{α},χ)}^{u(𝔭_{α},χ)}Z_{α}(λ), Z_{α}(λ)) ≅ Ext_{u(𝔩_{α},χ)}^{1}(ind_{u(𝔩_{α},χ)}^{u(𝔭_{α}^{+},χ)}Z_{α}(λ), Z_{α}(λ)) ≠ 0.

Let A = u(𝔤, χ). If ζ_{α} represents the extension class in Ext_{u(𝔤,χ)}^{1}(Z(λ), Z(λ)) given by (–2–), then res_{A,u(𝔩_{α},χ)}(ζ_{α}) ≠ 0.

On the other hand, if one restricts the sequence (–2–) to u(𝔩_{β}, χ) where β is another simple root and β ≠ α, then the sequence (–2–) can be expressed as

(3.2.4) 0 → ind_{u(𝔷,χ)}^{u(𝔭_{β}^{+},χ)}Z_{α}(λ) → ind_{u(𝔷,χ)}^{u(𝔭_{β}^{+},χ)}K → ind_{u(𝔷,χ)}^{u(𝔭_{β}^{+},χ)}Z_{α}(λ) → 0.

where 𝔷 = 𝔱 ⊕ ⟨x_{α}⟩ ⊕ ⟨x_{−β}⟩. The sequence above splits because the action of ⟨x_{−β}⟩ on Z_{α}(λ) is trival and Z_{α}(λ) is a projective u(𝔱 ⊕ ⟨x_{α}⟩, χ) -module. Therefore, res_{A,u(𝔩_{β},χ)}(ζ_{α}) = 0.

Let S = {α_{1}, α_{2}, …, α_{r}} denote the set of simple roots where r is the rank of 𝔤. Moreover, let {ζ_{α_{1}}, ζ_{α_{2}}, … ζ_{α_{r}}} be the classes in Ext_{A}^{1}(Z(λ), Z(λ)) as constructed above. Suppose that ζ = c_{1}ζ_{α_{1}} + c_{2}ζ_{α_{2}} + … c_{r}ζ_{α_{r}} = 0 where c_{j} ∈ k for j = 1, 2, … r. It follows that

res_{A,u(𝔩_{α},χ)}(ζ) = c_{j}res_{A,u(𝔩_{α},χ)}(ζ_{α_{j}}) = 0,

thus c_{j} = 0 for all j. Consequently, the set {ζ_{α_{1}}, ζ_{α_{2}}, … ζ_{α_{r}}} is linearly independent in Ext_{A}^{1}(Z(λ), Z(λ)) and

dim_{k} Ext_{u(𝔤,χ)}^{1}(Z(λ), Z(λ)) ≥ rank 𝔤 ≥ 3.

Hence, u(𝔤, χ) has wild representation type. _{□}

4 Rank two algebras with regular nilpotent characters

Throughout this section we will assume that χ is a regular nilpotent character. In this situation every block of u(𝔤, χ) has one simple module. The blocks are symmetric local algebras and results presented in [Erd1] can be applied.

4.1 Let 𝔤 be of type A_{2}. Moreover, let S = L(0, 0) and P = P (0, 0) the projective over of S. Set Γ = B_{(0,0)}. The module P cannot be uniserial. Therefore, P has one of two possible radical structures:

S  S
S S S S  S S
S  S S
S

because Γ is a symmetric algebra. In the first case dim_{k} Ext_{Γ}^{1}(S, S) = 4 and Γ is wild. Otherwise, Γ has an Ext^{1}-quiver of the form

The basic algebra Γ_{0} associated to Γ has dimension equal to 6, and is not of the form 4s for s ∈ ℕ. Hence, by [Erd1] Γ must have wild representation type.

4.2 Now assume that 𝔤 is of type G_{2}. Let λ be a weight on precisely one wall in X_{1}(T) and let L(λ) be the corresponding simple module. The projective cover of L(λ), namely P (λ), has precisely |W/Stab_{W} λ| = 12/2 composition factors all of which are isomorphic to L(λ). The argument given in the previous section for A_{2} can now be applied to show that u(𝔤, χ) is of wild representation type.

4.3 Let 𝔤 be now of type B_{2} and let λ be a weight on precisely one wall. The projective cover P (λ) of L(λ) has four copies of L(λ) appearing as composition factors. The radical series of P (λ) is given by

Now according to [Erd1, III.3 Prop] the block B_{λ} is of tame representation type. Therefore, we need to consider the block B_{(0,0)}. There are 8 composition factors of P (0, 0) with the following radical structure.

¿¿¿DETERMINE WHETHER B_{(0,0)} IS TAME OR WILD

5 Rank two and three algebras with sub-regular nilpotent characters

Let 𝔤 = ∑_{α∈R} 𝔤_{α} ⊕𝔱 be a rank two classical Lie algebra and let α_{1}, α_{2} be the simple roots in R^{−} . Set 𝔭 = ⊕_{α∈R^{+}} ⊕ 𝔱 ⊕ ke_{α_{1}} be the parabolic subalgebra whose Levi component is of type A_{1}.

5.1 Let 𝔤 be of type A_{2} with χ is standard Levi form with χ(e_{α_{1}}) = 1 and χ(e_{α_{2}}) = 0. From the results of Panov [?], the modules L(λ) = u(𝔤, χ) ⊗_{u(𝔭)} L(λ) are precisely the simple u(𝔤, χ)-modules. We have two extensions of u(𝔭)

0 → L(0, 0) → V_{j} → L(0, p − 2) → 0 (5.1.1)

for j = 1, 2. These extensions yield two inequivalent extensions of L(0, 0) and L(0, p− 2).

0 → L(0, 0) → u(𝔤, χ) ⊗_{u(𝔭)} V_{j} → L(0, p − 2) → 0 (5.1.2)

Since there are more that two simple modules in the block B_{(0,0)} and the block is a symmetric algebras there exists a simple module S such that quiver given in Figure (*) appears as a subquiver of the Ext^{1}-quiver of B_{(0,0)}.

Hence, by [Erd1, I 10.8], u(𝔤, χ) has wild representation type.

5.2 Let 𝔤 be of type B_{2} and let χ(e_{α_{1}}) = 1 and χ(e_{α_{2}}) = 0 where α_{1} is the short root. A direct calculation shows that L(λ) ≅ u(𝔤, χ)⊗_{u(𝔭)}L(λ). Note that this is consistent with the statement of the Kac-Weisfeiler conjecture, proved by Premet [?]. The same argument used in the last section for type A_{2} shows that there is a subquiver of the form (8) appearing as a subquiver. Hence, u(𝔤, χ) has wild representation type.

Now we have to consider the situation where χ(e_{α_{1}}) = 0 and χ(e_{α_{2}}) = 1. The positive roots of B_{2} are {e_{α_{1}}, e_{α_{2}}, e_{α_{1}+α_{2}}, e_{2α_{1}+α_{2}}}. Let 𝔩 = 𝔱 ⊕ ⟨e_{±α_{2}}, e_{±(2α_{1}+α_{2})}⟩. Set 𝔞 = ⟨e_{±α_{1}}, e_{±(α+β)}⟩. Then 𝔤 ≅ 𝔩 ⊕ 𝔞 where [𝔩, 𝔞] ⊂ 𝔞. Observe that 𝔩 is isomorphic to A_{1} × A_{1}. Moreover, u(𝔩, χ) ≅ u(A_{1}, χ_{1}) ⊗ u(A_{1}, χ_{2}) where u(A_{1}, χ) ≅ k[x, y]/ and u(A_{1}, χ_{2}) ≅ k[z]/(z^{p} = 0). This shows that u(𝔩, χ) has wild representation type and by Theorem—, so does u(𝔤, χ).

5.3 Finally, let 𝔤 be of type G_{2} and let χ(e_{α_{1}}) = 1 and χ(e_{α_{2}}) = 0 where α_{1} is the short root. The positive roots for G_{2} are {e_{α_{1}}, e_{α_{2}}, e_{α_{1}+α_{2}}, e_{2α_{1}+α_{2}}, e_{3α_{1}+α_{2}}, e_{3α_{1}+2α_{2}}}. Let 𝔩 = 𝔱⊕⟨e_{±α_{1}}, e_{±(3α_{1}+2α_{2})}⟩. Once again, 𝔤 = 𝔩⊕𝔞 where 𝔞 = ⟨e_{±α_{2}}, e_{±(α_{1}+α_{2})}, e_{±(2α_{1}+α_{2})}, e_{±(3α_{1}+2α_{2})}⟩. Since 𝔩 ≅ A_{1}×A_{1} the same argument as in the previous case for B_{2} shows that u(𝔤, χ) has wild representation type.

On the other hand, if χ(e_{α_{1}}) = 0 and χ(e_{α_{2}}) = 1 then we can set 𝔩 = 𝔱 ⊕ ⟨e_{±α_{2}}, e_{±(2α_{1}+α_{2})}⟩, and

𝔞 = ⟨e_{±α_{1}}, e_{±(α_{1}+α_{2})}, e_{±(3α_{1}+α_{2})}, e_{±(3α_{1}+2α_{2})}⟩

. We have 𝔤 = 𝔩 ⊕ 𝔞 with [𝔩, 𝔞] ⊂ 𝔞. Therefore, by Theorem—-, u(𝔤, χ) has wild representation type.

6 Characters on the exceptional algebras F_{4} and G_{2}

6.1

6.2

6.3 With all the information provided we will now present the proofs of Theorem (A) and (B).

Proof. (Theorem (A)) Let 𝔤 be a classical Lie algebra of types A_{n}, B_{n}, Cn or D_{n} and χ a nilpotent character. The case in which χ = 0 was proved by Pollack [Po]. Assume that χ ≠ 0. Let 𝔩 be the subalgebra constructed in Section on which χ is regular nilpotent. Let Z(𝔩) be the center of 𝔩. If the rank of 𝔩/Z(𝔩) is greater than three then by Theorem 3 u(𝔩, χ) has wild representation type. Since the subalgebra 𝔩 satisfies the assumptions of Theorem it follows that u(𝔤, χ) has wild representation type for rank 𝔩 ≥ 3.

From the preceding paragraph we may now assume that 𝔩/Z(𝔩) is of type A_{1}, A_{1} × A_{1}, B_{2} or A_{2}. We have seen that for the cases A_{1} × A_{1} and A_{2} (Section 2) the algebra u(𝔩, χ) has wild representation type, thus u(𝔤, χ) has wild representation type. On the other hand, if 𝔩 is A_{1} and 𝔤 is not A_{1}, then 𝔩 corresponds to Levi subalgebra obtained by taking one of simple roots. In this case we can find a larger Levi subalgebra 𝔩′ containing 𝔩 by adding an adjacent simple root. Then 𝔩′/Z(′) is either A_{2} or B_{2} with χ being subregular on 𝔩′. According to 2, 2, u(𝔩′) has wild representation type and by Theorem u(𝔤, χ) has wild representation type. Similarly, if 𝔩 is B_{2} and 𝔤 is not B_{2}, then 𝔩 is a Levi subalgebra obtained by taking two adjacent simple roots. Once again there exists a larger Levi subalgebra 𝔩′ containing 𝔩 which is obtained by taking one additional simple roots that is adjacent to one of the simple roots for 𝔩. The algebra 𝔩′/Z(𝔩′) is either B_{3} or C_{3}. In Section 3 and 3 it was shown that u(𝔩, χ) is of wild representation type. Therefore, u(𝔤, χ) is of wild representation type by Theorem .

Finally, if 𝔤 is A_{1} and χ is regular nilpotent. Each non-semisimple block is Morita equivalent to the algebra k[x]/x^{p}, which is of finite representation type. Now if 𝔤 is B_{2} and χ is regular nilpotent then by 2 it follows that u(𝔤, χ) has tame representation type.

□

Proof. (Theorem (B)) Let p ≥ 3h − 3 and let 𝔤 be a classical exceptional algebra. We may assume that χ ≠ 0 because this case is handled by Pollack’s results [Po]. If χ is regular nilpotent then by Theorem 3 u(𝔤, χ) is of wild representation type for 𝔤 of types E_{6}, E_{7}, E_{8} and F_{4}. If 𝔤 is of type G_{2}, then u(𝔤, χ) has wild representation type by 2.

Let X be the Bala-Carter label for χ. The regular nilpotent case was handled in the preceding paragraph so we may assume that X ≠ E_{6}, E_{7}, E_{8}, F_{4}, G_{2}. Therefore, there are three possibilities that might occur:

(1) X has a component of type E_{j}(σ) for j = 6, 7, 8 and σ = a_{j} or b_{j};

(2) X is G_{2}(a_{1}) or F_{4}(a_{j}) for j = 1, 2, 3, 4

(3) there exists a proper Levi subalgebra 𝔩 of 𝔤 such that the restriction of χ to 𝔩 will correspond to the nilpotent orbit X in a classical Lie algebra;

The first case is ruled out by hypothesis. For case (2), u(𝔤, χ) has wild representation type by 2 and 4. For the final case (3), if 𝔩 is not of type A_{1} or B_{2}, then u(𝔩, χ) is of wild representation type, and by Theorem u(𝔤, χ) is of wild representation type. On the other hand, if 𝔩 is of type A_{1}, then 𝔩 is obtained by taking one simple root. As in the argument for Theorem (A) there exists a Levi subalgebra 𝔩′ such that 𝔩′/z(𝔩′) is of type A_{2}. The character on 𝔩′ is subregular. By 2 u(𝔩′, χ) is of wild representation type, and by Theorem u(𝔤, χ) is of wild representation type. A similar argument for when 𝔩 is of type B_{2} uses the coinduction argument by locating a subalgebra 𝔩′ of type B_{3} on which the character is subregular. This case will only occur for the algebra F_{4}.

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References

[CM] D.H. Collingwood, W.M. McGovern, Nilpotent orbits in semisimple Lie algebras, Van Nostrand Reinhold, New York 1993.

[DR] V. Dlab, C. M. Ringel, Indecomposable representations of graphs and algebras, Memoirs of A.M.S 173 (1976).

[Erd1] K. Erdmann, Blocks of tame representation type, (Lecture Notes in Math. 1428), Springer-Verlag, New York 1990.

[Erd2] K. Erdmann, Schur algebras of finite type, Quart. J. Math. Oxford (2) 44 (1993), 17-41.

[FP1] E.M. Friedlander, B.J. Parshall, Modular representation theory of Lie algebras, Amer. J. Math 110 (1988), 1055-1094.

[FP2] E.M. Friedlander, B.J. Parshall, Deformations of Lie algebra representations, Amer. J. Math 112 (1990), 375-395.

[FP3] E.M. Friedlander, B.J. Parshall, Induction, deformation and specialization, Math. Ann. 290 (1991), 473-489.

[KW1] V. Kac, B. Weisfeiler, The irreducible representations of Lie p-algebras, Functional Anal. Appl. 5 (1971), 471-503.

[KW2] V. Kac, B. Weisfeiler, Coadjoint action of a semisimple algebraic group and the center of the enveloping algebras in characteristic p, Indag. Math. 38 (1976), 135-151.

[Po] R.D. Pollack, Restricted Lie algebras of bounded type, Bull of AMS, 74 (2) (1968), 326-331.

[P] R. S. Pierce, Associative algebras, Springer-Verlag, New York 1982.

Mathematisches Seminar der Univerisitat, Bundesstr. 55, W-2000 Hamburg 13, Federal Republic of Germany

E-mail address: ms3a016@GEOMAT.math.uni-hamburg.de

Department of Mathematics and Statistics, Utah St. University, Logan, UT. 84322 U.S.A.

E-mail address: nakano@sunfs.math.usu.edu