REPRESENTATION TYPE OF THE BLOCKS OF CATEGORY O

VYACHESLAV FUTORNY, DANIEL K. NAKANO, AND R. DAVID POLLACK

Abstract. In this paper the authors classify the blocks of the category 0 for complex semisimple Lie algebra having finite, tame or wild representation type.

1 Introduction

1.0 Acknowledgements. The first and third authors would like to acknowledge the Department of Mathematics and Statistics at Utah State University for their hospitality and support during the 1997-1998 academic year. The second author would like to thank the Mathematisk Institute at Aarhus Universitet for their hospitality during the preparation of this paper. The authors are greatfull to R.Irving, Yu. Drozd, V.Dlab, J. A. de la Peña and V.Bekkert for usefull discussions.

2 Preliminaries

2.0 Type of finite-dimensional algebras Let Λ be a basic finite-dimensional algebra over an algebraically closed filed K. We will denote by mod(Λ) the category of all finite-dimensional Λ-modules and by ind(Λ) the set of all indecomposable modules in mod(Λ). The algebra Λ is said to be of finite representation type if ind(Λ) contains finite many isomorfism classes. Otherwise, we say that Λ has infinite type.

The algebra Λ of infinite type is said to be tame if for any positive integer d there exist finitely many finitely-generated Λ−K[x]-bimodules M_{1}, …, M_{m} free as right K[x]-modules and such that for any module M ∈ ind(Λ) \ {finitenumberofmodules} with dim M = d,

M ≃ M_{i} ⊗_{K[x]} N

for some i and some simple K[x]-module N. Otherwise, Λ is wild.

Date: June 1998.

Research of the first author partially supported by CRDF grant; Regular Associate ICTP .

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

Research of the third author was supported by Queen’s sabbatical grant.

Theorem (Drozd). Every finite-dimensional algebra of infinite type is either tame or wild.

We will also use the following simple fact (cf. [Erd]).

Proposition . (i) Let I ⊂ Λ be an ideal and Λ/I is of infinite (wild) type

then Λ is of infinite (wild) type. (ii) Let e ∈ Λ be an idempotent and eΛe is of infinite (wild) type then Λ is

of infinite (wild) type.

Let L_{1}, … L_{n} are all non-isomorphic simple modules in mod(Λ) and P_{1}, …, P_{n} are corresponding projective covers. Here P_{i} ≃ e_{i}Λ, onde e_{1}, …, e_{n} is a complete set of primitive orthogonal idempotents.

The quiver Q(Λ) of a basic algebra Λ has the set of verticies identified with the set {L_{i}} and the number of arrows between verticies i and j equal dimExt^{1}(L_{i}, L_{j}) = dimHom_{Λ}(P_{j}, P_{i}).

Given a quiver Q one constructs the path algebra KQ such that the category of representations R(Q) of Q and the category mod(KQ) are equivalent.

By the theorem of Gabriel every basic finite-dimensional algebra Λ is isomorphic to Q(Λ)/I for some ideal I, hence the category mod(Λ) is equivalent to the category of representations of some quiver with relations. In the case when I = 0 we get hereditary algebras.

Denote by R the radical of Λ, R = ∑_{i≠j} e_{i}Λe_{j}. Suppose that R^{2} = 0 and let Q′ be the separated quiver of Λ. It consist of 2n verticies 1, …, n, 1′, … n′ and theres is an arrow from i to j′ if Hom_{Λ}(P_{j}, P_{i}) ≠ 0. It follows from [Au] that the representation type of Λ in this case corresponds to the representation type of Q′.

2.0 Quasi-hereditary algebras and highest weight categories. For details on quasi-hereditary algebras see [DK] or [D]. Fix an order on a complete set of primitive orthogonal idempotents e_{1}, …, e_{n} of Λ. Denote ϵ_{i} = e_{i} + … + e_{n} for 1 ≤ i ≤ n, ϵ_{n+1} = 0. Consider standard Λ-modules Δ_{i} = e_{Λ}/e_{i}radΛϵ_{i+1}Λ, i = 1, …, n.

The algebra Λ is said to be quasi-hereditary (with respect to the order on the idempotents) if Λ as a regular module has a filtration with subquotients Δ_{i} and Hom_{K}(Δ_{i}, Δ_{i}) ≃ K for all i.

Let C be an abelian category and S be a finite poset in bijective correspondence with the isomorfism classes of simple objects in C. Again for s ∈ S denote by L_{s} a simple module from the isomorfism class corresponding to s. For an object V ∈ C, (V : L_{s}) denotes the multiplicity of L_{s} in a composition series of V. A collection {M_{s}} of non-isomorphic objects of C is called a choice of Verma modules if M_{s}/radM_{s} ≃ L_{s}, (M_{s} : L_{s}) = 1 and (M_{s} : L_{t}) = 0 unless t ≤ s. The category C is called highest weight category (with respect to the choice of Verma modules) if evry projective indecomposable module in C has a filtration with quotiens from the set {M_{s}}, and if V ∈ C is such that V/radV ≃ L_{s} and all its other composition subquotients are of the form L_{t}, t ≤ s, then V is a homomorphic image of M_{s}. It was shown by Cline, Parshal and Scott [CPS] that an algebra Λ is quasi-hereditary if and only if mod(Λ) is highest weight category with respect to choice of standard modules Δ_{i}.

2.0 Highest weight modules and category O.

2.0 Blocks of category O.

3 Induction and restriction

4 Wild blocks: codimension 2 case

4.0 Let 𝔤 be simple of rank equal to two and Φ_{μ} = ∅. By analyzing the structure of M(λ) we have Ext_{O_{μ}}^{1}(L(σ), L(s_{α′} · σ)) ≠ 0 for s_{α′} a simple reflection. Let Π = {α, β} be the simple roots where α is short and β is long. Set t_{α} = s_{α}s_{β} and t_{β} = s_{β}s_{α}. The extension information can be used to prove the block O_{μ} has a subquiver of the form given in Figure 1 in each of the rank two cases. The correspondence between simple modules and the vertices of the quiver is given via the following table.

[figure: X_{1}; •; X_{2}; •; X_{r}; •; Y_{1}; •; Y_{2}; •; Y_{r}; •; ⋯; ⋯; •; •; A; B] Figure 1

vertex Type A_{2}  Type B_{2}  Type G_{2}
A  L(λ)  L(λ)  L(λ)
B  L((s_{α}t_{β}).λ) L(t_{β}^{2}.λ)  L(t_{β}^{3}.λ)
X_{1}  L(s_{α}.λ)  L(s_{α}.λ)  L(s_{α}.λ)
X_{2}  L(t_{β}.λ)  L(t_{β}.λ)  L(t_{β}.λ)
X_{3}  –  L((s_{α}t_{β}).λ) L((s_{α}t_{β}).λ)
X_{4}  –  –  L(t_{β}^{2}.λ)
X_{5}  –  –  L((s_{α}t_{β}^{2}).λ)
Y_{1}  L(s_{β}.λ)  L(s_{β}.λ)  L(s_{β}.λ)
Y_{2}  L(t_{α}.λ)  L(t_{α}.λ)  L(t_{α}.λ)
Y_{3}  –  L((s_{β}t_{α}).λ) L((s_{β}t_{α}).λ)
Y_{4}  –  –  L(t_{α}^{2}.λ)
Y_{5}  –  –  L((s_{β}t_{α}^{2}).λ)

Lemma . Let 𝔤 be of type A_{2}, B_{2} or G_{2} and Φ_{μ} = ∅. Then O_{μ} has wild representation type.

Proof. It is enough to show that a subquiver of the Ext^{1}-quiver yields wild representation type. For each of the cases above, the separated quiver is not a union of Dynkin or extended Dynkin diagrams, thus the separated Ext^{1}-quiver is not a union of Dynkin or extended Dynkin diagrams. Hence, the block O_{μ} is of wild representation type when Φ_{μ} = ∅ and 𝔤 is simple of rank two. _{□}

4.0 Let 𝔤 be of type A_{1} ×A_{1} with Φ_{μ} = ∅. Let Π = {α_{1}, α_{2}} be the simple roots. Set X = L(λ), Y_{1} = L(s_{α_{1}}.λ), Y_{2} = L(s_{α_{2}}.λ), and Z = L((s_{α_{1}}s_{α_{2}}).λ). The structure of the Verma module M(λ) is provided in Figure 2.

[figure: M(λ); X; Y_{1}; Y_{2}; Z] Figure 2

The structures of the other Verma modules are uniserial and the composition factors are given by the following diagram.

M(s_{α_{1}}.λ):  M(s_{α_{2}}.λ):  M((s_{α_{1}}s_{α_{2}}).λ):
Y_{1}  Y_{2}  Z
Z  Z

By using reciprocity law one can compute the structures of the projectives for O_{μ}. These module structure are provided in Figure 3.

[figure: P (X); X; Y_{1}; Y_{2}; Z; P (Y_{1}); Y_{1}; Z; X; Y_{1}; Y_{2}; Z; P (Y_{2}); Y_{2}; X; Z; Y_{1}; Y_{2}; Z; P (Z); Z; Y_{1}; Y_{2}; Z; X; Z; Y_{1}; Y_{2}; Z] Figure 3

Lemma . Let 𝔤 be of type A_{1} × A_{1} and Φ_{μ} = ∅. Then O_{μ} has wild representation type.

Proof. The Ext^{1}-quiver for O_{μ} is given in Figure 4. Let e = 1_{X} + 1_{Z} and consider the algebra eO_{μ}e. The quiver for this algebra has the form given in Figure 5. According to [Erd, I 10.8], this implies that eO_{μ}e has wild representation type, thus O_{μ} has wild representation type.

[figure: X; Y_{1}; Y_{2}; Z; •; •; •; •] Figure 4

[figure: •; eX; •; eZ] Figure 5
□

4.0 We can now prove that the block O_{μ} has wild representation type if the codimension of subroot system Φ_{μ} in Φ is greater than or equal to two.

Proposition . Let 𝔤 be a complex simple finite-dimensional Lie algebra and μ ∈ X_{+}. If rank(𝔤) − rank(Φ_{μ}) ≥ 2 then O_{μ} has wild representation type.

Proof. Suppose that rank(𝔤)−rank(Φ_{μ}) ≥ 2. Without a loss of generality we may assume that the base of Φ_{μ} is contained in Π. Under these assumptions there exists two simple roots α_{1} and α_{2} such that (μ + ρ, β^{∨}) ≠ 0 for all roots β in the subroot system generated by α_{1} and α_{2}. Let 𝔩 be the Levi subalgebra of 𝔤 obtained by taking {α_{1}, α_{2}} ⊂ Π. Note that 𝔩/Z(𝔩) is of type A_{1} × A_{1}, A_{2}, B_{2} or G_{2}. According to Lemma 4.1 and 4.2, O_{μ}^{𝔩} has wild representation type. Therefore, by Proposition 3.1, O_{μ}^{𝔤} := O_{μ} has wild representation type. _{□} 5 Wild blocks: codimension 1 case

5.0 We will first consider the case when 𝔤 is of rank three and Φ_{μ} is a subroot system of type A_{1} × A_{1}. Once again by analyzing the structure of M(λ) we have Ext_{O_{μ}}^{1}(L(σ), L(s_{α′} · σ) =≠ 0 for α′ a simple root. Let Π = {α, β, γ} where β and γ are long for type B_{3} and α is long for type C_{3}. Set t_{1} = s_{γ}s_{β}, t_{2} = s_{β}s_{α} and t_{3} = s_{α}s_{β}. For type A_{3}, B_{3} and C_{3} one can locate a subquiver of the Ext^{1}-quiver of the form given in Figure 6. The correspondence between the vertices and the simple modules is given in the table below.

[figure: X_{1}; •; X_{2}; •; X_{r}; •; Y_{1}; •; Y_{2}; •; Y_{r}; •; ⋯; ⋯; •; •; B; C; •; •; A; D] Figure 6

vertex Type A_{3}  Type B_{3} and Type C_{3}
A  L(λ)  L(λ)
B  L(s_{β}.λ)  L(s_{β}.λ)
C  L((s_{γ}t_{1}).λ) L((t_{3}t_{1}t_{3}).λ)
D  L((s_{β}s_{γ}t_{1}).λ) L((s_{β}t_{3}t_{1}t_{3}).λ)
X_{1}  L(t_{1}.λ)  L(t_{1}.λ)
X_{2}  –  L((s_{α}t_{1}).λ)
X_{3}  –  L((t_{2}t_{1}).λ)
X_{4}  –  L((s_{α}t_{2}t_{1}).λ)
Y_{1}  L(t_{3}.λ)  L(t_{3}.λ)
Y_{2}  –  L((s_{β}t_{3}).λ)
Y_{3}  –  L((t_{1}t_{3}).λ)
Y_{4}  –  L((s_{β}t_{1}t_{3}).λ)

With this information one can prove that the block O_{μ} in these three cases has wild representation type.

Lemma . Let 𝔤 be of type A_{3}, B_{3}, or C_{3} and Φ_{μ} of type A_{1} × A_{1}. Then O_{μ} has wild representation type.

Proof. From our analysis above, the Ext^{1}-quiver of O_{μ} contains a subquiver that does not separate into a disjoint union of Dynkin or extended Dynkin diagrams. Consequently, the block O_{μ} must be of wild representation type.

□

5.0 The results in the previous section can be used to show that the block O_{μ} has wild representation type in the case when the codimension of the subroot system Φ_{μ} in Φ is one and Φ_{μ} is not irreducible.

Proposition . Let 𝔤 be a complex simple finite-dimensional Lie algebra and μ ∈ X_{+}. If rank(𝔤) − rank(Φ_{μ}) = 1 and Φ_{μ} is not irreducible then O_{μ} has wild representation type.

Proof. Suppose that Φ_{μ} has codimension one in Φ. We can assume that a base of Φ_{μ} is contained in Π. Let Π = {α_{1}, α_{2}, …, α_{n}}. Since Φ_{μ} is not irreducible there exists a j such that 1 < j < n with (λ + ρ, α_{j}^{∨}) ≠ 0 and (λ + ρ, α_{i}^{∨}) = 0 for i ≠ j. Let 𝔩 be the Levi subalgebra in 𝔤 corresponding to taking {α_{j−1}, α_{j}, α_{j+1}}. Then 𝔩/Z(𝔩) is of type A_{3}, B_{3} or C_{3}. According to Lemma 5.1, O_{μ}^{𝔩} has wild representation type, thus O_{μ}^{𝔤} := O_{μ} has wild representation type by Proposition 3.1. _{□}

5.0

For 𝔤 of type A_{4} and Φ_{μ} of type A_{3}, there exists an idempotent e such that eO_{μ}e has a subquiver of the form given in Figure 10. This shows immediately that eO_{μ}e is of wild representation type. Therefore, O_{μ} is of wild representation type.

[figure: •; •; •] Figure 7

The next result provides sufficient conditions for blocks O_{μ} with codimension one irreducible root systems Φ_{μ} in algebras of type A to be of wild representation type.

???

Proposition . Let 𝔤 be of type A_{n} and Φ_{μ} of type A_{n−1} for n ≥ 3. Then O_{μ} has wild representation type.

Proof. Let 𝔤 be of type A_{n} and Φ_{μ} be of type A_{n−1} for n ≥ 3. Without a loss of generality we may assume that Π = {α_{1}, α_{2}, …, α_{n}} and Φ_{μ} has base {α_{1}, α_{2}, …, α_{n−1}}. Let 𝔩 be the Levi subalgebra corresponding to {α_{n−2}, α_{n−1}, α_{n}}. Then 𝔩/Z(𝔩) is of type A_{3}. From our analysis of this case above, O_{μ}^{𝔩} has wild representation type. Consequently, O_{μ}^{𝔤} := O_{μ} has wild representation type by Proposition 3.1. _{□}

5.0 Let 𝔤 be of type B_{2} or G_{2} with Φ_{μ} of type A_{1}. In this case the Verma module M(λ) is uniserial and has composition factors given as follows.

M(λ):  X_{1}

X_{2}

⋮ X_{r}

Let Π = {α, β} be the simple roots where β is the long root. Set t_{α} = s_{α}s_{β} and t_{β} = s_{β}s_{α}. The table below gives the labeling of the simple modules.

simples B_{2} (A_{1} short) B_{2} (A_{1} long) G_{2} (A_{1} short) G_{2} (A_{1} long)
X_{1}  L(λ)  L(λ)  L(λ)  L(λ)
X_{2}  L(s_{β}.λ)  L(s_{α}.λ)  L(s_{β}.λ)  L(s_{α}.λ)
X_{3}  L(t_{α}.λ)  L(t_{β}.λ)  L(t_{α}.λ)  L(t_{β}.λ)
X_{4}  L((s_{β}t_{α}).λ) L((s_{α}t_{β}).λ) L((s_{β}t_{α}).λ)  L((s_{α}t_{β}).λ)
X_{5}  –  –  L((t_{α}^{2}).λ)  L((t_{β}^{2}).λ)
X_{6}  –  –  L((s_{β}t_{α}^{2}).λ)  L((s_{α}t_{β}^{2}).λ)

Lemma . Let 𝔤 be of type B_{2} and G_{2} and Φ_{μ} of type A_{1}. Then O_{μ} has wild representation type.

Proof. The Verma module M(λ) is uniserial of length r = 4 (resp. 6) for 𝔤 of type B_{2} (resp. G_{2}) and Φ_{μ} of type A_{1}. The block O_{μ} will be Morita equivalent to a block for 𝔤 of type A_{n} and Φ_{μ} of type A_{n−1} where n = 3 or 5. Therefore, by Proposition 5.3, O_{μ} must be of wild representation type. _{□}

5.0 Let 𝔤 be of type B_{3} or C_{3} and Φ_{μ} be of type A_{2} or B_{2}. Moreover, let Π = {α, β, γ} where β and γ are long for type B_{3} and α is long for type C_{3}. Set w_{1} = s_{α}s_{β}s_{α}, w_{2} = s_{γ}s_{β}s_{α}, and w_{3} = s_{α}s_{β}s_{γ}. First consider the case when Φ_{μ} is of type A_{2}. By analyzing the structure of M(λ) one can see that Ext_{O_{λ}}^{1}(L(σ), L(s_{α′}.σ)) ≠ 0 for α′ a simple root. Therefore, the Ext^{1}-quiver has a subquiver of the form given in Figure 5. The correspondence between simple modules and the vertices is provided in a table below.

[figure: X; •; Y; •; •; •; C; D; •; •; B; E; •; •; A; F] Figure 8

10 VYACHESLAV FUTORNY, DANIEL K. NAKANO, AND R. DAVID POLLACK

vertex Type B_{3} and Type C_{3}
A  L(λ)
B  L(s_{α}.λ)
C  L((s_{β}s_{α}).λ)
D  L((s_{γ}w_{1}).λ)
E  L((s_{β}s_{γ}w_{1}).λ)
F  L((s_{α}s_{β}s_{γ}w_{1}).λ)
X  L(w_{1}.λ)
Y  L(w_{2}).λ)

Now consider the case when Φ_{μ} is of type B_{2}. The Verma module M(λ) is uniserial with six composition factors:

M(λ):  L(λ)
L(s_{γ}.λ)
L((s_{β}s_{γ}).λ)
L(w_{3}.λ)
L((s_{β}w_{3}).λ)
L((s_{γ}s_{β}w_{3}).λ)

Lemma . Let 𝔤 be of type B_{3} and C_{3} and Φ_{μ} be of type A_{2} or B_{2}. Then O_{μ} has wild representation type.

Proof. Let 𝔤 be of type B_{3} or C_{3}. If Φ_{μ} is of type A_{2}, then the Ext^{1}-quiver of O_{μ} has a subquiver which cannot be separated into the disjoint union of Dynkin or extended Dynkin diagrams. Therefore, O_{μ} is wild. On the other hand, if Φ_{μ} is of type B_{2}, then O_{μ} is Morita equivalent to a block for type A_{6} with stabilizer A_{5}. Therefore, by Proposition 5.3, O_{μ} must be wild. _{□}

5.0 The following result gives sufficient condition to insure when a block O_{μ} has wild representation type in the case that Φ_{μ} is irreducible and has codimension one in Φ.

Proposition . Let 𝔤 be a complex simple finite-dimensional Lie algebra and μ ∈ X_{+}. If rank(𝔤) − rank(Φ_{μ}) = 1, Φ_{μ} is irreducible, and 𝔤 be not of type A_{1} or A_{2} then O_{μ} has wild representation type.

Proof. Let Φ_{μ} have codimension one in Φ and assume that Φ_{μ} is irreducible. Again one may assume that the base of Φ_{μ} is contained in Π. Let Π = {α_{1}, α_{2}, …, α_{n}}. By assumption the rank of 𝔤 must be greater than two. The irreducibility of Π implies that either (μ + ρ, α_{j}) ≠ 0 for j = 1 or n and (μ+ρ, α_{i}) ≠ 0 for j ≠ i. If rank(𝔤) = 2 and 𝔤 is not type A_{2}, then by Lemma 5.4, O_{μ} has wild representation type. Now assume that rank(𝔤) ≥ 3. Let 𝔩 be the Levi subalgebra corresponding to the simple roots {α_{1}, α_{2}, α_{3}} if (μ + ρ, α_{1}) ≠ 0 or {α_{n−2}, α_{n−1}, α_{n}} if (μ + ρ, α_{n}) ≠ 0. Then 𝔩/Z(𝔩) is of type A_{3}, B_{3} or C_{3}. Proposition 5.3 and Lemma 5.5 indicate that O_{μ}^{𝔩} has wild representation type in these cases. Hence, by Proposition 3.1, O_{μ}^{𝔤} := O_{μ} has wild representation type. _{□}

6 Representation type

6.0 Finite type: Let 𝔤 = 𝔰𝔩_{2}(ℂ) and μ ∈ X_{+} such that Φ_{μ} = ∅. The largest Verma module M(λ) is uniserial and has two composition factors L(λ) and L(s_{α}.λ). By using reciprocity the projective covers of L(λ) and L(s_{α}.λ) are both uniserial and have the following structures:

P (λ) :  L(λ)  P (s_{α}.λ) :  L(s_{α}.λ)
L(s_{α}.λ)  L(λ)
L(s_{α}.λ)

The quiver for O_{μ} is given by Figure 12 with relations αβ = 0 and βαβ = 0.

[figure: •; L(λ); •; L(s_{α}.λ); α; β] Figure 9

Proposition . Let 𝔤 be of type A_{1} and Φ_{μ} = ∅. Then O_{μ} has finite representation type.

Proof. Since P (s_{α}.λ) is self-dual, it is both projective and injective for O_{μ}. The other projective indecomposable module P (λ) has radical length two. The two facts imply that the only indecomposable module of radical length three in O_{μ} is P (s_{α}.λ). Therefore, O_{μ} has the same representation type as O_{μ}/(rad O_{μ})^{2}. The latter algebra is two-nilpotent with the same quiver as O_{μ}. This quiver separates into a union of two quivers of type A_{1}. Hence, O_{μ}/(rad O_{μ})^{2} has finite representation type and so does O_{μ}. _{□}

Consider the Lie algebra 𝔤 of type A_{2} with Φ_{μ} of type A_{1}. Let Π = {α_{1}, α_{2}} and set A = L(λ), B = L(s_{α_{2}}.λ) C = L((s_{α_{1}}s_{α_{2}}).λ). The Verma module M(λ) is uniserial and has the following composition series:

M(λ):  A
B
C

This information along with reciprocity can be used to construct the projective indecomposable modules. Their structures are given in Figure 13. The algebra O_{μ} has quiver given in Figure 14 with relations α_{1}β_{1} = 0, α_{2}β_{2} = β_{1}α_{1}. This algebra has dimension 14.

Proposition . Let 𝔤 be of type A_{2} and Φ_{μ} be of type A_{1}. Then O_{μ} has finite representation type.

[figure: P (A); A; B; C; P (B); B; A; C; B; C; P (C); C; B; A; C; B; C] Figure 10

[figure: •; A; •; B; •; C; α_{1}; β_{1}; α_{2}; β_{2}] Figure 11

Proof. Follows from [Fi]. _{□}

6.0 Tame type:

We will consider next the case when 𝔤 is of type A_{3} and Φ_{μ} is of type A_{2}. We may assume that Π = {α_{1}, α_{2}, α_{3}} with Φ_{μ} = {±α_{1}, ±α_{2}, ±(α_{1} + α_{2})}. Set A = L(λ), B = L(s_{α_{3}}.λ), C = L((s_{α_{1}}s_{α_{3}}).λ) and D = L((s_{α_{2}}s_{α_{1}}s_{α_{3}}).λ). The Verma modules are all uniserial and have the following structures.

M(λ)  M(s_{α_{3}}.λ):  M((s_{α_{1}}s_{α_{3}}).λ):  M((s_{α_{2}}s_{α_{1}}s_{α_{3}}).λ):
A  B  C  D
B  C  D
C  D
D

The structures of the projectives can be obtained by using reciprocity. These structures are given via the module diagrams in Figure 7. The Ext^{1}-quiver is given in Figure 8, with relations α_{2}β_{2} = β_{1}α_{1}, β_{2}α_{2} = α_{3}β_{3}, α_{1}β_{1} = 0.

Remark . Let e = 1_{B} +1_{D}. One can quickly observe that eO_{μ}e has infinite representation type (and also O_{λ}) because eO_{μ}e has a subquiver of the form given in Figure 9.

[figure: P (A); B; C; D; A; P (B); B; A; C; B; D; C; D; P (C); C; B; D; A; C; B; D; C; D; P (D); D; C; B; D; A; C; B; D; C; D] Figure 12

[figure: •; A; •; B; •; C; •; D; α_{1}; β_{1}; α_{2}; β_{2}; α_{3}; β_{3}] Figure 13

[figure: •; eX; •; eZ] Figure 14

We will use the covering technique to investigate the type of our algebra. The covering technique (Galois covering) was introduced by Bongartz and Gabriel [BG] and then developed by Gabriel [Ga], Riedtmann [Ri], Martinez and de la Peña [MP] for the study of algebras of finite type. The Galois coverings of algebras of infinite type were studied in [DS1, DS2 ] were the classification problem of indecomposables was reduced to the corresponding problem for locally bounded categories.

Our algebra Λ is a bound quiver algebra KQ/I where Q is given on Figure 8 and ideal I is generated by α_{2}β_{2} −β_{1}α_{1}, β_{2}α_{2} −α_{3}β_{3}, α_{1}β_{1}. Then Λ admits a strongly simply connected Galois covering Φ : R → R/G ≃ Λ where R = KQ̃/Ĩ is a locally bounded K-category defined by the quiver Q̃ on Figure ?, Ĩ is generated by all elements of the form α_{2}β_{2}−β_{1}α_{1}, β_{2}α_{2}−α_{3}β_{3}, α_{1}β_{1} and G is the infinite cyclic group generated by the obvious shift g of R. Since R does not contain a convex subcategory which is hypercritical ([Le], [Un], [Wi]) or pg-critical ([NS]) it implies that Λ is of tame type by [Sk, Theorem 2.4].

???Picture

6.0 Main result: The following theorem provides a complete classification of the representation type of the blocks for category O.

Theorem . Let 𝔤 be a complex simple finite-dimensional Lie algebra and λ ∈ X_{+}.

(a) O_{μ} is semisimple if and only if Φ_{μ} = Φ;

(b) O_{μ} has finite representation type if and only if 𝔤 is of type A_{1} and Φ_{μ} = ∅ or 𝔤 is of type A_{2} and Φ_{μ} is of type A_{1};

(c) O_{μ} has tame representation type if and only if 𝔤 is of type A_{3} and Φ_{μ} is of type A_{2}. In all other cases, the block O_{μ} has wild representation type.

Proof. (a) This is immediate because the largest Verma module P (λ) = M(λ) in O_{μ} has |W |/|Stab_{W} μ| composition factors, and |W |/|Stab_{W} μ| = 1 if and only if Φ_{μ} = Φ.

???(b) Suppose that rank(𝔤)−rank(Φ_{μ}) ≥ 2. Then O_{μ} has wild representation type by Proposition 4.3. Now suppose that rank(𝔤) − rank(Φ_{μ}) = 1. If Φ_{μ} is not irreducible then O_{μ} has wild representation type by Proposition 5.2. On the other hand, if Φ_{μ} is irreducible then O_{μ} has wild representation type unless 𝔤 is of type A_{1} or A_{2} by Proposition 5.6. In the case when 𝔤 is of type A_{1}, O_{μ} has finite representation type by Proposition 6.1. If 𝔤 is of type A_{2} then O_{μ} has tame representation type by Proposition 6.2. Finally note that if rank(𝔤) − rank(Φ_{μ}) = 0, then Φ = Φ_{μ} and this case is covered in part (a). _{□}

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Department of Mathematics, University of São Paulo, Caixa Postal 66281, CEP 05351-970, São Paulo, Brasil

   E-mail address: futorny@ime.usp.br

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

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

Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario, Canada

   E-mail address: pollackd@post.queensu.ca