Reduction Method for Representations of Queer Lie Superalgebras 3

Home , Superalgebra

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BGG category for queer Lie superalgebra. In particular, we will prove equivalence of categories between certain blocks for q(n) and gl(ℓ|n − ℓ). Throughout the paper we denote by g the queer Lie superalgebra q(n) with the standard Cartan subalgebra h for a fixed integer n ≥ 1. Let Og denote the BGG category (see, e.g., [Fr, Section 3]) of finitely generated g-modules which are locally Og finite b-modules and semisimple h0¯-modules. Note that morphisms in are even. For a finite direct sum of queer Lie superalgebras and reductive Lie algebras, we have analogous notation of its BGG category. Let m ∈ Z+, 0 ≤ ℓ ≤ m and s ∈ C. If m ≥ 1, Cm let Λsℓ (m) ⊂ :

(1) λi ≡ s mod Z for 1 ≤ i ≤ ℓ, (1.1) Λ ℓ (m) := λ = (λ , . . . , λ ) | . s 1 m (2) λ ≡−s mod Z for ℓ + 1 ≤ i ≤ n. i ∗ We define q(0) and Λsℓ (0) to be 0 and the empty set, respectively. For each λ ∈ h0¯, we shall assign a specific irreducible module L(λ) of highest weight λ and then define Og the corresponding block λ, see the definitions in Section 2.2. The following theorem is the first main result of this paper.

∗ Og Ol Theorem 1.1. Let λ ∈ h0¯. Then λ is equivalent to a block µ of a Levi subalgebra k l = q(n1) × q(n2) ×···× q(nk) ⊆ g with i=1 ni = n and the weight µ of the form

(1.2) µ ∈ Λ ℓ1 (n1) × Λ ℓP2 (n2) ×···× Λ ℓk (nk), s1 s2 sk such that si 6≡ ±sj mod Z for all i 6= j.

Accordingly, the study of blocks of Og is reduced to the study of blocks of the following three types: (i) (s = 0) a BGG category On,Z of the q(n)-modules of integer Z 1 O weights, see, e.g., [Br2]. (ii) (s ∈ + 2 ) a BGG category n, 1 +Z of the q(n)-modules Z 2 O of half-integer weights, see, e.g., [CK], [CKW]. (iii) (s∈ / /2) a BGG category n,sℓ of the q(n)-modules of ”±s-weights”, see the deﬁnition in Section 4.2.

1.2. Let gl(ℓ|n − ℓ) be the general linear Lie superalgebra with the standard Cartan subalgebra hℓ|n−ℓ for 1 ≤ ℓ ≤ n. Another main result of the present paper is to establish an equivalence between a block Fλ of certain maximal parabolic categories F for q(n) and certain block of atypicality-one of the finite-dimensional module category Fℓ|n−ℓ for gl(ℓ|n − ℓ), see the definitions in Sections 4.1 and 4.2. Their identical linkage principle (see Lemma 4.1) is the first piece of evidence to support such an equivalence. ∗ ∗ For a weight λ ∈ h0¯, or λ ∈ hℓ|n−ℓ, we denote by ♯λ the atypicality degree of λ (see, e.g. [CW, Definitions 2.29, 2.49]). According to [Ser98, Theorem 2.6] and F [BS12, Theorem 1.1] the blocks ( ℓ|n−ℓ)λ (see Section 4.1) for all ℓ,n − ℓ with the same ♯λ are equivalent. More precisely, the endomorphism ring of projective generator F ∞ of ( ℓ|n−ℓ)λ is isomorphic to the opposite ring of the diagram algebra K♯λ (see, e.g., ∞ [BS12, Introduction]). In particular, K1 is the path algebra of the infinite quiver REDUCTION METHOD FOR REPRESENTATIONS OF QUEER LIE SUPERALGEBRAS 3

xi−1 xi xi+1 ···•←−−→ •←−−→ • ←−−→ • ··· yi−1 yi yi+1 modulo the relations xi ·yi = yi−1 ·xi−1 and xi ·xi+1 = yi+1 ·yi = 0 for all i ∈ Z. We are now in a position to state the following theorem, which provides a Morita equivalence between Fλ and (F1|1)0. Theorem 1.2. Let L(λ) ∈ F with ♯λ = 1. Then the endomorphism ring of the projec- F ∞ op tive generator of λ is isomorphic to (K1 ) . 1.3. The paper is organized as follows. In Section 2, we recall deﬁnitions of queer Lie superalgebras, general linear Lie superalgebras and their categories of modules. In Section 3, an approach of reduction similar to [CMW] is established for queer Lie superalgebras. Equivalences of blocks via twisting functors and parabolic induction functors are established. In addition, a description of decomposition of blocks of O is given in Theorem 3.8. In Section 4, we recall the category of ﬁnite-dimensional modules for gl(ℓ|n − ℓ) and introduce certain maximal parabolic category for q(n). A correspondence preserving linkage principles between their irreducibles is established. Finally, we compute the endomorphism ring of projective generator to obtain Theorem 1.2.

Acknowledgments. The author is very grateful to Shun-Jen Cheng for numerous helpful comments and suggestions.

2. Preliminaries 2.1. Lie superalgebras gl and q. For positive integers m,n ≥ 1, let Cm|n be the complex superspace of dimension (m|n). Let {v1¯, . . . , vm¯ } be an ordered basis for the m|0 0|n even subspace C and {v1, . . . , vn} be an ordered basis for the odd subspace C so that the general linear Lie superalgebra gl(m|n) may be realized as (m + n) × (m + n) complex matrices indexed by I(m|n) := {1¯ < · · · < m<¯ 1 < · · ·

Let Eab be the elementary matrix in gl(m|n) with (a, b)-entry 1 and other entries 0, for a, b ∈ I(m|n). Then {eij, e¯ij|1 ≤ i, j ≤ n} is a linear basis for g, where eij = E¯i¯j +Eij ande ¯ij = E¯ij + Ei¯j. Note that the even subalgebra g0¯ is spanned by {eij |1 ≤ i, j ≤ n}, which is isomorphic to the general linear Lie algebra gl(n). 4 CHEN

∗ Let hm|n and hm|n be respectively the standard Cartan subalgebra of gl(m|n) and its dual space, with linear bases {Eii|i ∈ I(m|n)} and {δi|i ∈ I(m|n)} such that δi(Ejj)= δi,j. Let h = h0¯ ⊕ h1¯ be the standard Cartan subalgebra of g, with linear bases {hi := ¯ eii|1 ≤ i ≤ n} and {hi :=e ¯ii|1 ≤ i ≤ n} of h0¯ and h1¯, respectively. Let {εi|1 ≤ i ≤ n} ∗ be the basis of h0¯ dual to {hi|1 ≤ i ≤ n}. We define a symmetric bilinear form (, ) on ∗ h0¯ by (εi, εj)= δij, for 1 ≤ i, j ≤ n. We denote by Φ, Φ0¯, Φ1¯ the sets of roots, even roots and odd roots of g, respectively. + + + Let Φ = Φ0¯ ⊔ Φ1¯ be the set of positive roots with respect to its standard Borel subalgebra b = b0¯ ⊕ b1¯, which consists of matrices of the form (2.2) with A and B upper triangular. Denote the set of negative roots by Φ− := Φ \ Φ+. Ignoring the + parity we have Φ0¯ =Φ1¯ = {εi − εj|1 ≤ i, j ≤ n} and Φ = {εi − εj|1 ≤ i < j ≤ n}. We ∗ + denote by ≤ the partial order on h0¯ defined by using Φ . The Weyl group W of g is defined to be the Weyl group of the reductive Lie algebra g0¯ and hence acts naturally ∗ + on h0¯ by permutation. We also denote by sα the reflection associated to a root α ∈ Φ . ∗ For a given root α = εi − εj ∈ Φ, letα ¯ := εi + εj. For each λ ∈ h0¯, we have the integral root system Φλ := {α ∈ Φ|(λ, α) ∈ Z} and the integral Weyl group Wλ defined to be the subgroup of W generated by all reflections sα, α ∈ Φλ.

2.2. Categories of modules. Let V = V0 ⊕ V1 be a superspace. For a given homoge- nous element v ∈ Vi (i ∈ Z2), we let v= i denote its parity. Let Π denote the parity change functor on the category of superspaces. Let Π0 be the identity functor. For a ∗ g-module M and µ ∈ h0¯, let Mµ := {m ∈ M|h · m = µ(h)m, for h ∈ h0¯} denote its µ-weight space. If M has a weight space decomposition M = ⊕ h∗ M , its character is µ∈ 0¯ µ µ given as usual by chM = h∗ dimMµe , where e is an indeterminate. In particular, µ∈ 0¯ we have the root space decomposition g = h ⊕ (⊕ g ) with respect to the adjoint P α∈Φ α representation of g. n ∗ ∗ Let λ = i=1 λiεi ∈ h0¯, and consider the symmetric bilinear form on h1¯ deﬁned by h·, ·i := λ([·, ·]). Let ℓ(λ) be the number of i’s with λ 6= 0 and δ(λ) = 0 (resp. λ P i δ(λ) = 1) if ℓ(λ) is even (resp. odd). Let 1 ≤ i1 < i2 < · · · < iℓ(λ) ≤ n such that

λi1 , λi2 , . . . , λiℓ(λ) are non-zero. Denote by ⌈·⌉ the ceiling function. Then the space

−λ ′ C ℓ(λ)−⌈ℓ(λ)/2⌉C i2k−1 (2.3) h1¯ := ⊕j6=i1,...,iℓ(λ) hj ⊕ ⊕k=1 (hi2k−1 + hi2k ) , λi2k ! p ′ p ′ C is a maximal isotropic subspace of h1¯ associated to h·, ·iλ. Put h = h0¯ ⊕ h1¯. Let vλ be ′ ¯ ′ the one-dimensional h -module with vλ = 0, h · vλ = λ(h)vλ and h · vλ = 0 for h ∈ h0¯, ′ ′ h C ⌈ℓ(λ)/2⌉ h ∈ h1¯. Then Iλ := Indh′ vλ is an irreducible h-module of dimension 2 (see, g e.g., [CW, Section 1.5.4]). We let M(λ) := IndbIλ be the Verma module, where Iλ is extended to a b-module in a trivial way, and deﬁne L(λ) to be the unique irreducible quotient of M(λ). REDUCTION METHOD FOR REPRESENTATIONS OF QUEER LIE SUPERALGEBRAS 5

Let Og denote the BGG category (see, e.g., [Fr, Section 3]) of ﬁnitely generated g- modules which are locally ﬁnite over b and semisimple over h0¯. Note that the morphisms in Og are even. It is known (see, e.g., [CW, Section 1.5.4]) that L(λ) =∼ ΠL(λ) if and only if δ(λ) = 1. Therefore we have the following.

∗ ∗ Lemma 2.1. {L(λ)|λ ∈ h0¯ with δ(λ) = 1} ∪ {L(λ), ΠL(λ)|λ ∈ h0¯ with δ(λ) = 0} is a complete set of irreducible g-modules in Og up to isomorphism.

We denote by Z(g) the center of U(g). As in the case of Lie algebras, the BGG category Og of g has a decomposition into subcategories corresponding to central characters C ∗ χλ : Z(g) → for λ ∈ h0¯. We have a reﬁned decomposition by the linkage principle (see, e.g., [CW, Section 2.3]) Og Og (2.4) = λ, λ∈h∗/∼ M0¯ ∗ where the equivalence relation ∼ on h0¯ is deﬁned by

(2.5) λ ∼ µ if and only if χλ = χµ and µ ∈ λ + ZΦ, Og Og and λ is the Serre subcategory of generated by simple objects with highest weight Og µ such that λ ∼ µ. The subcategories λ are decomposable in general. For a ﬁnite direct sum of queer Lie superalgebras and reductive Lie algebras, we have analogous notation and decomposition of its BGG category. When there is no Og O ∗ O confusion, we denote by . For λ ∈ h0¯, denote the block of containing L(λ) by Oλ. Namely, it is the Serre subcategory generated by the set of vertices in the connected component of the Ext-quiver for O containing L(λ).

3. Equivalences and Reductions for Blocks of Queer Lie Superalgebra 3.1. Equivalence using twisting functors. For a simple root α ∈ Φ+, we can defined the twisting functor Tα associated to α. The twisting functor was originally defined by Arkhipov in [Ar] and further investigated in more detail in [AS], [KM], [CMW], [AL], [MS], [GG13], [KM]. Recall the precise definition of Tα as follows. First, fix a non-zero root vector X ∈ (g0¯)−α. Since the adjoint action of X on g is nilpotent, by using a standard argument (see e.g. [MO00, Lemma 4.2]) we can form the Ore localization ′ Uα of U(g) with respect to the set of powers of X. Since X is not a zero divisor in ′ ′ U(g), U(g) can be viewed as an associative subalgebra of Uα. The quotient Uα/U(g) has the induced structure of a U(g)-U(g)-bimodule. Let ϕ = ϕα be an automorphism ¯ ¯ of g that maps (gi)β to (gi)sα(β) for all simple root β and i ∈ {0, 1}. Finally, consider ϕ the bimodule Uα, which is obtained from Uα by twisting the left action of U(g) by ϕ. We also have an analogous construction with respect to the subalgebra g0¯ to obtain ϕ 0¯ the U(g0¯)-U(g0¯)-bimodule Uα. Now we are in a position to define twisting functors: ¯ ¯ ϕ Og Og 0 ϕ 0 Og0¯ Og0¯ (3.1) Tα(−) := Uα ⊗− : → and Tα(−) := Uα ⊗− : → . 6 CHEN

0¯ 0¯ Then Tα and Tα have right adjoints Kα and Kα, respectively (see, e.g., [AS]). Let Db(O) and Db(Og0¯ ) be the bounded derived categories of O and Og0¯ , respectively. It 0¯ L L 0¯ is not hard to prove that Tα and Tα are right exact functors. Let iTα, iTα the 0¯ L 0¯ i-th left derived functors of Tα, Tα, respectively. It was proved in [AS] that iTα = 0 L 0¯ for i > 1 and 1Tα is isomorphic to the functor of taking the maximal submodule on which the action of g−α is locally nilpotent. Similarly, we have analogous deﬁnition for Ri Ri 0¯ 0¯ right derived endofunctors Kα and Kα of Kα and Kα, respectively. Furthermore, Ri 0¯ R1 0¯ Kα = 0 for i > 1 and Kα is isomorphic to the functor of taking the maximal subquotient on which the action of g−α is locally nilpotent. The star action ∗ of sα on weights had been introduced in [GG13, Introduction] and [CM]: sα ∗ λ := sαλ if (λ, α¯) 6= 0 and sα ∗ λ := sαλ − α if (λ, α¯) = 0. We call the former an α-typical weight and the later an α-atypical weight (also see [GG13, Section 1.2.3]). The following theorem is inspired by [CM, Proposition 8.6]. ∗ + Z Theorem 3.1. Let λ ∈ h0¯ and α ∈ Φ be a simple root such that (λ, α) ∈/ . Then i O O j O O Π ◦ Tα : λ → sαλ is an equivalence with inverse Π ◦ Kα : sαλ → λ for some i, j ∈ {0, 1}. O O Proof. We claim that Tα and Kα are exact functors on λ and sαλ, respectively. To L 0¯ R1 0¯ see this, we ﬁrst note that 1Tα and Kα vanish at each simple g0¯-module of highest Z g L weight µ with (µ, α) 6∈ (e.g., [Mar, Chapter 3]). Next we recall that Resg0¯ ◦ iTα = L 0¯ g g Ri Ri 0¯ g iTα ◦ Resg0¯ and Resg0¯ ◦ Kα = Kα ◦ Resg0¯ (e.g. [CM, Lemma 5.1]) for all i ≥ 0. L Ri ′ O ′ O This means that iTαM = KαM = 0 for all M ∈ λ, M ∈ sαλ and i ≥ 1. As O O ∗ a conclusion, Tα and Kα are exact functors on λ and sαλ, respectively. For µ ∈ h0¯ Z 2 with (µ, α) 6∈ , it is proved in [CM, Lemma 5.8] that TαL(µ) is simple with TαL(µ) ∈ {L(µ), ΠL(µ)}. By a similar argument we can show that KαL(µ) is also simple with 2 O KαL(µ) ∈ {L(µ), ΠL(µ)}. That is, that Tα and Kα preserve simple objects of λ and O sαλ, respectively. Finally recall that chTαM(µ) = chM(sαµ) [CM, Lemma 5.5] for ∗ O ′ O ′ all µ ∈ h0¯. From this together with the fact that Hom (TαL,L ) = Hom (L, KαL ) ′ for all simple objects L,L ∈ O, we conclude that Tα sends objects of Oλ to objects of O O iO sαλ and Kα sends objects of sαλ to objects of Π λ, for some i = 0, 1. Consequently, O iO the restrictions of Tα and Kα make Tα : λ → Π sαλ an equivalence with inverse O jO Kα : sαλ → Π λ, for some i, j ∈ {0, 1}.

Remark 3.2. Let λ, α be as in Theorem 3.1. It is worth pointing out that TαL(λ) only depends on whether λ is α-typical or α-atypical. That is, it was determined in [CM, Corollary 8.15]: By the classiﬁcation of simple q(2)-highest weight modules in [Mar10] i we have [TαL(λ) : Π L(sα ∗ λ)] 6= 0, for some i = 0, 1. This is also proved in [GG13, i Proposition 4.7.1]. As a consequence, we have TαL(λ) = Π L(sα ∗ λ) for some i = 0, 1. Example 3.3. Let n = 3 and λ := (−π,π, −π). Then by Theorem 3.1 and Remark i O Oe 3.2, Tǫ1−ǫ2 : λ → λ is an equivalence sending L(λ) to Π L(λ − (ǫ1 − ǫ2)) for some i = 0, 1, where λ = (π, −π, −π) . e e REDUCTION METHOD FOR REPRESENTATIONS OF QUEER LIE SUPERALGEBRAS 7

3.2. Equivalence using parabolic induction functor. The goal of this section is to show that the parabolic induction functors give equivalences of blocks under some suitable condition. For given integers ℓ,m with 1 ≤ ℓ ≤ m and s ∈ C, recall the set O ∗ Λsℓ (m) defined in 1.1. In this section, we consider blocks λ with the weight λ ∈ h0¯ of the following form 1 Z (3.2) λ ∈ Λ ℓ1 (n1) ×···× Λ ℓk (nk) such that s1 = 0, s2 = and si 6≡ ±sj mod , s1 sk 2 for all i 6= j. We define Φ := {α ∈ Φ|(λ, α¯) ∈ Z} Φ and l := h ⊕ ⊕ g to be λ λ λ α∈Φλ α the Levi subalgebra associated to λ. In this case, we denote by u the corresponding S λ nilradical. Furthermore, we have isomorphisms ∼ (3.3) Wλ = Sn1 × Sn2 × (Sℓ3 × Sn3−ℓ3 ) ×···× (Sℓk × Snk−ℓk ), ∼ and lλ = q(n1) × q(n2) ×···× q(nk). In order to prove that the parabolic induction functors are equivalences in this setting, we first recall the following well-known characterization of central characters (see, e.g., [CW, Theorem 2.48]). ∗ C Lemma 3.4. For λ, µ ∈ h0¯, χλ = χµ if and only if there exist w ∈ W , {kj }j ⊂ , and a subset of mutually orthogonal roots {αj}j such that µ = w(λ − j kjαj) and (λ, α ) = 0 for all j . j P ∗ ∗ Define a relation ≈ on h0¯ as follows. For λ, µ ∈ h0¯ we let λ ≈ µ if there exist w ∈ Wλ, Z {kj }⊂ , and a subset of mutually orthogonal roots {αj} such that µ = w(λ− j kjαj) and (λ, α ) = 0 for all j. The following lemma shows that ∼ and ≈ coincide in our j P setting.

∗ Lemma 3.5. Let λ ∈ h0¯ be of the form (3.2). Then µ ∼ λ if and only if µ ≈ λ. In i particular, if Π L(µ) ∈ Oλ, for some i = 1, 2, then µ ≈ λ. C Proof. Since χλ = χµ we have µ = w(λ − j kjαj) for some w ∈ W , {kj }j ⊂ and {α } ⊂ Φ such that (λ, α ) = 0 for all j by Lemma 3.4. Furthermore, we have j j j P λ ∈ µ + ZΦ. It follows that w ∈ Wλ and kj ∈ Z for all j. This completes the proof. The following theorem is inspired by [CMW, Proposition 3.6].

∗ Theorem 3.6. Let λ ∈ h0¯ be of the form (3.2). Let l := lλ, u := uλ. Then there i g Ol O are i, j ∈ {0, 1} such that the parabolic induction functor Π ◦ Indl+u : λ → λ is an j g O Ol u equivalence, with inverse equivalence Π ◦ Resl : λ → λ deﬁned by M 7→ M , where M u is the maximal trivial u-submodule of M.

g 0 Proof. As in the proof of [CMW, Propositon 3.6], it suﬃces to show that Indl+uLµ is 0 Ol irreducible for each irreducible l-module Lµ ∈ λ of highest weight µ. We ﬁrst assume ∗ g 0 that ζ ∈ h0¯ is a weight of a non-zero singular vector in Indl+uLµ. Then by Lemma 3.5 there exist w ∈ Wµ, {kj}j ⊂ Z, and a subset of mutually orthogonal roots {αj}j such that ζ = w(µ − j kjαj) and (µ, αj ) = 0 for all j (note that Φλ = Φµ). On the P 8 CHEN

g 0 Z other hand, by consideration of the weights of Indl+uLµ, we have ζ ∈ µ − α∈Φ+ ≥0α. g 0 Hence ζ ∈ µ− + Z≥0α by (3.3). This means that every subquotient of Ind L α∈Φµ∩Φ P l+u µ intersects L0 and so Indg L0 is irreducible. This completes the proof. µ P l+u µ ∗ Proof of Theorem 1.1. Let λ ∈ h0¯. We can ﬁrst apply a sequence of suitable twisting O Oe functors (see Theorem 3.1) to λ and obtain an equivalent block λ such that λ ∈ Z Λ ℓ1 (n1) × Λ ℓ2 (n2) ×···× Λ ℓk (nk) and si 6≡ ±sj mod for all i 6= j. Next we can s1 s2 sk apply the parabolic induction functor (see Theorem 3.6) to obtain an equivalent blocke of the desired Levi subalgebra. This completes the proof.

1 3 3 Example 3.7. Let λ := ( 5 , 1, −π, 2 , π, − 2 , −π). Then by applying a sequence of iα twisting functors Π ◦ Tα with some iα ∈ {0, 1} in Theorem 3.1 related to α-typical 1 3 3 weights, we may transform λ to the weight λ = (1, 5 , 2 , − 2 , −π,π, −π), which gives O Oe an equivalence from λ to λ sending L(λ) to L(λ). Then we apply the twisting e functor Πi ◦ T with some i = 0, 1 to obtain the weight λ = (1, 3 , − 3 , 1 , π, −π, −π) ε5−ε6 e 2 2 5 O O and an equivalence λ to e which sends L(λ) to L(λ − (ε5 − ε6)). Next we use λ e the parabolic induction functors. Deﬁne α := (ε5 − eε6). Note that λ, λ − α ∈ ∼ Λ00 (1) × Λ 1 1 (2) × Λ 1 0 (1) × Λπ1 (3) and le = le = q(1) × q(2) × q(1) × q(3). By 2 5 λ−α λ e e O Ol e e Theorem 3.6, there is an equivalence from λ to e sending L(λ) to the irreducible λ l-module with highest weight λ − α.

3.3. Description of blocks.e ∗ i O Theorem 3.8. Let λ, µ ∈ h0¯. Then Π L(µ) ∈ λ for some i = 0, 1 if and only if µ ≈ λ.

∗ Proof. First assume that λ ∈ h0¯ is of the form (3.2). Thanks to Lemma 3.5, it remains i to show that µ ≈ λ implies Π L(µ) ∈ Oλ for some i = 0, 1. Recall the fundamental lemma in [PS2, Proposition 2.1] by Penkov and Serganova. It follows from Homg(M(λ− j + i α), Π M(λ)) 6= 0 for some j = 0, 1, for all α ∈ Φ with (λ, α) = 0 that Π L(λ−α) ∈ Oλ for some i = 0, 1. Therefore we may assume that µ is of the form s(λ), for some reﬂection s ∈ Wλ corresponding to a simple root εi − εi+1. In this case, we have λi − λi+1 = k ∈ Z. Without loss of generality, assume that k > 0. Let vλ ∈ M(λ) bea k+1 highest weight vector, it is not hard to compute that ei,i+1ei+1,ivλ is a singular vector in i M(λ) of weight s(λ) (see, e.g., [CW, Lemma 2.39]). This means that Π L(µ) ∈ Oλ for ∗ ′ ∗ O O ′ some i = 0, 1. For arbitrary λ ∈ h0¯, there are λ ∈ h0¯ of the form (3.2) and T : λ → λ an equivalence constructed by using a sequence of twisting functors in Theorem 3.1. ′ ∗ ′ ′ For ζ,ζ ∈ h0¯ and simple reﬂection s ∈ W , note that s ∗ ζ ≈ s ∗ ζ if and only if ζ ≈ ζ . The theorem now follows by Remark 3.2.

Remark 3.9. If ℓ(λ) is odd, then Oλ is the Serre subcategory generated by {L(µ)|µ ≈ λ}. REDUCTION METHOD FOR REPRESENTATIONS OF QUEER LIE SUPERALGEBRAS 9

4. Equivalences of Certain Maximal Parabolic Subcategory In this section, we ﬁx non-negative integers n,ℓ with n ≥ ℓ and s 6∈ Z/2. The goal of this section is to establish an equivalence between certain block of atypicality-one of ﬁnite-dimensional category for gl(ℓ|n − ℓ) and some block of certain maximal parabolic subcategory for q(n).

4.1. Finite-dimensional representations of gl(ℓ|n − ℓ). We denote by Fℓ|n−ℓ the category of integral weight, ﬁnite-dimensional gl(ℓ|n−ℓ)-modules with even morphisms. a n Z e Let Λ := ⊕i=1 δi be the weight lattice. Recall that the set of all irreducible objects (up F a,+ a to parity) of ℓ|n−ℓ are parametrized by its highest weight λ in Λ := {λ ∈ Λ | λi ≥ n λi+1, for 1 ≤ i<ℓ and ℓ ≤ i

(resp. ΠFℓ|n−ℓ) is the full subcategory consisting of all M ∈ Fℓ|n−ℓ such that M = M+ e a,+ F F (resp. M = M−). For ζ ∈ Λ , denote by ( ℓ|n−ℓ)ζ the block of ℓ|n−ℓ containing the a e (unique) irreducible module Lζ of highest weight ζ. Namely, it is the Serre subcategory generated by the set of vertices in the connected component of the Ext-quiver for Fℓ|n−ℓ a containing Lζ . ∞ As we mentioned in Section 1, the diagram algebra K1 is the path algebra of a ∞ op certain inﬁnite quiver. Therefore we can identify (K1 ) as the associative algebra generated by elements {zi,xj,yk}i,j,k∈Z and relations

zic = czi = yiyj = xjxi = 0,

xiyi = zi+1, yixi = zi,

for all i, j ∈ Z, c ∈ {xs,yt}s,t∈Z.

4.2. Parabolic categories of q(n) and Equivalences. We define a bijection ·a : a Λsℓ (n) → Λ by n ℓ n a a (4.1) λ = λiεi ∈ Λsℓ (n) 7−→ λ := (λi − s)δi + (λi + s)δi − ρ ∈ Λ , Xi=1 Xi=1 i=Xℓ+1 ℓ n where ρ := i=1 −(ℓ − i + 1)δi + i=ℓ+1(i − ℓ)δi. Let χa be the central character of gl(ℓ|n − ℓ) corresponding to λ ∈ h∗ . We first λ P P ℓ|n−ℓ consider the linkage principles under this bijection. Denote by Ψ := {δi − δj|1 ≤ i 6= j ≤ n} the root system of gl(ℓ|n − ℓ). Recall that the linkage principle in (2.5) for q(n) defines an equivalence relation ∼. The following lemma follows from Lemma 3.5 and the proof of [CMW, Proposition 3.3]. a a a a Z Lemma 4.1. Let λ, µ ∈ Λsℓ (n). Then λ ∼ µ if and only if χλa = χµa and µ ∈ λ + Ψ. 10 CHEN

+ We define the set Λsℓ (n) := {λ ∈ Λsℓ (n)| λi > λi+1, for 1 ≤ i<ℓ and ℓ ≤ i < a,+ + a n}. Note that we have Λ = (Λsℓ (n)) . For arbitrary λ ∈ Λsℓ (n) we have a Levi ∼ subalgebra l := h⊕(⊕α∈Φλ gα) = q(ℓ)×q(n−ℓ) and the maximal parabolic subalgebras p + O p := h ⊕ ⊕α∈Φλ∪Φ gα . Let u be the corresponding nilradical of p. We denote by the maximal parabolic subcategory of (see, e.g., [Mar14, Section 3.1]) O with respect Op O to p. Namely, is the Serre subcategory of generated by p-locally finite, and l0¯- O O semisimple g-modules. We define n,sℓ to be the full subcategory of g-modules in F Op O with weights in Λsℓ (n) and := ∩ n,sℓ its maximal parabolic subcategory. For each M ∈ F, note that M is also l-semisimple since all weights of M are l-typical. As i F + a conclusion, if Π L(µ) ∈ for some i = 0, 1 then we have µ ∈ Λsℓ (n). + l Let λ ∈ Λsℓ (n). Note that every irreducible -module of highest weight λ can be extended to a p-module by letting u act trivially. We define L0(λ) to be the finite- dimensional irreducible l-module with highest weight space Iλ. Therefore the corre- g 0 sponding parabolic Verma module K(λ) := IndpL (λ) has the irreducible quotient L(λ). Furthermore, we note that K(λ) is p-locally finite and all the l-weights of K(λ) are l-typical. Therefore we have K(λ),L(λ) ∈ F. Consequently, F is the Serre subcat- O i + egory of generated by {Π L(λ)|λ ∈ Λsℓ (n), i ∈ {0, 1}}. + For λ ∈ Λsℓ (n). We also denote by P (λ) and U(λ) the projective cover of L(λ) and the tilting module corresponding to λ in Op, respectively. For their definitions and existences, we refer to [Mar14, Proposition 1,7] and [Mar14, Theorem 2]. Note that all weights of P (λ),U(λ) are in Λsℓ (n) since they are indecomposable (by definition). That is, P (λ),U(λ) ∈ F. + Z Let P be the free abelian group on basis {ǫa}a∈ . Let wt(·) : Λsℓ (n) → P be the weight function defined by (c.f. [Br2, Section 2-c])

ℓ n

(4.2) wt(λ) := ǫλi−s + (−ǫ−(λi+s)). Xi=1 i=Xℓ+1

By Lemma 3.4, we have χλ = χµ if and only if wt(λ) = wt(µ). By (2.4), we have decomposition F = ⊕ h∗ F = ⊕ F according to central characters χ with wt(λ)= γ. λ∈ 0¯ χλ γ∈P γ λ Let Cn|n and (Cn|n)∗ be the standard representation and its dual, respectively. De- F F note the projection functor from to γ by prγ. We deﬁne the translation functors Ea, Fa : F → F as follows Cn|n ∗ Cn|n (4.3) Ea(M) := prγ+(ǫa−ǫa+1)(M ⊗ ( ) ), Fa(M) := prγ−(ǫa−ǫa+1)(M ⊗ ), for all M ∈ Fγ, γ ∈ P , a ∈ Z. For each a ∈ Z , it is not hard to see that both Ea and + Fa are exact and bi-adjoint to each other. We write λ →a µ if λ, µ ∈ Λsℓ (n) and there exists 1 ≤ i ≤ ℓ such that λi = µi − 1 = a + s or there exists ℓ + 1 ≤ i ≤ n such that λi = µi − 1= −a − 1 − s, and in addition, λj = µj for all i 6= j. We have the following lemma. REDUCTION METHOD FOR REPRESENTATIONS OF QUEER LIE SUPERALGEBRAS 11

Lemma 4.2. Let + . Then both and have ﬂags of parabolic λ ∈ Λsℓ (n) EaK(λ) FaK(λ) Verma modules and we have the following formula:

(4.4) chEaK(λ) = 2 chK(µ), µX→aλ (4.5) chFaK(λ) = 2 chK(µ). λX→aµ Proof. We ﬁrst note that we have n|n 0 n|n (4.6) K(λ) ⊗ C =∼ U(g) ⊗p (L (λ) ⊗ C ).

n|n This implies that K(λ) ⊗ C (and so the summand of FaK(λ)) has a ﬁltration whose ∼ non-zero subquotients are parabolic Verma modules. Note the l0¯ = gl(ℓ)×gl(n−ℓ) with n C ∗ n C (1) ℓ the Cartan subalgebra h0¯ = ⊕i=1 eii and its dual h0¯ = ⊕i=1 εi. Let ζ := i=1 λiεi and ζ(1) := n λ ε , and ρℓ , ρn−ℓ be Weyl vectors for the general linear Lie algebras i=ℓ+1 i i 0¯ 0¯ P gl(ℓ) and gl(n − ℓ), respectively. Then the module in (4.6) has character P n Cn|n εi (4.7) ch K(λ) ⊗ = 2 · D · e · s(1)λ−ρℓ · s (1) n−ℓ . 0¯ λ −ρ0¯ Xi=1 (1) ℓ where s(1)λ−ρℓ and s (1) n−ℓ are respectively Schur functions corresponding to λ−ρ¯ 0¯ λ −ρ0¯ 0 (1) n−ℓ and λ − ρ0¯ , respectively, and 1+ e−εi+εj 1+ e−εi+εj (4.8) D = 2⌈n/2⌉ · , 1 − e−εi+εj 1 − e−εi+εj 1≤Yi

(4.9) D · s(1)ζ−ρℓ · s (1) n−ℓ = chK(ζ), 0¯ ζ −ρ0¯ + for all ζ ∈ Λsℓ (n) (c.f. [Pe, Theorem 2] and [CW, Section 3.1.3]). By the Pieri formula (see, e.g., [Mac95, Lemma 5.16]), we may conclude that ch K(λ) ⊗ Cn|n = 2 chK(µ), where the summation is over µ such that µ = λ + ε , for some 1 ≤ i ≤ n, µ i and µ > · · · >µ , µ > · · · >µ . From the deﬁnition of F , we obtain (4.5). P 1 ℓ ℓ+1 n a Formula (4.4) can be obtained by a similar argument.

+ Let λ ∈ Λsℓ (n). It is not hard to prove that both EaU(λ) and FaU(λ) are direct sums of tilting modules (see, e.g., [Br1, Corollary 4.27]). Furthermore, we have the following lemma. Lemma 4.3. Let + . Then the multiplicity of each non-zero tilting summand λ ∈ Λsℓ (n) of EaU(λ) and FaU(λ) is even.

Proof. We ﬁrst let U(ν1),...,U(νp) be the (not necessarily distinct) direct summands of FaU(λ). Suppose on the contrary that [FaU(λ) : U(νi)] is odd for some 1 ≤ i ≤ p. ′ Without loss of generality, we let 1 ≤ p ≤ p such that [FaU(λ) : U(νi)] are odd for all ′ ′ ′ i ≤ p , and [FaU(λ) : U(νi′ )] are even for all i > p , and in addition, ν1 is a maximal 12 CHEN

′ ∗ element in {νi|1 ≤ i ≤ p }. Since the coeﬃcients of {chK(µ)|µ ∈ h0¯} in chFaU(λ) are even by Lemma 4.2, we have p′ p q

(4.10) chU(νi) = chFaU(λ) − chU(νi) = 2 chK(µi), ′ Xi=1 i=Xp +1 Xi=1 ∗ ∗ for some (not necessarily distinct) µ1,µ2,...,µq ∈ h0¯. For a given ν ∈ h0¯, note that Z chU(ν) ∈ chK(ν)+ ζ<ν ≥0chK(ζ). This means that the coefficient of chK(ν1) in ′ each chU(νj)’s is zero for all 1 ≤ j ≤ p with νj 6= ν1. But [FaU(λ) : U(ν1)] = |{j|1 ≤ ′ ′ L p j ≤ p , νj = ν1}| is the coefficient of chK(ν1) in i=1 chU(νi) contradicting (4.10). The result for E U(λ) can be obtained by a similar argument. This completes the a P proof. By Lemma 4.3 and algorithm in [Br1, Procedure 3.20], we have the following lemma: Lemma 4.4. Let + with . There is an operation consisting of first λ ∈ Λsℓ (n) ♯λ = 1 Zλ applying the operators Xa’s coming from Brundan’s procedure [Br1, Procedure 3.20] and then taking a direct summand of a direct sum of two isomorphic copies such that − (4.11) chZλU(tλ)=chK(λ)+chK(λ ), where + is typical, and − , where + with and tλ ∈ Λsℓ (n) λ = w(λ − kα) α ∈ Φ (λ, α) = 0 k ∈ N is the smallest positive integer such that λ − kα is W -conjugate to an element in + , and is such that + . Λsℓ (n) w ∈ W w(λ − kα) ∈ Λsℓ (n) Remark . + − + + 4.5 Note that the map λ ∈ Λsℓ (n) 7→ λ ∈ Λsℓ (n) define a bijection on Λsℓ (n). + + + − We denote by λ the unique element in Λsℓ (n) such that (λ ) = λ. Remark 4.6. Since the translation functors are exact and both bi-adjoint to each other, they preserve projective and injective modules. In particular, ZλU(tλ) is both a projective cover and an injective hull.

Let τ : g → g be the anti-automorphism on g deﬁned by τ(eij)= eji and τ(eij)= eji (see, e.g., [Br3, Example 7.10]). Then τ induces a contravariant auto-equivalence on O and F (see, e.g., [Hum08, Section 3.2] and [Ger98, Section 2.1]). For a given M ∈ F, let M τ be the image of τ. Since chL(λ)τ = chL(λ), we have L(λ)τ ∈ {L(λ), ΠL(λ)}.

Note that ℓ(λ)= n for each λ ∈ Λsℓ (n), we thus have the following lemma:

Lemma 4.7. ( [Fr, Lemma 7] ) Let n be even and λ ∈ Λsℓ (n). Then we have ΠL(λ) if n ≡ 2 mod 4, (4.12) L(λ)τ =∼ L(λ) if n ≡ 0 mod 4. The following corollary is an immediate consequence by Lemma 4.7. Corollary 4.8. Let M ∈ F. If n ≡ 0 mod 4 then M τ and M have identical set of composition factors. If n ≡ 2 mod 4 then M τ and ΠM have identical set of composition factors. REDUCTION METHOD FOR REPRESENTATIONS OF QUEER LIE SUPERALGEBRAS 13

For a given M ∈ O, denote by radM and socM the radical and socle of M, respectively. We now can prove the following BGG reciprocity. Lemma 4.9. BGG reciprocity Let + . Then we have ( ) λ, µ ∈ Λsℓ (n) (4.13) (P (λ) : ΠiK(µ)) = [ΠiK(µ) : L(λ)], . for i = 0, 1. Proof. ′ ′ + ′ ∼ We ﬁrst assume n is even. Let λ ,µ ∈ Λsℓ (n) be arbitrary. Recall that L(λ ) =6 ΠL(λ′) in this case. By a similar arguments of consideration on (highest) weights as in [Hum08, Theorem 3.3.(c),(d)] we ﬁrst have that i ′ ′ τ (4.14) ExtF(Π K(λ ), K(µ ) ) = 0, i ′ ′ τ ′ ′ (4.15) dimHomF(Π K(λ ), K(µ ) ) 6= 0 implies that λ = µ . Furthermore, by Lemma 4.7 we have ΠL(µ′) if n ≡ 2 mod 4, (4.16) socK(µ′)τ = L(µ′)τ =∼ L(µ′) if n ≡ 0 mod 4. By a similar proof as in [Hum08, Theorem 3.3(c)], we may conclude that 0 if λ′ 6= µ′, i ′ ′ τ (4.17) dimHomF(Π K(λ ), (K(µ )) )= i if λ′ = µ′ and n ≡ 2 mod 4,  ′ ′  1 − i if λ = µ and n ≡ 0 mod 4.

 0 if λ′ 6= µ′, i ′ ′ τ (4.18) dimHomF(Π K(λ ), (ΠK(µ )) )= i if λ′ = µ′ and n ≡ 0 mod 4,  ′ ′  1 − i if λ = µ and n ≡ 2 mod 4. Since P (λ) has a flag of parabolic Verma modules, as a conclusion, we have i τ dimHomF(Π P (λ), (ΠK(µ)) ) if n ≡ 2 − 2i mod 4, (4.19) (ΠiP (λ) : K(µ)) = dimHomF(ΠiP (λ), K(µ)τ ) if n ≡ 2i mod 4. + F i i F Recall that dimHom (Π P (λ), M) = [M : Π L(λ)] for all M ∈ and λ ∈ Λsℓ (n) (see, e.g, [Hum08, Section 3.9]). By Corollary 4.8, we have [(ΠK(µ))τ : ΠiL(λ)] = [K(µ) : ΠiL(λ)] for n ≡ 2 mod 4 (resp. [K(µ)τ : ΠiL(λ)] = [K(µ) : ΠiL(λ)] for n ≡ 0 mod 4). The proof of this lemma follows provided that n is even. Recall that L(λ′) =∼ ΠL(λ′) if n is odd. By a similar argument, the proof for odd n can be obtained. This completes the proof. + i Z Let λ ∈ Λsℓ (n) with ♯λ = 1. Define Λλ := {λ |i ∈ } by the recursive relation i+1 i + 0 λ = (λ ) and λ = λ. We define irreducible modules Lλi and parabolic Verma Z modules Kλi for i ∈ as follows. First, Lλ0 := L(λ) and Kλ0 := K(λ). Then Lλi for i 6= 0 is defined by the recursive relation coming from the short exact sequences:

(4.20) 0 → Lλi−1 → Kλi → Lλi → 0, 14 CHEN

Z for i ∈ . Let Pλi be the projective cover of Lλi . Then by Lemma 4.9, we have short exact sequence

(4.21) 0 → Kλi+1 → Pλi → Kλi → 0, for i ∈ Z. Let Fλ be the block of F containing L(λ). Namely, it is the Serre subcategory generated by the set of vertices in the connected component of the Ext-quiver for F containing L(λ). Corollary 4.10. Let + with . Then and are in diﬀerent λ ∈ Λsℓ (n) ♯λ = 1 L(λ) ΠL(λ) blocks if and only if n is even. Furthermore, Fλ is the Serre subcategory of F generated by {Lµ|µ ∈ Λλ}.

Proof. By Lemma 4.4 and Lemma 4.9, the set {Lλi ,Lλi ,Lλi+1 ,Lλi−1 } is the set of all composition factors of Pλi . Recall that j ∼ 2 j (4.22) ExtF(Lλi , Π L(µ)) = HomF(radPλi /rad Pλi , Π L(µ)), Z + Z for all i ∈ , j = 0, 1 and µ ∈ Λsℓ (n). This means that {Lλi |i ∈ } is the complete set F Z of irreducible objects of λ by (4.20). Since for each i ∈ the object Lλi is of highest i weight λ if and only if i = 0 (i.e. λ = λ), we may conclude that ΠL(λ) 6∈ Fλ if and only if n is even by Lemma 2.1.

Example 4.11. (The Ext-quiver for q(2)) Let n = 2. Let s∈ / Z/2, α := ε1 − ε2, Z + h C s ∈ s + and λ := sα ∈ Λs1 (2). In this case, we have K(λ)= M(λ) and Iλ = Indh′ vλ ′ C ′ is a two dimensional irreducible h-module with h = h0¯ ⊕ (h1 + h2), where h1¯ acts C one vλ trivially. Fore a given irreducible h-module V of h0¯-weight λ − α, we have that ∼ ∼ V = Iλ−α (resp. V = ΠIλ−α ) if and only (h1 + h2)V0¯ = 0 (resp. (h1 − h2)V0¯ = 0). It is not hard to compute that u := e21h1vλ − se21vλ is a singular vector in M(λ) with h1u = −h2u (see, e.g., [CW, Lemma 2.44(2)]). Note that u = 1, therefore we have short exact sequence e (4.23) 0 → ΠL(λ − α) → K(λ) → L(λ) → 0.

Thus, ExtF(L(λ), ΠL(λ − α)) 6= 0. By Lemma 4.9 we may conclude that P (λ) has a Verma ﬂag (4.24) 0 → ΠK(λ + α) → P (λ) → K(λ) → 0, and so {L(λ), L(λ), ΠL(λ+α), ΠL(λ−α)} is the complete set of composition factors k of P (λ). Therefore the set of objects of Fλ is {Π L(λ + kα)|k ∈ Z}. Furthermore, if we apply Π ◦ τ to the short exact sequence (4.23), then it follows that ExtF(L(λ − α),L(λ)) 6= 0. Replace λ by arbitrary λ + kα, we may conclude that the Ext-quivers of Fλ and that of the principal block (F1|1)0 for gl(1|1) are the same. By a similar argument, we may generalize results in Example 4.11. More precisely, + let τ := Π ◦ τ (resp. τ := τ) if n ≡ 2 mod 4 (resp. n ≡ 0 mod 4). Let λ ∈ Λsℓ (n) with ♯λ = 1. By applying τ to the non-trivial short exact sequence (4.20), it follows from e e e REDUCTION METHOD FOR REPRESENTATIONS OF QUEER LIE SUPERALGEBRAS 15

2 ∼ (4.22) that radP (λ)/rad P (λ) = Lλ+ ⊕Lλ− . As a consequence, we obtain the following lemma.

Lemma 4.12. Let + with . Then there are exactly four distinct proper λ ∈ Λsℓ (n) ♯λ = 1 submodules of P (λ): ∼ 2 ∼ (4.25) Aλ = Kλ+ , Bλ, radP (λ)= Aλ + Bλ, rad P (λ) = socP (λ) = Lλ. Furthermore, we have the following short exact sequences:

(4.26) 0 → Aλ → radP (λ) → Lλ− → 0,

(4.27) 0 → Bλ → radP (λ) → Lλ+ → 0,

(4.28) 0 → Lλ → Aλ → Lλ+ → 0,

(4.29) 0 → Lλ → Bλ → Lλ− → 0. We are now in a position to state the main theorem in this section.

+ Theorem 4.13. Let with . Then F is equivalent to F a . λ ∈ Λsℓ (n) ♯λ = 1 λ ( ℓ|n−ℓ)λ

Proof. Our goal is to prove that the endomorphism algebra EndFλ (⊕µ∈Λλ Pµ) of pro- F ∞ op i jective generator ⊕µ∈Λλ Pµ for λ is isomorphic to (K1 ) . We ﬁx µ = λ ∈ Λλ. ′ ′ + − It follows from Lemma 4.12 that EndFλ (Pµ, Pµ ) = 0 if µ 6∈ {µ ,µ,µ }. Since ′ ′ ′ dimHomFλ (Lµ,Lµ ) = δµ,µ for all µ,µ ∈ Λλ, it is not hard to prove that (see, e.g, [Hum08, Section 3.9]) 1 in case µ′ ∈ {µ+,µ−} , (4.30) dimHomF (P , P ′ ) = [P ′ : L ]= λ µ µ µ µ 2 in case µ′ = µ . ′ ′ + − We now construct bases for HomFλ (Pµ, Pµ ), for µ ∈ {µ,µ ,µ }:

(1). A Basis for HomFλ (Pµ, Pµ): First note that the canonical epimorphism Pµ →

Lµ gives an endomorphism of zi ∈ EndFλ Pµ by mapping Pµ onto socPµ ⊂ Pµ. Let

1i ∈ EndFλ Pµ be the identity map, then we may conclude that EndFλ Pµ is generated by 1i, zi by (4.30). e − (2). A Basis for HomFλ (Pµ, Pµ ): Next note the composition of homomorphisms e ∼ , −yi−1 : Pµ → Pµ/Aµ = Aµ− ֒→ Pµ (4.31) gives a non-zero element yi−1 ∈ HomF (Pµ, P − ), and so Cyi−1 = HomF (Pµ, P − ) by e λ µ λ µ (4.30) again.

(3). A Basis for HomFeλ (Pµ, Pµ+ ): Note that dimHomFeλ (Pµ, Pµ+ ) = 1 by (4.30). + Let xi ∈ HomFλ (Pµ, Pµ ) be a non-zero element. Note that each composition factor of xi(Pµ) lies in {Lµ++ ,Lµ,Lµ+ } because xi(Pµ) ⊂ radPµ+ . Since every non-trivial quotiente of Pµ has composition factor Lµ, we may conclude that xi(Pµ)= Bµ+ . Con- sequently,e xi can be expressed as the followinge composition of homomorphisms e . + xi : Pµ → Pµ/Bµ =∼ B + ֒→ P (4.32) e µ µ e 16 CHEN

It is not hard to compute the relations xjxj+1 = yjyj+1 = 0 for all j ∈ Z by (4.31) and (4.32). Therefore we may conclude that zic = czi = yiyj = xixj = 0 for all i, j ∈ Z, c ∈ {xi, yi}i∈Z. Finally, we notee thate yixi eande xiyi can be expressed as the following composition of homomorphisms e e e e e e e e e e e e ∼ Aµ+ + Bµ+ ∼ (4.33) yixi : Pµ → Pµ/Bµ = Bµ+ → = socPµ ⊂ Pµ, Aµ+ ∼ Aµ + Bµ ∼ (4.34) xiyei e: Pµ+ → Pµ+ /Aµ+ = Aµ → = socPµ+ ⊂ Pµ+ . Bµ Z It is not harde e to see that yixi = zi and xiyi = zi+1 for all i ∈ . Therefore, we ∞ op have an isomorphism from EndFλ (⊕µ∈Λλ Pµ) to (K1 ) sending xi, yi, zi to xi,yi, zi, respectively. This completese thee proof.e e e e e e e

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Department of Mathematics, National Taiwan University, Taipei 10617, Taiwan E-mail address: [email protected]