The BGG Category O Over Generalized Weyl Algebras

The BGG Category O over generalized Weyl algebras

Apoorva Khare

Stanford University

Joint work with Akaki Tikaradze (U. Toledo) First results The endomorphism algebra of projectives Koszulity and categorication Outline

1 First results Classical and quantum generalized Weyl algebras Basic properties of Category O

2 The endomorphism algebra of projectives Projective resolutions and Ext's Young diagrams and projectives in the block

3 Koszulity and categorication Koszulity and presentation GWAs categorify Young diagrams

2 / 34 Introduced by Bernstein, Gelfand, and Gelfand in the 1970s; widely studied in the above settings (and others). Important connections to Geometry - ag manifold Algebra - primitive ideals, abelian ideals Mathematical physics Combinatorics - crystals, identities, Coxeter groups Categorication

First results Classical and quantum generalized Weyl algebras The endomorphism algebra of projectives Basic properties of Category Koszulity and categorication O Category O Fundamental category of representations, dened and studied over: Lie algebras - semisimple, Kac-Moody, (generalized) Virasoro. . . Quantum groups Generalized Weyl algebras Continuous and innitesimal Hecke algebras Cherednik algebras

3 / 34 Important connections to Geometry - ag manifold Algebra - primitive ideals, abelian ideals Mathematical physics Combinatorics - crystals, identities, Coxeter groups Categorication

3 / 34 First results Classical and quantum generalized Weyl algebras The endomorphism algebra of projectives Basic properties of Category Koszulity and categorication O Category O Fundamental category of representations, dened and studied over: Lie algebras - semisimple, Kac-Moody, (generalized) Virasoro. . . Quantum groups Generalized Weyl algebras Continuous and innitesimal Hecke algebras Cherednik algebras Introduced by Bernstein, Gelfand, and Gelfand in the 1970s; widely studied in the above settings (and others). Important connections to Geometry - ag manifold Algebra - primitive ideals, abelian ideals Mathematical physics Combinatorics - crystals, identities, Coxeter groups Categorication 3 / 34 the corresponding triangular Generalized Weyl algebra (GWA) is

W(H, θ, z0, z1) := Hhd, ui/(uh = θ(h)u, hd = dθ(h), ud = z0+dz1).

Goal: Study Category O over a triangular GWA. Understand the structure of projectives in a block, and the endomorphism algebra.

First results Classical and quantum generalized Weyl algebras The endomorphism algebra of projectives Basic properties of Category Koszulity and categorication O Triangular generalized Weyl algebras

Given a commutative F-algebra H, an algebra automorphism θ : H → H, and × elements z0 ∈ H and z1 ∈ H ,

4 / 34 Goal: Study Category O over a triangular GWA. Understand the structure of projectives in a block, and the endomorphism algebra.

First results Classical and quantum generalized Weyl algebras The endomorphism algebra of projectives Basic properties of Category Koszulity and categorication O Triangular generalized Weyl algebras

Given a commutative F-algebra H, an algebra automorphism θ : H → H, and × elements z0 ∈ H and z1 ∈ H , the corresponding triangular Generalized Weyl algebra (GWA) is

W(H, θ, z0, z1) := Hhd, ui/(uh = θ(h)u, hd = dθ(h), ud = z0+dz1).

4 / 34 First results Classical and quantum generalized Weyl algebras The endomorphism algebra of projectives Basic properties of Category Koszulity and categorication O Triangular generalized Weyl algebras

Given a commutative F-algebra H, an algebra automorphism θ : H → H, and × elements z0 ∈ H and z1 ∈ H , the corresponding triangular Generalized Weyl algebra (GWA) is

W(H, θ, z0, z1) := Hhd, ui/(uh = θ(h)u, hd = dθ(h), ud = z0+dz1).

Goal: Study Category O over a triangular GWA. Understand the structure of projectives in a block, and the endomorphism algebra.

4 / 34 Quantum algebra: Jing-Zhang studied non-commutative, non-cocommutative bialgebras that -deform . q U(gl2),U(sl2) Kac: dispin Lie superalgebra B[0, 1]. Combinatorics: BenkartRoby studied down, up operators on posets: (generalized) down-up algebras.

× For all of these algebras, H = F[h] 3 z0 and z1 ∈ F .

First results Classical and quantum generalized Weyl algebras The endomorphism algebra of projectives Basic properties of Category Koszulity and categorication O Triangular generalized Weyl algebras (cont.)

Triangular GWAs occur in many settings:

Representation theory: Smith studied deformations of sl2:

Che, f, hi/ ([h, e] = 2e, [h, f] = −2f, [e, f] = z0(h)). Mathematical physics: Witten introduced 7-parameter family

of deformations of U(sl2). Le Bruyn: Conformal sl2-algebras.

5 / 34 Combinatorics: BenkartRoby studied down, up operators on posets: (generalized) down-up algebras.

× For all of these algebras, H = F[h] 3 z0 and z1 ∈ F .

Triangular GWAs occur in many settings:

Representation theory: Smith studied deformations of sl2:

Che, f, hi/ ([h, e] = 2e, [h, f] = −2f, [e, f] = z0(h)). Mathematical physics: Witten introduced 7-parameter family

5 / 34 × For all of these algebras, H = F[h] 3 z0 and z1 ∈ F .

Triangular GWAs occur in many settings:

Representation theory: Smith studied deformations of sl2:

Che, f, hi/ ([h, e] = 2e, [h, f] = −2f, [e, f] = z0(h)). Mathematical physics: Witten introduced 7-parameter family

of deformations of U(sl2). Le Bruyn: Conformal sl2-algebras. Quantum algebra: Jing-Zhang studied non-commutative, non-cocommutative bialgebras that -deform . q U(gl2),U(sl2) Kac: dispin Lie superalgebra B[0, 1]. Combinatorics: BenkartRoby studied down, up operators on posets: (generalized) down-up algebras.

5 / 34 First results Classical and quantum generalized Weyl algebras The endomorphism algebra of projectives Basic properties of Category Koszulity and categorication O Triangular generalized Weyl algebras (cont.)

Triangular GWAs occur in many settings:

Representation theory: Smith studied deformations of sl2:

Che, f, hi/ ([h, e] = 2e, [h, f] = −2f, [e, f] = z0(h)). Mathematical physics: Witten introduced 7-parameter family

× For all of these algebras, H = F[h] 3 z0 and z1 ∈ F . 5 / 34 The classical algebras deform U(sl2); the quantum algebras deform Uq(sl2). Are there connections between these classical and quantum families of triangular GWAs? Yes: the classical families are classical limits of the quantum families:

First results Classical and quantum generalized Weyl algebras The endomorphism algebra of projectives Basic properties of Category Koszulity and categorication O Quantum examples of triangular GWAs

Families of quantum examples, with H a group algebra: ±1 Quantum sl2: H = F[K ]. ±1 ±1 Drinfeld double of positive part of Uq(sl2): H = F[K ,L ].

6 / 34 Yes: the classical families are classical limits of the quantum families:

First results Classical and quantum generalized Weyl algebras The endomorphism algebra of projectives Basic properties of Category Koszulity and categorication O Quantum examples of triangular GWAs

Families of quantum examples, with H a group algebra: ±1 Quantum sl2: H = F[K ]. ±1 ±1 Drinfeld double of positive part of Uq(sl2): H = F[K ,L ].

The classical algebras deform U(sl2); the quantum algebras deform Uq(sl2). Are there connections between these classical and quantum families of triangular GWAs?

6 / 34 First results Classical and quantum generalized Weyl algebras The endomorphism algebra of projectives Basic properties of Category Koszulity and categorication O Quantum examples of triangular GWAs

Families of quantum examples, with H a group algebra: ±1 Quantum sl2: H = F[K ]. ±1 ±1 Drinfeld double of positive part of Uq(sl2): H = F[K ,L ].

The classical algebras deform U(sl2); the quantum algebras deform Uq(sl2). Are there connections between these classical and quantum families of triangular GWAs? Yes: the classical families are classical limits of the quantum families:

6 / 34 Suppose q is transcendental over F, and l 6= 0 and m, n are integers. Dene the quantum algebra γ(1 − K) W (l, m, n) := W (q)[K±1], θ, qmKnz ( ), z , q F 0 l(q − 1) 1 where θ(K) = q−lK. Let R be the local subring of F(q) of rational functions regular at the point q = 1. Let R be the -subalgebra of Wq (l, m, n) R Wq(l, m, n) generated by u, d, K±1, (K − 1)/(q − 1). Then ∼ R R . W(F[h], θ, z0(h), z1) = Wq (l, m, n)/(q − 1)Wq (l, m, n)

First results Classical and quantum generalized Weyl algebras The endomorphism algebra of projectives Basic properties of Category Koszulity and categorication O Deformation-quantization equals quantization-deformation

Theorem (K., 2015) × Fix scalars γ ∈ F, z1 ∈ F , and a polynomial z0 = z0(h) ∈ F[h]. Consider the algebra W(F[h], θ, z0(h), z1), with θ(h) := h + γ.

7 / 34 Let R be the local subring of F(q) of rational functions regular at the point q = 1. Let R be the -subalgebra of Wq (l, m, n) R Wq(l, m, n) generated by u, d, K±1, (K − 1)/(q − 1). Then ∼ R R . W(F[h], θ, z0(h), z1) = Wq (l, m, n)/(q − 1)Wq (l, m, n)

7 / 34 Let R be the -subalgebra of Wq (l, m, n) R Wq(l, m, n) generated by u, d, K±1, (K − 1)/(q − 1). Then ∼ R R . W(F[h], θ, z0(h), z1) = Wq (l, m, n)/(q − 1)Wq (l, m, n)

7 / 34 Then ∼ R R . W(F[h], θ, z0(h), z1) = Wq (l, m, n)/(q − 1)Wq (l, m, n)

7 / 34 First results Classical and quantum generalized Weyl algebras The endomorphism algebra of projectives Basic properties of Category Koszulity and categorication O Deformation-quantization equals quantization-deformation

Theorem (K., 2015) × Fix scalars γ ∈ F, z1 ∈ F , and a polynomial z0 = z0(h) ∈ F[h]. Consider the algebra W(F[h], θ, z0(h), z1), with θ(h) := h + γ. Suppose q is transcendental over F, and l 6= 0 and m, n are integers. Dene the quantum algebra γ(1 − K) W (l, m, n) := W (q)[K±1], θ, qmKnz ( ), z , q F 0 l(q − 1) 1 where θ(K) = q−lK. Let R be the local subring of F(q) of rational functions regular at the point q = 1. Let R be the -subalgebra of Wq (l, m, n) R Wq(l, m, n) generated by u, d, K±1, (K − 1)/(q − 1). Then ∼ R R . W(F[h], θ, z0(h), z1) = Wq (l, m, n)/(q − 1)Wq (l, m, n) 7 / 34 Let R R . Then there is a W1 := Wq (l, m, n)/(q − 1)Wq (l, m, n) surjection of -algebras F γ(1 − K) π : W( [h], θ, z (h), z ) W (u 7→ u, d 7→ d, h 7→ ). F 0 1 1 l(q − 1) To show: π restricted to F[d] is an isomorphism. Enough to show this, after changing scalars to Fu, an uncountable eld extension of F. Now nd a Verma module Fu that is simple over M1 (λ) , hence innite-dimensional. Fu ⊗F W1 (Requires understanding the structure of Verma modules and blocks of Category O.)

First results Classical and quantum generalized Weyl algebras The endomorphism algebra of projectives Basic properties of Category Koszulity and categorication O Sketch of proof

8 / 34 To show: π restricted to F[d] is an isomorphism. Enough to show this, after changing scalars to Fu, an uncountable eld extension of F. Now nd a Verma module Fu that is simple over M1 (λ) , hence innite-dimensional. Fu ⊗F W1 (Requires understanding the structure of Verma modules and blocks of Category O.)

First results Classical and quantum generalized Weyl algebras The endomorphism algebra of projectives Basic properties of Category Koszulity and categorication O Sketch of proof

8 / 34 Now nd a Verma module Fu that is simple over M1 (λ) , hence innite-dimensional. Fu ⊗F W1 (Requires understanding the structure of Verma modules and blocks of Category O.)

First results Classical and quantum generalized Weyl algebras The endomorphism algebra of projectives Basic properties of Category Koszulity and categorication O Sketch of proof

Set R to be respectively, and R to be the W± R[u],R[d] W0 -subalgebra of ±1 generated by ±1 K−1 . R F(q)[K ] K , q−1 The multiplication map R R R : W− ⊗R W0 ⊗R W+ → Wq(l, m, n) is an R-algebra isomorphism. Let R R . Then there is a W1 := Wq (l, m, n)/(q − 1)Wq (l, m, n) surjection of -algebras F γ(1 − K) π : W( [h], θ, z (h), z ) W (u 7→ u, d 7→ d, h 7→ ). F 0 1 1 l(q − 1) To show: π restricted to F[d] is an isomorphism. Enough to show this, after changing scalars to Fu, an uncountable eld extension of F.

8 / 34 First results Classical and quantum generalized Weyl algebras The endomorphism algebra of projectives Basic properties of Category Koszulity and categorication O Sketch of proof

8 / 34 2 Category O is the full subcategory of modules that are: nitely generated, H-semisimple, with nite-dimensional H-weight spaces, and u acts locally nilpotently on each module.

3 Weights are characters (algebra maps) Hb := {λ : H → F}. 4 Given λ ∈ Hb, the λ-weight space of a module M is

Mλ := {m ∈ M : h · m = λ(h)m, ∀h ∈ H}.

5 The weights of a module M are wt M := {λ ∈ Hb : Mλ 6= 0}.

First results Classical and quantum generalized Weyl algebras The endomorphism algebra of projectives Basic properties of Category Koszulity and categorication O Properties of Category O

A := W(H, θ, z0, z1) = triangular GWA. 1 PBW property: The multiplication map : F[d] ⊗ H ⊗ F[u] → A is a vector space isomorphism.

9 / 34 3 Weights are characters (algebra maps) Hb := {λ : H → F}. 4 Given λ ∈ Hb, the λ-weight space of a module M is

Mλ := {m ∈ M : h · m = λ(h)m, ∀h ∈ H}.

5 The weights of a module M are wt M := {λ ∈ Hb : Mλ 6= 0}.

First results Classical and quantum generalized Weyl algebras The endomorphism algebra of projectives Basic properties of Category Koszulity and categorication O Properties of Category O

A := W(H, θ, z0, z1) = triangular GWA. 1 PBW property: The multiplication map : F[d] ⊗ H ⊗ F[u] → A is a vector space isomorphism. 2 Category O is the full subcategory of modules that are: nitely generated, H-semisimple, with nite-dimensional H-weight spaces, and u acts locally nilpotently on each module.

9 / 34 4 Given λ ∈ Hb, the λ-weight space of a module M is

Mλ := {m ∈ M : h · m = λ(h)m, ∀h ∈ H}.

5 The weights of a module M are wt M := {λ ∈ Hb : Mλ 6= 0}.

First results Classical and quantum generalized Weyl algebras The endomorphism algebra of projectives Basic properties of Category Koszulity and categorication O Properties of Category O

3 Weights are characters (algebra maps) Hb := {λ : H → F}.

9 / 34 First results Classical and quantum generalized Weyl algebras The endomorphism algebra of projectives Basic properties of Category Koszulity and categorication O Properties of Category O

3 Weights are characters (algebra maps) Hb := {λ : H → F}. 4 Given λ ∈ Hb, the λ-weight space of a module M is

Mλ := {m ∈ M : h · m = λ(h)m, ∀h ∈ H}.

5 The weights of a module M are wt M := {λ ∈ Hb : Mλ 6= 0}.

9 / 34 Important objects in Category O: Verma modules: M(λ) := A/(A · u + A · ker λ), λ ∈ H.b Each Verma module has a unique simple quotient L(λ). Also lies in O. All simple objects in O are of the form L(λ). What are the weights of M(λ)? What is the length of M(λ)? General fact: Category O is nite length, if and only if all Verma modules have nite length. How to compute submodules of M(λ)?

First results Classical and quantum generalized Weyl algebras The endomorphism algebra of projectives Basic properties of Category Koszulity and categorication O Verma modules

Technical assumption: For all weights λ ∈ Hb, n if n ∈ Z and λ ≡ λ ◦ θ on all of H, then n = 0. (So θ is an automorphism of innite order.)

10 / 34 Also lies in O. All simple objects in O are of the form L(λ). What are the weights of M(λ)? What is the length of M(λ)? General fact: Category O is nite length, if and only if all Verma modules have nite length. How to compute submodules of M(λ)?

First results Classical and quantum generalized Weyl algebras The endomorphism algebra of projectives Basic properties of Category Koszulity and categorication O Verma modules

Technical assumption: For all weights λ ∈ Hb, n if n ∈ Z and λ ≡ λ ◦ θ on all of H, then n = 0. (So θ is an automorphism of innite order.)

Important objects in Category O: Verma modules: M(λ) := A/(A · u + A · ker λ), λ ∈ H.b Each Verma module has a unique simple quotient L(λ).

10 / 34 What are the weights of M(λ)? What is the length of M(λ)? General fact: Category O is nite length, if and only if all Verma modules have nite length. How to compute submodules of M(λ)?

First results Classical and quantum generalized Weyl algebras The endomorphism algebra of projectives Basic properties of Category Koszulity and categorication O Verma modules

Technical assumption: For all weights λ ∈ Hb, n if n ∈ Z and λ ≡ λ ◦ θ on all of H, then n = 0. (So θ is an automorphism of innite order.)

10 / 34 How to compute submodules of M(λ)?

First results Classical and quantum generalized Weyl algebras The endomorphism algebra of projectives Basic properties of Category Koszulity and categorication O Verma modules

Technical assumption: For all weights λ ∈ Hb, n if n ∈ Z and λ ≡ λ ◦ θ on all of H, then n = 0. (So θ is an automorphism of innite order.)

Important objects in Category O: Verma modules: M(λ) := A/(A · u + A · ker λ), λ ∈ H.b Each Verma module has a unique simple quotient L(λ). Also lies in O. All simple objects in O are of the form L(λ). What are the weights of M(λ)? What is the length of M(λ)? General fact: Category O is nite length, if and only if all Verma modules have nite length.

10 / 34 First results Classical and quantum generalized Weyl algebras The endomorphism algebra of projectives Basic properties of Category Koszulity and categorication O Verma modules

Technical assumption: For all weights λ ∈ Hb, n if n ∈ Z and λ ≡ λ ◦ θ on all of H, then n = 0. (So θ is an automorphism of innite order.)

10 / 34 3 ∼ M(λ) = F[d] as free F[d]-modules.

First results Classical and quantum generalized Weyl algebras The endomorphism algebra of projectives Basic properties of Category Koszulity and categorication O Properties of Verma modules

A = W(H, θ, z0, z1). Facts: 1 wt M(λ) = {λ ◦ θn : n ≥ 0}.

−n 2 Given λ ∈ Hb and n ∈ Z, dene n ∗ λ := λ ◦ θ . Thus, wt M(λ) = {(−n) ∗ λ : n ≥ 0}. All (nonzero) weight multiplicities are 1.

11 / 34 First results Classical and quantum generalized Weyl algebras The endomorphism algebra of projectives Basic properties of Category Koszulity and categorication O Properties of Verma modules

A = W(H, θ, z0, z1). Facts: 1 wt M(λ) = {λ ◦ θn : n ≥ 0}.

−n 2 Given λ ∈ Hb and n ∈ Z, dene n ∗ λ := λ ◦ θ . Thus, wt M(λ) = {(−n) ∗ λ : n ≥ 0}. All (nonzero) weight multiplicities are 1.

3 ∼ M(λ) = F[d] as free F[d]-modules.

11 / 34 Proposition (K., 2015)

For all weights λ ∈ Hb, the Verma module M(λ) is uniserial, with unique composition series

M(λ) ⊃ M((−n1) ∗ λ) ⊃ M((−n2) ∗ λ) ⊃ · · · , where comprise the set . 0 < n1 < n2 < ··· {n > 0 : λ(zen) = 0} Moreover, [M(λ): L(µ)] ≤ 1 for all λ, µ ∈ Hb.

First results Classical and quantum generalized Weyl algebras The endomorphism algebra of projectives Basic properties of Category Koszulity and categorication O Properties of Verma modules (cont.)

For , dene distinguished elements : n ∈ N zen ∈ H n−1 n−1 0 Y i 0 X j 0 zn := θ (z1), z0 := 1, zen := θ (z0zn−1−j). i=0 j=0 Also dene for non-positive : zen n ∈ Z −n ze0 := 0, ze−n := θ (zen)(n > 0).

12 / 34 Moreover, [M(λ): L(µ)] ≤ 1 for all λ, µ ∈ Hb.

First results Classical and quantum generalized Weyl algebras The endomorphism algebra of projectives Basic properties of Category Koszulity and categorication O Properties of Verma modules (cont.)

For , dene distinguished elements : n ∈ N zen ∈ H n−1 n−1 0 Y i 0 X j 0 zn := θ (z1), z0 := 1, zen := θ (z0zn−1−j). i=0 j=0 Also dene for non-positive : zen n ∈ Z −n ze0 := 0, ze−n := θ (zen)(n > 0). Proposition (K., 2015)

For all weights λ ∈ Hb, the Verma module M(λ) is uniserial, with unique composition series

M(λ) ⊃ M((−n1) ∗ λ) ⊃ M((−n2) ∗ λ) ⊃ · · · , where comprise the set . 0 < n1 < n2 < ··· {n > 0 : λ(zen) = 0}

12 / 34 First results Classical and quantum generalized Weyl algebras The endomorphism algebra of projectives Basic properties of Category Koszulity and categorication O Properties of Verma modules (cont.)

For all weights λ ∈ Hb, the Verma module M(λ) is uniserial, with unique composition series

M(λ) ⊃ M((−n1) ∗ λ) ⊃ M((−n2) ∗ λ) ⊃ · · · , where comprise the set . 0 < n1 < n2 < ··· {n > 0 : λ(zen) = 0} Moreover, [M(λ): L(µ)] ≤ 1 for all λ, µ ∈ Hb.

12 / 34 Category O satises many desirable properties if we can obtain a block decomposition a M Hb = Ti, O = OTi , i∈I i∈I with Ti nite for all i. This partition denes an equivalence relation on Hb, whose classes are the Ti. For any such partition, and any indecomposable module M ∈ O (e.g., M = M(λ)), all simple factors of M lie in the same class. There exists a unique nest partition. Equivalence classes: λ [λ] := {(−n) ∗ λ : n ∈ Z, λ(zen) = 0} ⊂ H.b For many families of triangular GWAs in the literature, [λ] is nite for all λ ∈ Hb.

First results Classical and quantum generalized Weyl algebras The endomorphism algebra of projectives Basic properties of Category Koszulity and categorication O Blocks of Category O

Denition: Given T ⊂ Hb, let OT denote the full subcategory of objects, all of whose Jordan-Holder factors L(µ) satisfy: µ ∈ T .

13 / 34 This partition denes an equivalence relation on Hb, whose classes are the Ti. For any such partition, and any indecomposable module M ∈ O (e.g., M = M(λ)), all simple factors of M lie in the same class. There exists a unique nest partition. Equivalence classes: λ [λ] := {(−n) ∗ λ : n ∈ Z, λ(zen) = 0} ⊂ H.b For many families of triangular GWAs in the literature, [λ] is nite for all λ ∈ Hb.

First results Classical and quantum generalized Weyl algebras The endomorphism algebra of projectives Basic properties of Category Koszulity and categorication O Blocks of Category O

13 / 34 For any such partition, and any indecomposable module M ∈ O (e.g., M = M(λ)), all simple factors of M lie in the same class. There exists a unique nest partition. Equivalence classes: λ [λ] := {(−n) ∗ λ : n ∈ Z, λ(zen) = 0} ⊂ H.b For many families of triangular GWAs in the literature, [λ] is nite for all λ ∈ Hb.

First results Classical and quantum generalized Weyl algebras The endomorphism algebra of projectives Basic properties of Category Koszulity and categorication O Blocks of Category O

13 / 34 There exists a unique nest partition. Equivalence classes: λ [λ] := {(−n) ∗ λ : n ∈ Z, λ(zen) = 0} ⊂ H.b For many families of triangular GWAs in the literature, [λ] is nite for all λ ∈ Hb.

First results Classical and quantum generalized Weyl algebras The endomorphism algebra of projectives Basic properties of Category Koszulity and categorication O Blocks of Category O

13 / 34 For many families of triangular GWAs in the literature, [λ] is nite for all λ ∈ Hb.

First results Classical and quantum generalized Weyl algebras The endomorphism algebra of projectives Basic properties of Category Koszulity and categorication O Blocks of Category O

Denition: Given T ⊂ Hb, let OT denote the full subcategory of objects, all of whose Jordan-Holder factors L(µ) satisfy: µ ∈ T . Category O satises many desirable properties if we can obtain a block decomposition a M Hb = Ti, O = OTi , i∈I i∈I with Ti nite for all i. This partition denes an equivalence relation on Hb, whose classes are the Ti. For any such partition, and any indecomposable module M ∈ O (e.g., M = M(λ)), all simple factors of M lie in the same class. There exists a unique nest partition. Equivalence classes: λ [λ] := {(−n) ∗ λ : n ∈ Z, λ(zen) = 0} ⊂ H.b

13 / 34 First results Classical and quantum generalized Weyl algebras The endomorphism algebra of projectives Basic properties of Category Koszulity and categorication O Blocks of Category O

O[λ] is a nite length, abelian category with enough projectives and injectives.

The indecomposable projectives in O[λ] are the projective covers P (µ) of the simple modules {L(µ): µ ∈ [λ]}.

There is a exact, contravariant duality endofunctor F of O[λ] that xes L(µ), and sends P (µ) to the injective hull of L(µ). ∼ O[λ] = A[λ]-Mod for a nite-dimensional, quasi-hereditary algebra n op. A[λ] := EndO(⊕j=1P (λj))

14 / 34 O[λ] is a nite length, abelian category with enough projectives and injectives.

The indecomposable projectives in O[λ] are the projective covers P (µ) of the simple modules {L(µ): µ ∈ [λ]}.

First results Classical and quantum generalized Weyl algebras The endomorphism algebra of projectives Basic properties of Category Koszulity and categorication O Blocks with nitely many simples Henceforth, assume: 1 There do not exist λ ∈ Hb and n > 0 such that λ ≡ λ ◦ θn. 2 The set [λ] is nite for all λ ∈ Hb. Theorem (K., 2015) Under the above assumptions, O = L O is a direct sum of [λ]⊂Hb [λ] blocks. Now x λ ∈ Hb and suppose [λ] = {λ1, . . . , λn}. Then,

14 / 34 The indecomposable projectives in O[λ] are the projective covers P (µ) of the simple modules {L(µ): µ ∈ [λ]}.

O[λ] is a nite length, abelian category with enough projectives and injectives.

14 / 34 There is a exact, contravariant duality endofunctor F of O[λ] that xes L(µ), and sends P (µ) to the injective hull of L(µ). ∼ O[λ] = A[λ]-Mod for a nite-dimensional, quasi-hereditary algebra n op. A[λ] := EndO(⊕j=1P (λj))

O[λ] is a nite length, abelian category with enough projectives and injectives.

The indecomposable projectives in O[λ] are the projective covers P (µ) of the simple modules {L(µ): µ ∈ [λ]}.

14 / 34 ∼ O[λ] = A[λ]-Mod for a nite-dimensional, quasi-hereditary algebra n op. A[λ] := EndO(⊕j=1P (λj))

O[λ] is a nite length, abelian category with enough projectives and injectives.

The indecomposable projectives in O[λ] are the projective covers P (µ) of the simple modules {L(µ): µ ∈ [λ]}.

There is a exact, contravariant duality endofunctor F of O[λ] that xes L(µ), and sends P (µ) to the injective hull of L(µ).

14 / 34 First results Classical and quantum generalized Weyl algebras The endomorphism algebra of projectives Basic properties of Category Koszulity and categorication O Blocks with nitely many simples Henceforth, assume: 1 There do not exist λ ∈ Hb and n > 0 such that λ ≡ λ ◦ θn. 2 The set [λ] is nite for all λ ∈ Hb. Theorem (K., 2015) Under the above assumptions, O = L O is a direct sum of [λ]⊂Hb [λ] blocks. Now x λ ∈ Hb and suppose [λ] = {λ1, . . . , λn}. Then,

O[λ] is a nite length, abelian category with enough projectives and injectives.

The indecomposable projectives in O[λ] are the projective covers P (µ) of the simple modules {L(µ): µ ∈ [λ]}.

3 Understand all Ext's between simples, Vermas, and projectives.

4 Quadratic presentation of the algebra A[λ].

5 Quadratic dual of A[λ].

First results Projective resolutions and Ext's The endomorphism algebra of projectives Young diagrams and projectives in the block Koszulity and categorication Future goals

Rest of the talk: Work in a nite block O[λ].

1 Understand the detailed structure of projective objects (e.g., classify all submodules), and maps between them.

15 / 34 3 Understand all Ext's between simples, Vermas, and projectives.

4 Quadratic presentation of the algebra A[λ].

5 Quadratic dual of A[λ].

First results Projective resolutions and Ext's The endomorphism algebra of projectives Young diagrams and projectives in the block Koszulity and categorication Future goals

Rest of the talk: Work in a nite block O[λ].

1 Understand the detailed structure of projective objects (e.g., classify all submodules), and maps between them.

2 Understand tilting objects in the block.

15 / 34 4 Quadratic presentation of the algebra A[λ].

5 Quadratic dual of A[λ].

First results Projective resolutions and Ext's The endomorphism algebra of projectives Young diagrams and projectives in the block Koszulity and categorication Future goals

Rest of the talk: Work in a nite block O[λ].

1 Understand the detailed structure of projective objects (e.g., classify all submodules), and maps between them.

2 Understand tilting objects in the block.

3 Understand all Ext's between simples, Vermas, and projectives.

15 / 34 First results Projective resolutions and Ext's The endomorphism algebra of projectives Young diagrams and projectives in the block Koszulity and categorication Future goals

Rest of the talk: Work in a nite block O[λ].

1 Understand the detailed structure of projective objects (e.g., classify all submodules), and maps between them.

2 Understand tilting objects in the block.

3 Understand all Ext's between simples, Vermas, and projectives.

4 Quadratic presentation of the algebra A[λ].

5 Quadratic dual of A[λ].

15 / 34 Dually, every Pj has a nite ltration

Pj ⊃ Pj+1 ⊃ · · · ⊃ Pn ⊃ 0,

with successive subquotients Mk for j ≤ k ≤ n.

First results Projective resolutions and Ext's The endomorphism algebra of projectives Young diagrams and projectives in the block Koszulity and categorication Verma ag of projectives

Notation: Verma modules are uniserial, so suppose

M(λn) ⊃ M(λn−1) ⊃ · · · ⊃ M(λ1) ⊃ 0,

with subquotients L(λn),...,L(λ1) respectively. Thus, λn > λn−1 > ··· > λ1. Now dene Mj := M(λj),Lj := L(λj),Pj := P (λj). Proposition (Khare-Tikaradze, 2015)

For all 1 ≤ j ≤ n, Mj has a nite ltration

Mj ⊃ Mj−1 ⊃ · · · ⊃ M1 ⊃ 0,

with successive subquotients Lk for 1 ≤ k ≤ j.

16 / 34 First results Projective resolutions and Ext's The endomorphism algebra of projectives Young diagrams and projectives in the block Koszulity and categorication Verma ag of projectives

Notation: Verma modules are uniserial, so suppose

M(λn) ⊃ M(λn−1) ⊃ · · · ⊃ M(λ1) ⊃ 0,

with subquotients L(λn),...,L(λ1) respectively. Thus, λn > λn−1 > ··· > λ1. Now dene Mj := M(λj),Lj := L(λj),Pj := P (λj). Proposition (Khare-Tikaradze, 2015)

For all 1 ≤ j ≤ n, Mj has a nite ltration

Mj ⊃ Mj−1 ⊃ · · · ⊃ M1 ⊃ 0,

with successive subquotients Lk for 1 ≤ k ≤ j. Dually, every Pj has a nite ltration

Pj ⊃ Pj+1 ⊃ · · · ⊃ Pn ⊃ 0,

with successive subquotients Mk for j ≤ k ≤ n.

16 / 34 If 0 = j < k ≤ n, then the Verma module Mk has a projective resolution:

0 → Pk+1 → Pk → Mk → 0, ∀1 ≤ k ≤ n.

The proofs use the explicit construction of the projective module

Pj, as the [λ]-direct summand of the A-module λn−λj +1 . (Has .) A/(Au + A · ker(λj)) ∈ O 1Pj ∈ (Pj)λj

Also use standard facts in the highest weight category O[λ]: dim HomO(Pj, −) = [− : Lj], dim HomO(Pj,Lk) = δj,k.

First results Projective resolutions and Ext's The endomorphism algebra of projectives Young diagrams and projectives in the block Koszulity and categorication Resolution of highest weight modules

Theorem (Khare-Tikaradze, 2015) Suppose 0 < j < k ≤ n. Then the following is a projective resolution of the highest weight module Mk/Mj in O:

0 → Pj+1 → Pj ⊕ Pk+1 → Pk → Mk/Mj → 0,

with the understanding that Pn+1 = 0.

17 / 34 The proofs use the explicit construction of the projective module

Pj, as the [λ]-direct summand of the A-module λn−λj +1 . (Has .) A/(Au + A · ker(λj)) ∈ O 1Pj ∈ (Pj)λj

Also use standard facts in the highest weight category O[λ]: dim HomO(Pj, −) = [− : Lj], dim HomO(Pj,Lk) = δj,k.

First results Projective resolutions and Ext's The endomorphism algebra of projectives Young diagrams and projectives in the block Koszulity and categorication Resolution of highest weight modules

Theorem (Khare-Tikaradze, 2015) Suppose 0 < j < k ≤ n. Then the following is a projective resolution of the highest weight module Mk/Mj in O:

0 → Pj+1 → Pj ⊕ Pk+1 → Pk → Mk/Mj → 0,

with the understanding that Pn+1 = 0. If 0 = j < k ≤ n, then the Verma module Mk has a projective resolution:

0 → Pk+1 → Pk → Mk → 0, ∀1 ≤ k ≤ n.

17 / 34 Also use standard facts in the highest weight category O[λ]: dim HomO(Pj, −) = [− : Lj], dim HomO(Pj,Lk) = δj,k.

First results Projective resolutions and Ext's The endomorphism algebra of projectives Young diagrams and projectives in the block Koszulity and categorication Resolution of highest weight modules

Theorem (Khare-Tikaradze, 2015) Suppose 0 < j < k ≤ n. Then the following is a projective resolution of the highest weight module Mk/Mj in O:

0 → Pj+1 → Pj ⊕ Pk+1 → Pk → Mk/Mj → 0,

with the understanding that Pn+1 = 0. If 0 = j < k ≤ n, then the Verma module Mk has a projective resolution:

0 → Pk+1 → Pk → Mk → 0, ∀1 ≤ k ≤ n.

The proofs use the explicit construction of the projective module

Pj, as the [λ]-direct summand of the A-module λn−λj +1 . (Has .) A/(Au + A · ker(λj)) ∈ O 1Pj ∈ (Pj)λj

17 / 34 First results Projective resolutions and Ext's The endomorphism algebra of projectives Young diagrams and projectives in the block Koszulity and categorication Resolution of highest weight modules

Theorem (Khare-Tikaradze, 2015) Suppose 0 < j < k ≤ n. Then the following is a projective resolution of the highest weight module Mk/Mj in O:

0 → Pj+1 → Pj ⊕ Pk+1 → Pk → Mk/Mj → 0,

with the understanding that Pn+1 = 0. If 0 = j < k ≤ n, then the Verma module Mk has a projective resolution:

0 → Pk+1 → Pk → Mk → 0, ∀1 ≤ k ≤ n.

The proofs use the explicit construction of the projective module

Pj, as the [λ]-direct summand of the A-module λn−λj +1 . (Has .) A/(Au + A · ker(λj)) ∈ O 1Pj ∈ (Pj)λj

Also use standard facts in the highest weight category O[λ]: dim HomO(Pj, −) = [− : Lj], dim HomO(Pj,Lk) = δj,k. 17 / 34 2 l dim ExtO(Pj/Pk,Mr/Ms) = δl,01(s < j ≤ r) + δl,11(s < k ≤ r). 3 For all 1 ≤ j, k ≤ n and l > 0,  , if |j − k| = l = 0; F  if ; l F, |j − k| = l = 1 ExtO(Lj,Lk) = F, if j = k 6= 1 and l = 2;  0, otherwise.

Uses construction and Jordan-Holder factors of Pj, and homological arguments.

First results Projective resolutions and Ext's The endomorphism algebra of projectives Young diagrams and projectives in the block Koszulity and categorication Ext-formulas

18 / 34 3 For all 1 ≤ j, k ≤ n and l > 0,  , if |j − k| = l = 0; F  if ; l F, |j − k| = l = 1 ExtO(Lj,Lk) = F, if j = k 6= 1 and l = 2;  0, otherwise.

Uses construction and Jordan-Holder factors of Pj, and homological arguments.

First results Projective resolutions and Ext's The endomorphism algebra of projectives Young diagrams and projectives in the block Koszulity and categorication Ext-formulas

18 / 34 Uses construction and Jordan-Holder factors of Pj, and homological arguments.

First results Projective resolutions and Ext's The endomorphism algebra of projectives Young diagrams and projectives in the block Koszulity and categorication Ext-formulas

Can compute lot of homological information about the block: Theorem (Khare-Tikaradze, 2015) Fix 1 ≤ j < k ≤ n + 1 and 0 ≤ s < r ≤ n. Then, 1 l . dim ExtO(Mr,Pj/Pk) = δl,01(r < k) + δl,11(r < j) 2 l dim ExtO(Pj/Pk,Mr/Ms) = δl,01(s < j ≤ r) + δl,11(s < k ≤ r). 3 For all 1 ≤ j, k ≤ n and l > 0,  , if |j − k| = l = 0; F  if ; l F, |j − k| = l = 1 ExtO(Lj,Lk) = F, if j = k 6= 1 and l = 2;  0, otherwise.

18 / 34 First results Projective resolutions and Ext's The endomorphism algebra of projectives Young diagrams and projectives in the block Koszulity and categorication Ext-formulas

Uses construction and Jordan-Holder factors of Pj, and homological arguments. 18 / 34 First results Projective resolutions and Ext's The endomorphism algebra of projectives Young diagrams and projectives in the block Koszulity and categorication Projectives, Vermas, and Young diagrams

Highest weight modules (or their composition series) can be represented by Young diagrams:

Pj/Pk = k k−1 k−2 ··· j k−1 . k−. 2 ···. ··· j−. 1 ...... j+1 Mk/Mj = 4 3 2 1 3 2 1 2 1 k k−1 ··· j+1 F (Mk/Mj) = 1

19 / 34 First study the larger algebra

op M Ag[λ] = EndO(Pg[λ]) , where Pg[λ] = Pj/Pk. 1≤j

Proposition (Khare-Tikaradze, 2015) Given integers 1 ≤ j ≤ k ≤ n, we have the following short exact sequence in the block O[λ]:

++ fj,k 0 → Pj/Pk −→ Pj+1/Pk+1 → F (Mk/Mj) → 0.

In pictures, ++ adds a (topmost) row to the diagram. fjk

Want to study the algebra

op M A[λ] = EndO(P[λ]) , where P[λ] = Pj. 1≤j≤n

20 / 34 Proposition (Khare-Tikaradze, 2015) Given integers 1 ≤ j ≤ k ≤ n, we have the following short exact sequence in the block O[λ]:

++ fj,k 0 → Pj/Pk −→ Pj+1/Pk+1 → F (Mk/Mj) → 0.

In pictures, ++ adds a (topmost) row to the diagram. fjk

Want to study the algebra

op M A[λ] = EndO(P[λ]) , where P[λ] = Pj. 1≤j≤n First study the larger algebra

op M Ag[λ] = EndO(Pg[λ]) , where Pg[λ] = Pj/Pk. 1≤j

20 / 34 In pictures, ++ adds a (topmost) row to the diagram. fjk

Want to study the algebra

op M A[λ] = EndO(P[λ]) , where P[λ] = Pj. 1≤j≤n First study the larger algebra

op M Ag[λ] = EndO(Pg[λ]) , where Pg[λ] = Pj/Pk. 1≤j

Proposition (Khare-Tikaradze, 2015) Given integers 1 ≤ j ≤ k ≤ n, we have the following short exact sequence in the block O[λ]:

++ fj,k 0 → Pj/Pk −→ Pj+1/Pk+1 → F (Mk/Mj) → 0.

20 / 34 First results Projective resolutions and Ext's The endomorphism algebra of projectives Young diagrams and projectives in the block Koszulity and categorication Graded maps between quotients of projectives

Want to study the algebra

op M A[λ] = EndO(P[λ]) , where P[λ] = Pj. 1≤j≤n First study the larger algebra

op M Ag[λ] = EndO(Pg[λ]) , where Pg[λ] = Pj/Pk. 1≤j

Proposition (Khare-Tikaradze, 2015) Given integers 1 ≤ j ≤ k ≤ n, we have the following short exact sequence in the block O[λ]:

++ fj,k 0 → Pj/Pk −→ Pj+1/Pk+1 → F (Mk/Mj) → 0.

In pictures, ++ adds a (topmost) row to the diagram. fjk 20 / 34 Theorem (Khare-Tikaradze, 2015)

1 Fix integers 1 ≤ {r, s} ≤ j ≤ k ≤ n + 1. Then the image of the vector λj −λs λj −λr d u 1Pr/Pk ∈ Pr/Pk generates the submodule Ps/Ps+k−j of Pj/Pk ,→ Pr/Pk. 2 The maps ++ −• •− generate the -algebra fjk , fjk , fjk F op Ag[λ] = EndO(Pg[λ]) .

More examples of maps in Ag[λ]: −• •− fjk : Pj/Pk ,→ Pj−1/Pk, fjk : Pj/Pk Pj/Pk−1. Add the rightmost column, and remove the leftmost column, respectively.

21 / 34 2 The maps ++ −• •− generate the -algebra fjk , fjk , fjk F op Ag[λ] = EndO(Pg[λ]) .

More examples of maps in Ag[λ]: −• •− fjk : Pj/Pk ,→ Pj−1/Pk, fjk : Pj/Pk Pj/Pk−1. Add the rightmost column, and remove the leftmost column, respectively.

Theorem (Khare-Tikaradze, 2015)

1 Fix integers 1 ≤ {r, s} ≤ j ≤ k ≤ n + 1. Then the image of the vector λj −λs λj −λr d u 1Pr/Pk ∈ Pr/Pk generates the submodule Ps/Ps+k−j of Pj/Pk ,→ Pr/Pk.

21 / 34 First results Projective resolutions and Ext's The endomorphism algebra of projectives Young diagrams and projectives in the block Koszulity and categorication Graded maps between quotients of projectives (cont.)

More examples of maps in Ag[λ]: −• •− fjk : Pj/Pk ,→ Pj−1/Pk, fjk : Pj/Pk Pj/Pk−1. Add the rightmost column, and remove the leftmost column, respectively.

Theorem (Khare-Tikaradze, 2015)

21 / 34 Dene (t) ϕ(r,s),(j,k) := −• −• ++ ++ •− •− fj+1,k ◦ · · · ◦ fk−t,k ◦ fk−t−1,k−1 ◦ · · · ◦ fr,r+t ◦ fr,r+t+1 ◦ · · · ◦ fr,s . | {z } | {z } | {z } k−j−t k−r−t s−r−t Theorem (Khare-Tikaradze, 2015)

1 (t) {ϕ(r,s),(j,k) : r < s, j < k, t ≤ min(s − r, k − r, k − j)} is a -graded basis of . Z+ Ag[λ] 2 Under this grading on , ++ −• , and Ag[λ] deg fjk = deg fjk = 1 •− (t) deg fjk = 0, deg ϕ(r,s),(j,k) = 2(k − t) − r − j.

3 If 1 ≤ a 0)ϕ(r,s),(a,b) .

22 / 34 Theorem (Khare-Tikaradze, 2015)

3 If 1 ≤ a 0)ϕ(r,s),(a,b) .

First results Projective resolutions and Ext's The endomorphism algebra of projectives Young diagrams and projectives in the block Koszulity and categorication Graded maps between quotients of projectives (cont.) Produce a -graded basis of op? Dene Z+ Ag[λ] = EndO(Pg[λ]) (t) ϕ(r,s),(j,k) := −• −• ++ ++ •− •− fj+1,k ◦ · · · ◦ fk−t,k ◦ fk−t−1,k−1 ◦ · · · ◦ fr,r+t ◦ fr,r+t+1 ◦ · · · ◦ fr,s . | {z } | {z } | {z } k−j−t k−r−t s−r−t

22 / 34 2 Under this grading on , ++ −• , and Ag[λ] deg fjk = deg fjk = 1 •− (t) deg fjk = 0, deg ϕ(r,s),(j,k) = 2(k − t) − r − j.

3 If 1 ≤ a 0)ϕ(r,s),(a,b) .

1 (t) {ϕ(r,s),(j,k) : r < s, j < k, t ≤ min(s − r, k − r, k − j)} is a -graded basis of . Z+ Ag[λ]

22 / 34 3 If 1 ≤ a 0)ϕ(r,s),(a,b) .

22 / 34 First results Projective resolutions and Ext's The endomorphism algebra of projectives Young diagrams and projectives in the block Koszulity and categorication Graded maps between quotients of projectives (cont.) Produce a -graded basis of op? Dene Z+ Ag[λ] = EndO(Pg[λ]) (t) ϕ(r,s),(j,k) := −• −• ++ ++ •− •− fj+1,k ◦ · · · ◦ fk−t,k ◦ fk−t−1,k−1 ◦ · · · ◦ fr,r+t ◦ fr,r+t+1 ◦ · · · ◦ fr,s . | {z } | {z } | {z } k−j−t k−r−t s−r−t Theorem (Khare-Tikaradze, 2015)

3 If 1 ≤ a 0)ϕ(r,s),(a,b) . 22 / 34 Theorem (Khare-Tikaradze, 2015)

(t) 1 The maps form a {ϕ(r,n+1),(j,n+1) : t ≤ n + 1 − max(r, j)} -graded basis of . (Dimension 2 2.) Z A[λ] = 1 + ··· + n 2 The Ext-quiver of A[λ] is the double An of the An-quiver [1] → [2] → · · · → [n].

3 Label the arrows as γi :[i + 1] → [i] and δi :[i] → [i + 1]. Then −• , ++ , and op is isomorphic to the γi = fi+1,n−1 δi = fi,n+1 A[λ] path algebra of the quiver An with relations

δi ◦ γi = γi+1 ◦ δi+1 ∀0 < i < n − 1, δn−1 ◦ γn−1 = 0.

First results Koszulity and presentation The endomorphism algebra of projectives GWAs categorify Young diagrams Koszulity and categorication Presentation of endomorphism algebra

Provides complete description of algebra EndO(⊕j

23 / 34 2 The Ext-quiver of A[λ] is the double An of the An-quiver [1] → [2] → · · · → [n].

3 Label the arrows as γi :[i + 1] → [i] and δi :[i] → [i + 1]. Then −• , ++ , and op is isomorphic to the γi = fi+1,n−1 δi = fi,n+1 A[λ] path algebra of the quiver An with relations

δi ◦ γi = γi+1 ◦ δi+1 ∀0 < i < n − 1, δn−1 ◦ γn−1 = 0.

First results Koszulity and presentation The endomorphism algebra of projectives GWAs categorify Young diagrams Koszulity and categorication Presentation of endomorphism algebra

Provides complete description of algebra EndO(⊕j

(t) 1 The maps form a {ϕ(r,n+1),(j,n+1) : t ≤ n + 1 − max(r, j)} -graded basis of . (Dimension 2 2.) Z A[λ] = 1 + ··· + n

23 / 34 Then −• , ++ , and op is isomorphic to the γi = fi+1,n−1 δi = fi,n+1 A[λ] path algebra of the quiver An with relations

δi ◦ γi = γi+1 ◦ δi+1 ∀0 < i < n − 1, δn−1 ◦ γn−1 = 0.

First results Koszulity and presentation The endomorphism algebra of projectives GWAs categorify Young diagrams Koszulity and categorication Presentation of endomorphism algebra

Provides complete description of algebra EndO(⊕j

3 Label the arrows as γi :[i + 1] → [i] and δi :[i] → [i + 1].

23 / 34 First results Koszulity and presentation The endomorphism algebra of projectives GWAs categorify Young diagrams Koszulity and categorication Presentation of endomorphism algebra

Provides complete description of algebra EndO(⊕j

3 Label the arrows as γi :[i + 1] → [i] and δi :[i] → [i + 1]. Then −• , ++ , and op is isomorphic to the γi = fi+1,n−1 δi = fi,n+1 A[λ] path algebra of the quiver An with relations

δi ◦ γi = γi+1 ◦ δi+1 ∀0 < i < n − 1, δn−1 ◦ γn−1 = 0.

23 / 34 Regardless of the GWA, blocks with same number of simples are Morita equivalent. Proof: n X 2u−j−k Hilbert matrix of A[λ]: H(A[λ], t)j,k = t . u=max(j,k) Hilbert matrix of • : E(A[λ]) = ExtO(P[λ], P[λ]) 2 2 −1 H(E(A[λ]), t) = Toeplitz(1+t , t, 0,..., 0)−t E11 = H(A[λ], t) . is graded, quadratic; . A[λ] A[λ][0] = spanF{idPj : 1 ≤ j ≤ n} Now use numerical criterion for Koszulity [BGS].

First results Koszulity and presentation The endomorphism algebra of projectives GWAs categorify Young diagrams Koszulity and categorication Koszulity