![Page 1: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/1.jpg)
Category O for quantum groups(joint work with Henning Haahr Andersen)
Volodymyr Mazorchuk(Uppsala University)
Mathematical Physics and Developments in AlgebraJuly 6, 2012, Krakow, POLAND
Volodymyr Mazorchuk Category O for quantum groups
![Page 2: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/2.jpg)
Various quantum groups
g — simple finite dimensional Lie algebra over C
U — the universal enveloping algebra of g (generators: ei , fi , hi )
Uv — the corresp. quantum group over Q(v) (generators Ei , Fi , K±1i )
UA — the A = Z[v , v−1]-form of Uv (generators E(r)i , F
(r)i , K±1i )
Uq — the specialization UA ⊗A C, v 7→ q, q ∈ C
uq — the “small” quantum group (subalg. of Uq generated by Ei , Fi , Ki )
Volodymyr Mazorchuk Category O for quantum groups
![Page 3: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/3.jpg)
Various quantum groups
g — simple finite dimensional Lie algebra over C
U — the universal enveloping algebra of g (generators: ei , fi , hi )
Uv — the corresp. quantum group over Q(v) (generators Ei , Fi , K±1i )
UA — the A = Z[v , v−1]-form of Uv (generators E(r)i , F
(r)i , K±1i )
Uq — the specialization UA ⊗A C, v 7→ q, q ∈ C
uq — the “small” quantum group (subalg. of Uq generated by Ei , Fi , Ki )
Volodymyr Mazorchuk Category O for quantum groups
![Page 4: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/4.jpg)
Various quantum groups
g — simple finite dimensional Lie algebra over C
U — the universal enveloping algebra of g (generators: ei , fi , hi )
Uv — the corresp. quantum group over Q(v) (generators Ei , Fi , K±1i )
UA — the A = Z[v , v−1]-form of Uv (generators E(r)i , F
(r)i , K±1i )
Uq — the specialization UA ⊗A C, v 7→ q, q ∈ C
uq — the “small” quantum group (subalg. of Uq generated by Ei , Fi , Ki )
Volodymyr Mazorchuk Category O for quantum groups
![Page 5: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/5.jpg)
Various quantum groups
g — simple finite dimensional Lie algebra over C
U — the universal enveloping algebra of g (generators: ei , fi , hi )
Uv — the corresp. quantum group over Q(v) (generators Ei , Fi , K±1i )
UA — the A = Z[v , v−1]-form of Uv (generators E(r)i , F
(r)i , K±1i )
Uq — the specialization UA ⊗A C, v 7→ q, q ∈ C
uq — the “small” quantum group (subalg. of Uq generated by Ei , Fi , Ki )
Volodymyr Mazorchuk Category O for quantum groups
![Page 6: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/6.jpg)
Various quantum groups
g — simple finite dimensional Lie algebra over C
U — the universal enveloping algebra of g (generators: ei , fi , hi )
Uv — the corresp. quantum group over Q(v) (generators Ei , Fi , K±1i )
UA — the A = Z[v , v−1]-form of Uv (generators E(r)i , F
(r)i , K±1i )
Uq — the specialization UA ⊗A C, v 7→ q, q ∈ C
uq — the “small” quantum group (subalg. of Uq generated by Ei , Fi , Ki )
Volodymyr Mazorchuk Category O for quantum groups
![Page 7: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/7.jpg)
Various quantum groups
g — simple finite dimensional Lie algebra over C
U — the universal enveloping algebra of g (generators: ei , fi , hi )
Uv — the corresp. quantum group over Q(v) (generators Ei , Fi , K±1i )
UA — the A = Z[v , v−1]-form of Uv (generators E(r)i , F
(r)i , K±1i )
Uq — the specialization UA ⊗A C, v 7→ q, q ∈ C
uq — the “small” quantum group (subalg. of Uq generated by Ei , Fi , Ki )
Volodymyr Mazorchuk Category O for quantum groups
![Page 8: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/8.jpg)
Various quantum groups
g — simple finite dimensional Lie algebra over C
U — the universal enveloping algebra of g (generators: ei , fi , hi )
Uv — the corresp. quantum group over Q(v) (generators Ei , Fi , K±1i )
UA — the A = Z[v , v−1]-form of Uv (generators E(r)i , F
(r)i , K±1i )
Uq — the specialization UA ⊗A C, v 7→ q, q ∈ C
uq — the “small” quantum group (subalg. of Uq generated by Ei , Fi , Ki )
Volodymyr Mazorchuk Category O for quantum groups
![Page 9: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/9.jpg)
Various quantum groups
g — simple finite dimensional Lie algebra over C
U — the universal enveloping algebra of g (generators: ei , fi , hi )
Uv — the corresp. quantum group over Q(v) (generators Ei , Fi , K±1i )
UA — the A = Z[v , v−1]-form of Uv (generators E(r)i , F
(r)i , K±1i )
Uq — the specialization UA ⊗A C, v 7→ q, q ∈ C
uq — the “small” quantum group (subalg. of Uq generated by Ei , Fi , Ki )
Volodymyr Mazorchuk Category O for quantum groups
![Page 10: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/10.jpg)
Tools
Triangular decompositions:
U = U−U0U+ Uq = U−q U0qU
+q uq = u−q u0qu
+q
q — root of unity of odd degree k (k 6= 3 if g has type G2)
Quantum Frobenius: Frq : Uq → U (roughly E(mk)i 7→ e
(m)i )
The pullback of Frq: −[k] : U-mod→ Uq-mod
(Integral) weight modules: M ∈ Uq-mod such that M = ⊕λ∈ZnMλ,
Mλ = {v ∈ M : K±1i v = q±diλi v}
Volodymyr Mazorchuk Category O for quantum groups
![Page 11: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/11.jpg)
Tools
Triangular decompositions:
U = U−U0U+ Uq = U−q U0qU
+q uq = u−q u0qu
+q
q — root of unity of odd degree k (k 6= 3 if g has type G2)
Quantum Frobenius: Frq : Uq → U (roughly E(mk)i 7→ e
(m)i )
The pullback of Frq: −[k] : U-mod→ Uq-mod
(Integral) weight modules: M ∈ Uq-mod such that M = ⊕λ∈ZnMλ,
Mλ = {v ∈ M : K±1i v = q±diλi v}
Volodymyr Mazorchuk Category O for quantum groups
![Page 12: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/12.jpg)
Tools
Triangular decompositions:
U = U−U0U+ Uq = U−q U0qU
+q uq = u−q u0qu
+q
q — root of unity of odd degree k (k 6= 3 if g has type G2)
Quantum Frobenius: Frq : Uq → U (roughly E(mk)i 7→ e
(m)i )
The pullback of Frq: −[k] : U-mod→ Uq-mod
(Integral) weight modules: M ∈ Uq-mod such that M = ⊕λ∈ZnMλ,
Mλ = {v ∈ M : K±1i v = q±diλi v}
Volodymyr Mazorchuk Category O for quantum groups
![Page 13: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/13.jpg)
Tools
Triangular decompositions:
U = U−U0U+ Uq = U−q U0qU
+q uq = u−q u0qu
+q
q — root of unity of odd degree k (k 6= 3 if g has type G2)
Quantum Frobenius: Frq : Uq → U (roughly E(mk)i 7→ e
(m)i )
The pullback of Frq: −[k] : U-mod→ Uq-mod
(Integral) weight modules: M ∈ Uq-mod such that M = ⊕λ∈ZnMλ,
Mλ = {v ∈ M : K±1i v = q±diλi v}
Volodymyr Mazorchuk Category O for quantum groups
![Page 14: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/14.jpg)
Tools
Triangular decompositions:
U = U−U0U+ Uq = U−q U0qU
+q uq = u−q u0qu
+q
q — root of unity of odd degree k (k 6= 3 if g has type G2)
Quantum Frobenius: Frq : Uq → U (roughly E(mk)i 7→ e
(m)i )
The pullback of Frq: −[k] : U-mod→ Uq-mod
(Integral) weight modules: M ∈ Uq-mod such that M = ⊕λ∈ZnMλ,
Mλ = {v ∈ M : K±1i v = q±diλi v}
Volodymyr Mazorchuk Category O for quantum groups
![Page 15: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/15.jpg)
Tools
Triangular decompositions:
U = U−U0U+ Uq = U−q U0qU
+q uq = u−q u0qu
+q
q — root of unity of odd degree k (k 6= 3 if g has type G2)
Quantum Frobenius: Frq : Uq → U (roughly E(mk)i 7→ e
(m)i )
The pullback of Frq: −[k] : U-mod→ Uq-mod
(Integral) weight modules: M ∈ Uq-mod such that M = ⊕λ∈ZnMλ,
Mλ = {v ∈ M : K±1i v = q±diλi v}
Volodymyr Mazorchuk Category O for quantum groups
![Page 16: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/16.jpg)
Tools
Triangular decompositions:
U = U−U0U+ Uq = U−q U0qU
+q uq = u−q u0qu
+q
q — root of unity of odd degree k (k 6= 3 if g has type G2)
Quantum Frobenius: Frq : Uq → U (roughly E(mk)i 7→ e
(m)i )
The pullback of Frq: −[k] : U-mod→ Uq-mod
(Integral) weight modules: M ∈ Uq-mod such that M = ⊕λ∈ZnMλ,
Mλ = {v ∈ M : K±1i v = q±diλi v}
Volodymyr Mazorchuk Category O for quantum groups
![Page 17: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/17.jpg)
Category O
Category Oq:
I full subcategory of the category of finitely generated Uq-modules;
I integral weight;
I U+-locally finite.
Oint — integral block of category O for U
Properties of Oint :
I decomposes into blocks of the form A-mod, A – fin. dim. assoc. alg.
I every object has finite length;
I A is self-dual in many senses (opposite, Ringel, Koszul)
I Cartan matrix of A is given by Kazhdan-Lusztig combinatorics
Main question: What about Oq?
Volodymyr Mazorchuk Category O for quantum groups
![Page 18: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/18.jpg)
Category O
Category Oq:
I full subcategory of the category of finitely generated Uq-modules;
I integral weight;
I U+-locally finite.
Oint — integral block of category O for U
Properties of Oint :
I decomposes into blocks of the form A-mod, A – fin. dim. assoc. alg.
I every object has finite length;
I A is self-dual in many senses (opposite, Ringel, Koszul)
I Cartan matrix of A is given by Kazhdan-Lusztig combinatorics
Main question: What about Oq?
Volodymyr Mazorchuk Category O for quantum groups
![Page 19: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/19.jpg)
Category O
Category Oq:
I full subcategory of the category of finitely generated Uq-modules;
I integral weight;
I U+-locally finite.
Oint — integral block of category O for U
Properties of Oint :
I decomposes into blocks of the form A-mod, A – fin. dim. assoc. alg.
I every object has finite length;
I A is self-dual in many senses (opposite, Ringel, Koszul)
I Cartan matrix of A is given by Kazhdan-Lusztig combinatorics
Main question: What about Oq?
Volodymyr Mazorchuk Category O for quantum groups
![Page 20: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/20.jpg)
Category O
Category Oq:
I full subcategory of the category of finitely generated Uq-modules;
I integral weight;
I U+-locally finite.
Oint — integral block of category O for U
Properties of Oint :
I decomposes into blocks of the form A-mod, A – fin. dim. assoc. alg.
I every object has finite length;
I A is self-dual in many senses (opposite, Ringel, Koszul)
I Cartan matrix of A is given by Kazhdan-Lusztig combinatorics
Main question: What about Oq?
Volodymyr Mazorchuk Category O for quantum groups
![Page 21: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/21.jpg)
Category O
Category Oq:
I full subcategory of the category of finitely generated Uq-modules;
I integral weight;
I U+-locally finite.
Oint — integral block of category O for U
Properties of Oint :
I decomposes into blocks of the form A-mod, A – fin. dim. assoc. alg.
I every object has finite length;
I A is self-dual in many senses (opposite, Ringel, Koszul)
I Cartan matrix of A is given by Kazhdan-Lusztig combinatorics
Main question: What about Oq?
Volodymyr Mazorchuk Category O for quantum groups
![Page 22: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/22.jpg)
Category O
Category Oq:
I full subcategory of the category of finitely generated Uq-modules;
I integral weight;
I U+-locally finite.
Oint — integral block of category O for U
Properties of Oint :
I decomposes into blocks of the form A-mod, A – fin. dim. assoc. alg.
I every object has finite length;
I A is self-dual in many senses (opposite, Ringel, Koszul)
I Cartan matrix of A is given by Kazhdan-Lusztig combinatorics
Main question: What about Oq?
Volodymyr Mazorchuk Category O for quantum groups
![Page 23: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/23.jpg)
Category O
Category Oq:
I full subcategory of the category of finitely generated Uq-modules;
I integral weight;
I U+-locally finite.
Oint — integral block of category O for U
Properties of Oint :
I decomposes into blocks of the form A-mod, A – fin. dim. assoc. alg.
I every object has finite length;
I A is self-dual in many senses (opposite, Ringel, Koszul)
I Cartan matrix of A is given by Kazhdan-Lusztig combinatorics
Main question: What about Oq?
Volodymyr Mazorchuk Category O for quantum groups
![Page 24: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/24.jpg)
Category O
Category Oq:
I full subcategory of the category of finitely generated Uq-modules;
I integral weight;
I U+-locally finite.
Oint — integral block of category O for U
Properties of Oint :
I decomposes into blocks of the form A-mod, A – fin. dim. assoc. alg.
I every object has finite length;
I A is self-dual in many senses (opposite, Ringel, Koszul)
I Cartan matrix of A is given by Kazhdan-Lusztig combinatorics
Main question: What about Oq?
Volodymyr Mazorchuk Category O for quantum groups
![Page 25: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/25.jpg)
Category O
Category Oq:
I full subcategory of the category of finitely generated Uq-modules;
I integral weight;
I U+-locally finite.
Oint — integral block of category O for U
Properties of Oint :
I decomposes into blocks of the form A-mod, A – fin. dim. assoc. alg.
I every object has finite length;
I A is self-dual in many senses (opposite, Ringel, Koszul)
I Cartan matrix of A is given by Kazhdan-Lusztig combinatorics
Main question: What about Oq?
Volodymyr Mazorchuk Category O for quantum groups
![Page 26: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/26.jpg)
Category O
Category Oq:
I full subcategory of the category of finitely generated Uq-modules;
I integral weight;
I U+-locally finite.
Oint — integral block of category O for U
Properties of Oint :
I decomposes into blocks of the form A-mod, A – fin. dim. assoc. alg.
I every object has finite length;
I A is self-dual in many senses (opposite, Ringel, Koszul)
I Cartan matrix of A is given by Kazhdan-Lusztig combinatorics
Main question: What about Oq?
Volodymyr Mazorchuk Category O for quantum groups
![Page 27: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/27.jpg)
Category O
Category Oq:
I full subcategory of the category of finitely generated Uq-modules;
I integral weight;
I U+-locally finite.
Oint — integral block of category O for U
Properties of Oint :
I decomposes into blocks of the form A-mod, A – fin. dim. assoc. alg.
I every object has finite length;
I A is self-dual in many senses (opposite, Ringel, Koszul)
I Cartan matrix of A is given by Kazhdan-Lusztig combinatorics
Main question: What about Oq?
Volodymyr Mazorchuk Category O for quantum groups
![Page 28: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/28.jpg)
Category O
Category Oq:
I full subcategory of the category of finitely generated Uq-modules;
I integral weight;
I U+-locally finite.
Oint — integral block of category O for U
Properties of Oint :
I decomposes into blocks of the form A-mod, A – fin. dim. assoc. alg.
I every object has finite length;
I A is self-dual in many senses (opposite, Ringel, Koszul)
I Cartan matrix of A is given by Kazhdan-Lusztig combinatorics
Main question: What about Oq?
Volodymyr Mazorchuk Category O for quantum groups
![Page 29: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/29.jpg)
Comparing Oint and Oq
The full subcat. of Oint of all fin. dim. modules is semi-simple.
The full subcat. of Oq of all fin. dim. module is not semi-simple.
Each indecomposable block of Oint has finitely many simples.
Oq has blocks with infinitely many simples.
Each indecomposable block of Oint has one indec. proj.-inj. module.The endomorphism algebra of this module is commutative.
Oq has blocks with infinitely many indec. proj.-inj. modules.The end. alg. of the direct sum of these modules is not commutative.
Volodymyr Mazorchuk Category O for quantum groups
![Page 30: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/30.jpg)
Comparing Oint and Oq
The full subcat. of Oint of all fin. dim. modules is semi-simple.
The full subcat. of Oq of all fin. dim. module is not semi-simple.
Each indecomposable block of Oint has finitely many simples.
Oq has blocks with infinitely many simples.
Each indecomposable block of Oint has one indec. proj.-inj. module.The endomorphism algebra of this module is commutative.
Oq has blocks with infinitely many indec. proj.-inj. modules.The end. alg. of the direct sum of these modules is not commutative.
Volodymyr Mazorchuk Category O for quantum groups
![Page 31: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/31.jpg)
Comparing Oint and Oq
The full subcat. of Oint of all fin. dim. modules is semi-simple.
The full subcat. of Oq of all fin. dim. module is not semi-simple.
Each indecomposable block of Oint has finitely many simples.
Oq has blocks with infinitely many simples.
Each indecomposable block of Oint has one indec. proj.-inj. module.The endomorphism algebra of this module is commutative.
Oq has blocks with infinitely many indec. proj.-inj. modules.The end. alg. of the direct sum of these modules is not commutative.
Volodymyr Mazorchuk Category O for quantum groups
![Page 32: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/32.jpg)
Comparing Oint and Oq
The full subcat. of Oint of all fin. dim. modules is semi-simple.
The full subcat. of Oq of all fin. dim. module is not semi-simple.
Each indecomposable block of Oint has finitely many simples.
Oq has blocks with infinitely many simples.
Each indecomposable block of Oint has one indec. proj.-inj. module.The endomorphism algebra of this module is commutative.
Oq has blocks with infinitely many indec. proj.-inj. modules.The end. alg. of the direct sum of these modules is not commutative.
Volodymyr Mazorchuk Category O for quantum groups
![Page 33: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/33.jpg)
Comparing Oint and Oq
The full subcat. of Oint of all fin. dim. modules is semi-simple.
The full subcat. of Oq of all fin. dim. module is not semi-simple.
Each indecomposable block of Oint has finitely many simples.
Oq has blocks with infinitely many simples.
Each indecomposable block of Oint has one indec. proj.-inj. module.The endomorphism algebra of this module is commutative.
Oq has blocks with infinitely many indec. proj.-inj. modules.The end. alg. of the direct sum of these modules is not commutative.
Volodymyr Mazorchuk Category O for quantum groups
![Page 34: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/34.jpg)
Comparing Oint and Oq
The full subcat. of Oint of all fin. dim. modules is semi-simple.
The full subcat. of Oq of all fin. dim. module is not semi-simple.
Each indecomposable block of Oint has finitely many simples.
Oq has blocks with infinitely many simples.
Each indecomposable block of Oint has one indec. proj.-inj. module.The endomorphism algebra of this module is commutative.
Oq has blocks with infinitely many indec. proj.-inj. modules.The end. alg. of the direct sum of these modules is not commutative.
Volodymyr Mazorchuk Category O for quantum groups
![Page 35: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/35.jpg)
Comparing Oint and Oq
The full subcat. of Oint of all fin. dim. modules is semi-simple.
The full subcat. of Oq of all fin. dim. module is not semi-simple.
Each indecomposable block of Oint has finitely many simples.
Oq has blocks with infinitely many simples.
Each indecomposable block of Oint has one indec. proj.-inj. module.The endomorphism algebra of this module is commutative.
Oq has blocks with infinitely many indec. proj.-inj. modules.The end. alg. of the direct sum of these modules is not commutative.
Volodymyr Mazorchuk Category O for quantum groups
![Page 36: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/36.jpg)
Comparing Oint and Oq
The full subcat. of Oint of all fin. dim. modules is semi-simple.
The full subcat. of Oq of all fin. dim. module is not semi-simple.
Each indecomposable block of Oint has finitely many simples.
Oq has blocks with infinitely many simples.
Each indecomposable block of Oint has one indec. proj.-inj. module.The endomorphism algebra of this module is commutative.
Oq has blocks with infinitely many indec. proj.-inj. modules.The end. alg. of the direct sum of these modules is not commutative.
Volodymyr Mazorchuk Category O for quantum groups
![Page 37: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/37.jpg)
The special block of Oq
λ — integral weight
Lq(λ) — simple highest weight module with highest weight λ
Every simple in Oq has the form Lq(λ)
ρ — half of the sum of all positive roots
Define the special block Ospecq as the Serre subcategory of Oq generated
by Lq(λ), λ ∈ kZn + (k − 1)ρ.
Theorem. The categories Oint and Ospecq are equivalent.
The equivalence is given by first applying −[k] and then tensoring with
the Steinberg module (simple h. w. module with h. w. (k − 1)ρ)
Volodymyr Mazorchuk Category O for quantum groups
![Page 38: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/38.jpg)
The special block of Oq
λ — integral weight
Lq(λ) — simple highest weight module with highest weight λ
Every simple in Oq has the form Lq(λ)
ρ — half of the sum of all positive roots
Define the special block Ospecq as the Serre subcategory of Oq generated
by Lq(λ), λ ∈ kZn + (k − 1)ρ.
Theorem. The categories Oint and Ospecq are equivalent.
The equivalence is given by first applying −[k] and then tensoring with
the Steinberg module (simple h. w. module with h. w. (k − 1)ρ)
Volodymyr Mazorchuk Category O for quantum groups
![Page 39: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/39.jpg)
The special block of Oq
λ — integral weight
Lq(λ) — simple highest weight module with highest weight λ
Every simple in Oq has the form Lq(λ)
ρ — half of the sum of all positive roots
Define the special block Ospecq as the Serre subcategory of Oq generated
by Lq(λ), λ ∈ kZn + (k − 1)ρ.
Theorem. The categories Oint and Ospecq are equivalent.
The equivalence is given by first applying −[k] and then tensoring with
the Steinberg module (simple h. w. module with h. w. (k − 1)ρ)
Volodymyr Mazorchuk Category O for quantum groups
![Page 40: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/40.jpg)
The special block of Oq
λ — integral weight
Lq(λ) — simple highest weight module with highest weight λ
Every simple in Oq has the form Lq(λ)
ρ — half of the sum of all positive roots
Define the special block Ospecq as the Serre subcategory of Oq generated
by Lq(λ), λ ∈ kZn + (k − 1)ρ.
Theorem. The categories Oint and Ospecq are equivalent.
The equivalence is given by first applying −[k] and then tensoring with
the Steinberg module (simple h. w. module with h. w. (k − 1)ρ)
Volodymyr Mazorchuk Category O for quantum groups
![Page 41: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/41.jpg)
The special block of Oq
λ — integral weight
Lq(λ) — simple highest weight module with highest weight λ
Every simple in Oq has the form Lq(λ)
ρ — half of the sum of all positive roots
Define the special block Ospecq as the Serre subcategory of Oq generated
by Lq(λ), λ ∈ kZn + (k − 1)ρ.
Theorem. The categories Oint and Ospecq are equivalent.
The equivalence is given by first applying −[k] and then tensoring with
the Steinberg module (simple h. w. module with h. w. (k − 1)ρ)
Volodymyr Mazorchuk Category O for quantum groups
![Page 42: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/42.jpg)
The special block of Oq
λ — integral weight
Lq(λ) — simple highest weight module with highest weight λ
Every simple in Oq has the form Lq(λ)
ρ — half of the sum of all positive roots
Define the special block Ospecq as the Serre subcategory of Oq generated
by Lq(λ), λ ∈ kZn + (k − 1)ρ.
Theorem. The categories Oint and Ospecq are equivalent.
The equivalence is given by first applying −[k] and then tensoring with
the Steinberg module (simple h. w. module with h. w. (k − 1)ρ)
Volodymyr Mazorchuk Category O for quantum groups
![Page 43: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/43.jpg)
The special block of Oq
λ — integral weight
Lq(λ) — simple highest weight module with highest weight λ
Every simple in Oq has the form Lq(λ)
ρ — half of the sum of all positive roots
Define the special block Ospecq as the Serre subcategory of Oq generated
by Lq(λ), λ ∈ kZn + (k − 1)ρ.
Theorem. The categories Oint and Ospecq are equivalent.
The equivalence is given by first applying −[k] and then tensoring with
the Steinberg module (simple h. w. module with h. w. (k − 1)ρ)
Volodymyr Mazorchuk Category O for quantum groups
![Page 44: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/44.jpg)
The special block of Oq
λ — integral weight
Lq(λ) — simple highest weight module with highest weight λ
Every simple in Oq has the form Lq(λ)
ρ — half of the sum of all positive roots
Define the special block Ospecq as the Serre subcategory of Oq generated
by Lq(λ), λ ∈ kZn + (k − 1)ρ.
Theorem. The categories Oint and Ospecq are equivalent.
The equivalence is given by first applying −[k] and then tensoring with
the Steinberg module (simple h. w. module with h. w. (k − 1)ρ)
Volodymyr Mazorchuk Category O for quantum groups
![Page 45: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/45.jpg)
The special block of Oq
λ — integral weight
Lq(λ) — simple highest weight module with highest weight λ
Every simple in Oq has the form Lq(λ)
ρ — half of the sum of all positive roots
Define the special block Ospecq as the Serre subcategory of Oq generated
by Lq(λ), λ ∈ kZn + (k − 1)ρ.
Theorem. The categories Oint and Ospecq are equivalent.
The equivalence is given by first applying −[k] and then tensoring with
the Steinberg module (simple h. w. module with h. w. (k − 1)ρ)
Volodymyr Mazorchuk Category O for quantum groups
![Page 46: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/46.jpg)
First consequences
Uq is a Hopf algebra
Oq has “projective endofunctors” given by tensoring with fin. dim.modules
Corollaries:
I Oq has enough projectives;
I Oq has enough injectives;
I Every object in Oq has finite length;
I Oq has the double centralizer property with respect to projectiveinjective modules, i.e. for any projective P there is an exact sequence
0→ P → X → Y
with both X and Y projective injective.
Volodymyr Mazorchuk Category O for quantum groups
![Page 47: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/47.jpg)
First consequences
Uq is a Hopf algebra
Oq has “projective endofunctors” given by tensoring with fin. dim.modules
Corollaries:
I Oq has enough projectives;
I Oq has enough injectives;
I Every object in Oq has finite length;
I Oq has the double centralizer property with respect to projectiveinjective modules, i.e. for any projective P there is an exact sequence
0→ P → X → Y
with both X and Y projective injective.
Volodymyr Mazorchuk Category O for quantum groups
![Page 48: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/48.jpg)
First consequences
Uq is a Hopf algebra
Oq has “projective endofunctors” given by tensoring with fin. dim.modules
Corollaries:
I Oq has enough projectives;
I Oq has enough injectives;
I Every object in Oq has finite length;
I Oq has the double centralizer property with respect to projectiveinjective modules, i.e. for any projective P there is an exact sequence
0→ P → X → Y
with both X and Y projective injective.
Volodymyr Mazorchuk Category O for quantum groups
![Page 49: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/49.jpg)
First consequences
Uq is a Hopf algebra
Oq has “projective endofunctors” given by tensoring with fin. dim.modules
Corollaries:
I Oq has enough projectives;
I Oq has enough injectives;
I Every object in Oq has finite length;
I Oq has the double centralizer property with respect to projectiveinjective modules, i.e. for any projective P there is an exact sequence
0→ P → X → Y
with both X and Y projective injective.
Volodymyr Mazorchuk Category O for quantum groups
![Page 50: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/50.jpg)
First consequences
Uq is a Hopf algebra
Oq has “projective endofunctors” given by tensoring with fin. dim.modules
Corollaries:
I Oq has enough projectives;
I Oq has enough injectives;
I Every object in Oq has finite length;
I Oq has the double centralizer property with respect to projectiveinjective modules, i.e. for any projective P there is an exact sequence
0→ P → X → Y
with both X and Y projective injective.
Volodymyr Mazorchuk Category O for quantum groups
![Page 51: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/51.jpg)
First consequences
Uq is a Hopf algebra
Oq has “projective endofunctors” given by tensoring with fin. dim.modules
Corollaries:
I Oq has enough projectives;
I Oq has enough injectives;
I Every object in Oq has finite length;
I Oq has the double centralizer property with respect to projectiveinjective modules, i.e. for any projective P there is an exact sequence
0→ P → X → Y
with both X and Y projective injective.
Volodymyr Mazorchuk Category O for quantum groups
![Page 52: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/52.jpg)
First consequences
Uq is a Hopf algebra
Oq has “projective endofunctors” given by tensoring with fin. dim.modules
Corollaries:
I Oq has enough projectives;
I Oq has enough injectives;
I Every object in Oq has finite length;
I Oq has the double centralizer property with respect to projectiveinjective modules, i.e. for any projective P there is an exact sequence
0→ P → X → Y
with both X and Y projective injective.
Volodymyr Mazorchuk Category O for quantum groups
![Page 53: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/53.jpg)
First consequences
Uq is a Hopf algebra
Oq has “projective endofunctors” given by tensoring with fin. dim.modules
Corollaries:
I Oq has enough projectives;
I Oq has enough injectives;
I Every object in Oq has finite length;
I Oq has the double centralizer property with respect to projectiveinjective modules, i.e.
for any projective P there is an exact sequence
0→ P → X → Y
with both X and Y projective injective.
Volodymyr Mazorchuk Category O for quantum groups
![Page 54: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/54.jpg)
First consequences
Uq is a Hopf algebra
Oq has “projective endofunctors” given by tensoring with fin. dim.modules
Corollaries:
I Oq has enough projectives;
I Oq has enough injectives;
I Every object in Oq has finite length;
I Oq has the double centralizer property with respect to projectiveinjective modules, i.e. for any projective P there is an exact sequence
0→ P → X → Y
with both X and Y projective injective.
Volodymyr Mazorchuk Category O for quantum groups
![Page 55: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/55.jpg)
Ringel self-duality
∆q(λ) — quantum Verma module with highest weight λ
Titling module: a self dual module with Verma flag
Lemma: Oq is a highest weight category.
Corollary: There is a bijection between indecomposable tilting modulesand simple modules in Oq.
Theorem: Oq is Ringel self-dual, i.e. the endomorphism algebra of abasic projective module in Oq is isomorphic to the endomorphism algebraof a basic tilting module in Oq.
Recall: Blocks of Oq might have infinitely many simples.
Volodymyr Mazorchuk Category O for quantum groups
![Page 56: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/56.jpg)
Ringel self-duality
∆q(λ) — quantum Verma module with highest weight λ
Titling module: a self dual module with Verma flag
Lemma: Oq is a highest weight category.
Corollary: There is a bijection between indecomposable tilting modulesand simple modules in Oq.
Theorem: Oq is Ringel self-dual, i.e. the endomorphism algebra of abasic projective module in Oq is isomorphic to the endomorphism algebraof a basic tilting module in Oq.
Recall: Blocks of Oq might have infinitely many simples.
Volodymyr Mazorchuk Category O for quantum groups
![Page 57: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/57.jpg)
Ringel self-duality
∆q(λ) — quantum Verma module with highest weight λ
Titling module: a self dual module with Verma flag
Lemma: Oq is a highest weight category.
Corollary: There is a bijection between indecomposable tilting modulesand simple modules in Oq.
Theorem: Oq is Ringel self-dual, i.e. the endomorphism algebra of abasic projective module in Oq is isomorphic to the endomorphism algebraof a basic tilting module in Oq.
Recall: Blocks of Oq might have infinitely many simples.
Volodymyr Mazorchuk Category O for quantum groups
![Page 58: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/58.jpg)
Ringel self-duality
∆q(λ) — quantum Verma module with highest weight λ
Titling module: a self dual module with Verma flag
Lemma: Oq is a highest weight category.
Corollary: There is a bijection between indecomposable tilting modulesand simple modules in Oq.
Theorem: Oq is Ringel self-dual, i.e. the endomorphism algebra of abasic projective module in Oq is isomorphic to the endomorphism algebraof a basic tilting module in Oq.
Recall: Blocks of Oq might have infinitely many simples.
Volodymyr Mazorchuk Category O for quantum groups
![Page 59: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/59.jpg)
Ringel self-duality
∆q(λ) — quantum Verma module with highest weight λ
Titling module: a self dual module with Verma flag
Lemma: Oq is a highest weight category.
Corollary: There is a bijection between indecomposable tilting modulesand simple modules in Oq.
Theorem: Oq is Ringel self-dual, i.e. the endomorphism algebra of abasic projective module in Oq is isomorphic to the endomorphism algebraof a basic tilting module in Oq.
Recall: Blocks of Oq might have infinitely many simples.
Volodymyr Mazorchuk Category O for quantum groups
![Page 60: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/60.jpg)
Ringel self-duality
∆q(λ) — quantum Verma module with highest weight λ
Titling module: a self dual module with Verma flag
Lemma: Oq is a highest weight category.
Corollary: There is a bijection between indecomposable tilting modulesand simple modules in Oq.
Theorem: Oq is Ringel self-dual, i.e. the endomorphism algebra of abasic projective module in Oq is isomorphic to the endomorphism algebraof a basic tilting module in Oq.
Recall: Blocks of Oq might have infinitely many simples.
Volodymyr Mazorchuk Category O for quantum groups
![Page 61: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/61.jpg)
Ringel self-duality
∆q(λ) — quantum Verma module with highest weight λ
Titling module: a self dual module with Verma flag
Lemma: Oq is a highest weight category.
Corollary: There is a bijection between indecomposable tilting modulesand simple modules in Oq.
Theorem: Oq is Ringel self-dual, i.e. the endomorphism algebra of abasic projective module in Oq is isomorphic to the endomorphism algebraof a basic tilting module in Oq.
Recall: Blocks of Oq might have infinitely many simples.
Volodymyr Mazorchuk Category O for quantum groups
![Page 62: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/62.jpg)
Ringel self-duality
∆q(λ) — quantum Verma module with highest weight λ
Titling module: a self dual module with Verma flag
Lemma: Oq is a highest weight category.
Corollary: There is a bijection between indecomposable tilting modulesand simple modules in Oq.
Theorem: Oq is Ringel self-dual, i.e. the endomorphism algebra of abasic projective module in Oq is isomorphic to the endomorphism algebraof a basic tilting module in Oq.
Recall: Blocks of Oq might have infinitely many simples.
Volodymyr Mazorchuk Category O for quantum groups
![Page 63: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/63.jpg)
Decomposition theorem for structural modules
write λ = λ0 + kλ1, 0 ≤ λ0i < k.
Proposition (Lusztig). Lq(λ) ∼= L(λ1)[k] ⊗ Lq(λ0).
if L(λ) is finite dimensional, denote by Qq(λ) the projective cover of L(λ)in the category of finite dimensional Uq-modules.
Theorem. Pq(λ) ∼= P(λ1)[k] ⊗ Qq(λ0).
Theorem. Iq(λ) ∼= I (λ1)[k] ⊗ Qq(λ0).
Theorem. Tq(λ) ∼= T (λ1 − ρ)[k] ⊗ Qq(kρ+ wo · λ0).
Volodymyr Mazorchuk Category O for quantum groups
![Page 64: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/64.jpg)
Decomposition theorem for structural modules
write λ = λ0 + kλ1, 0 ≤ λ0i < k .
Proposition (Lusztig). Lq(λ) ∼= L(λ1)[k] ⊗ Lq(λ0).
if L(λ) is finite dimensional, denote by Qq(λ) the projective cover of L(λ)in the category of finite dimensional Uq-modules.
Theorem. Pq(λ) ∼= P(λ1)[k] ⊗ Qq(λ0).
Theorem. Iq(λ) ∼= I (λ1)[k] ⊗ Qq(λ0).
Theorem. Tq(λ) ∼= T (λ1 − ρ)[k] ⊗ Qq(kρ+ wo · λ0).
Volodymyr Mazorchuk Category O for quantum groups
![Page 65: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/65.jpg)
Decomposition theorem for structural modules
write λ = λ0 + kλ1, 0 ≤ λ0i < k .
Proposition (Lusztig). Lq(λ) ∼= L(λ1)[k] ⊗ Lq(λ0).
if L(λ) is finite dimensional, denote by Qq(λ) the projective cover of L(λ)in the category of finite dimensional Uq-modules.
Theorem. Pq(λ) ∼= P(λ1)[k] ⊗ Qq(λ0).
Theorem. Iq(λ) ∼= I (λ1)[k] ⊗ Qq(λ0).
Theorem. Tq(λ) ∼= T (λ1 − ρ)[k] ⊗ Qq(kρ+ wo · λ0).
Volodymyr Mazorchuk Category O for quantum groups
![Page 66: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/66.jpg)
Decomposition theorem for structural modules
write λ = λ0 + kλ1, 0 ≤ λ0i < k .
Proposition (Lusztig). Lq(λ) ∼= L(λ1)[k] ⊗ Lq(λ0).
if L(λ) is finite dimensional, denote by Qq(λ) the projective cover of L(λ)in the category of finite dimensional Uq-modules.
Theorem. Pq(λ) ∼= P(λ1)[k] ⊗ Qq(λ0).
Theorem. Iq(λ) ∼= I (λ1)[k] ⊗ Qq(λ0).
Theorem. Tq(λ) ∼= T (λ1 − ρ)[k] ⊗ Qq(kρ+ wo · λ0).
Volodymyr Mazorchuk Category O for quantum groups
![Page 67: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/67.jpg)
Decomposition theorem for structural modules
write λ = λ0 + kλ1, 0 ≤ λ0i < k .
Proposition (Lusztig). Lq(λ) ∼= L(λ1)[k] ⊗ Lq(λ0).
if L(λ) is finite dimensional, denote by Qq(λ) the projective cover of L(λ)in the category of finite dimensional Uq-modules.
Theorem. Pq(λ) ∼= P(λ1)[k] ⊗ Qq(λ0).
Theorem. Iq(λ) ∼= I (λ1)[k] ⊗ Qq(λ0).
Theorem. Tq(λ) ∼= T (λ1 − ρ)[k] ⊗ Qq(kρ+ wo · λ0).
Volodymyr Mazorchuk Category O for quantum groups
![Page 68: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/68.jpg)
Decomposition theorem for structural modules
write λ = λ0 + kλ1, 0 ≤ λ0i < k .
Proposition (Lusztig). Lq(λ) ∼= L(λ1)[k] ⊗ Lq(λ0).
if L(λ) is finite dimensional, denote by Qq(λ) the projective cover of L(λ)in the category of finite dimensional Uq-modules.
Theorem. Pq(λ) ∼= P(λ1)[k] ⊗ Qq(λ0).
Theorem. Iq(λ) ∼= I (λ1)[k] ⊗ Qq(λ0).
Theorem. Tq(λ) ∼= T (λ1 − ρ)[k] ⊗ Qq(kρ+ wo · λ0).
Volodymyr Mazorchuk Category O for quantum groups
![Page 69: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/69.jpg)
Decomposition theorem for structural modules
write λ = λ0 + kλ1, 0 ≤ λ0i < k .
Proposition (Lusztig). Lq(λ) ∼= L(λ1)[k] ⊗ Lq(λ0).
if L(λ) is finite dimensional, denote by Qq(λ) the projective cover of L(λ)in the category of finite dimensional Uq-modules.
Theorem. Pq(λ) ∼= P(λ1)[k] ⊗ Qq(λ0).
Theorem. Iq(λ) ∼= I (λ1)[k] ⊗ Qq(λ0).
Theorem. Tq(λ) ∼= T (λ1 − ρ)[k] ⊗ Qq(kρ+ wo · λ0).
Volodymyr Mazorchuk Category O for quantum groups
![Page 70: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/70.jpg)
Decomposition theorem for structural modules
write λ = λ0 + kλ1, 0 ≤ λ0i < k .
Proposition (Lusztig). Lq(λ) ∼= L(λ1)[k] ⊗ Lq(λ0).
if L(λ) is finite dimensional, denote by Qq(λ) the projective cover of L(λ)in the category of finite dimensional Uq-modules.
Theorem. Pq(λ) ∼= P(λ1)[k] ⊗ Qq(λ0).
Theorem. Iq(λ) ∼= I (λ1)[k] ⊗ Qq(λ0).
Theorem. Tq(λ) ∼= T (λ1 − ρ)[k] ⊗ Qq(kρ+ wo · λ0).
Volodymyr Mazorchuk Category O for quantum groups
![Page 71: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/71.jpg)
Characters and multiplicities
Proposition (BGG-reciprocity). (Pq(λ) : ∆q(µ)) = [∆q(µ) : Lq(λ)]
Px,y — Kazhdan-Lusztig polynomials; Qx,y — their “inverses”
Theorem. The formal character of Lq(λ) equals∑y∈W , z∈Wl
(−1)l(yw)+l(zx)Py ,w (1)Pz,x(1)ch∆q
(l(yw−1 · λ1) + zx−1 · λ0)
)
Theorem. For regular λ the multiplicity (Tq(λ) : ∆q(µ)) equals∑y ,z
Py ,w (1)Qz,x(1)
where the sum runs over those y ∈W , z ∈Wl for whichµ = lyw−1 · (λ1 − ρ) + zx−1 · λ0.
Volodymyr Mazorchuk Category O for quantum groups
![Page 72: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/72.jpg)
Characters and multiplicities
Proposition (BGG-reciprocity). (Pq(λ) : ∆q(µ)) = [∆q(µ) : Lq(λ)]
Px,y — Kazhdan-Lusztig polynomials; Qx,y — their “inverses”
Theorem. The formal character of Lq(λ) equals∑y∈W , z∈Wl
(−1)l(yw)+l(zx)Py ,w (1)Pz,x(1)ch∆q
(l(yw−1 · λ1) + zx−1 · λ0)
)
Theorem. For regular λ the multiplicity (Tq(λ) : ∆q(µ)) equals∑y ,z
Py ,w (1)Qz,x(1)
where the sum runs over those y ∈W , z ∈Wl for whichµ = lyw−1 · (λ1 − ρ) + zx−1 · λ0.
Volodymyr Mazorchuk Category O for quantum groups
![Page 73: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/73.jpg)
Characters and multiplicities
Proposition (BGG-reciprocity). (Pq(λ) : ∆q(µ)) = [∆q(µ) : Lq(λ)]
Px,y — Kazhdan-Lusztig polynomials; Qx,y — their “inverses”
Theorem. The formal character of Lq(λ) equals∑y∈W , z∈Wl
(−1)l(yw)+l(zx)Py ,w (1)Pz,x(1)ch∆q
(l(yw−1 · λ1) + zx−1 · λ0)
)
Theorem. For regular λ the multiplicity (Tq(λ) : ∆q(µ)) equals∑y ,z
Py ,w (1)Qz,x(1)
where the sum runs over those y ∈W , z ∈Wl for whichµ = lyw−1 · (λ1 − ρ) + zx−1 · λ0.
Volodymyr Mazorchuk Category O for quantum groups
![Page 74: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/74.jpg)
Characters and multiplicities
Proposition (BGG-reciprocity). (Pq(λ) : ∆q(µ)) = [∆q(µ) : Lq(λ)]
Px,y — Kazhdan-Lusztig polynomials; Qx,y — their “inverses”
Theorem. The formal character of Lq(λ) equals∑y∈W , z∈Wl
(−1)l(yw)+l(zx)Py ,w (1)Pz,x(1)ch∆q
(l(yw−1 · λ1) + zx−1 · λ0)
)
Theorem. For regular λ the multiplicity (Tq(λ) : ∆q(µ)) equals∑y ,z
Py ,w (1)Qz,x(1)
where the sum runs over those y ∈W , z ∈Wl for whichµ = lyw−1 · (λ1 − ρ) + zx−1 · λ0.
Volodymyr Mazorchuk Category O for quantum groups
![Page 75: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/75.jpg)
Characters and multiplicities
Proposition (BGG-reciprocity). (Pq(λ) : ∆q(µ)) = [∆q(µ) : Lq(λ)]
Px,y — Kazhdan-Lusztig polynomials; Qx,y — their “inverses”
Theorem. The formal character of Lq(λ) equals∑y∈W , z∈Wl
(−1)l(yw)+l(zx)Py ,w (1)Pz,x(1)ch∆q
(l(yw−1 · λ1) + zx−1 · λ0)
)
Theorem. For regular λ the multiplicity (Tq(λ) : ∆q(µ)) equals∑y ,z
Py ,w (1)Qz,x(1)
where the sum runs over those y ∈W , z ∈Wl for whichµ = lyw−1 · (λ1 − ρ) + zx−1 · λ0.
Volodymyr Mazorchuk Category O for quantum groups
![Page 76: Category O for quantum groupsmazor/PREPRINTS/DOKLADY/krakow.pdf · Category Ofor quantum groups (joint work with Henning Haahr Andersen) Volodymyr Mazorchuk (Uppsala University) Mathematical](https://reader033.vdocuments.us/reader033/viewer/2022060504/5f1d548f23b13d661333897b/html5/thumbnails/76.jpg)
Characters and multiplicities
Proposition (BGG-reciprocity). (Pq(λ) : ∆q(µ)) = [∆q(µ) : Lq(λ)]
Px,y — Kazhdan-Lusztig polynomials; Qx,y — their “inverses”
Theorem. The formal character of Lq(λ) equals∑y∈W , z∈Wl
(−1)l(yw)+l(zx)Py ,w (1)Pz,x(1)ch∆q
(l(yw−1 · λ1) + zx−1 · λ0)
)
Theorem. For regular λ the multiplicity (Tq(λ) : ∆q(µ)) equals∑y ,z
Py ,w (1)Qz,x(1)
where the sum runs over those y ∈W , z ∈Wl for whichµ = lyw−1 · (λ1 − ρ) + zx−1 · λ0.
Volodymyr Mazorchuk Category O for quantum groups