Injective Modules for Quantum Algebras

Andersen, Henning Haahr; Polo, Patrick; Kexin, Wen

doi:10.2307/2374770

Cited by 46 publications

(64 citation statements)

References 2 publications

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“…where Hom a (A, N ) is an infinite-dimensional vector space with left A-action given by (x · f )(y) = f (S(x) · y); see, e.g., [APW2]. It is clear that the functor Ind A a is the right adjoint to the obvious restriction functor Res Let T ⊂ B be the maximal torus and the Borel subgroup corresponding to the Lie algebras t ⊂ b, respectively.…”

Section: Smooth Coinductionmentioning

confidence: 99%

Quantum groups, the loop Grassmannian, and the Springer resolution

Arkhipov

Bezrukavnikov

Ginzburg

2004

J. Amer. Math. Soc.

184

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We establish equivalences of the following three triangulated categories: \[ D quantum ( g ) ⟷ D coherent G ( N ~ ) ⟷ D perverse ( G r ) . D_\text {quantum}(\mathfrak {g})\enspace \longleftrightarrow \enspace D^G_\text {coherent}(\widetilde {{\mathcal N}})\enspace \longleftrightarrow \enspace D_\text {perverse}(\mathsf {Gr}). \] Here, D quantum ( g ) D_\text {quantum}(\mathfrak {g}) is the derived category of the principal block of finite-dimensional representations of the quantized enveloping algebra (at an odd root of unity) of a complex semisimple Lie algebra g \mathfrak {g} ; the category D coherent G ( N ~ ) D^G_\text {coherent}(\widetilde {{\mathcal N}}) is defined in terms of coherent sheaves on the cotangent bundle on the (finite-dimensional) flag manifold for G G ( = = semisimple group with Lie algebra g \mathfrak {g} ), and the category D perverse ( G r ) D_\text {perverse}({\mathsf {Gr}}) is the derived category of perverse sheaves on the Grassmannian G r {\mathsf {Gr}} associated with the loop group L G ∨ LG^\vee , where G ∨ G^\vee is the Langlands dual group, smooth along the Schubert stratification. The equivalence between D quantum ( g ) D_\text {quantum}(\mathfrak {g}) and D coherent G ( N ~ ) D^G_\text {coherent}(\widetilde {{\mathcal N}}) is an “enhancement” of the known expression (due to Ginzburg and Kumar) for quantum group cohomology in terms of nilpotent variety. The equivalence between D perverse ( G r ) D_\text {perverse}(\mathsf {Gr}) and D coherent G ( N ~ ) D^G_\text {coherent}(\widetilde {{\mathcal N}}) can be viewed as a “categorification” of the isomorphism between two completely different geometric realizations of the (fundamental polynomial representation of the) affine Hecke algebra that has played a key role in the proof of the Deligne-Langlands-Lusztig conjecture. One realization is in terms of locally constant functions on the flag manifold of a p p -adic reductive group, while the other is in terms of equivariant K K -theory of a complex (Steinberg) variety for the dual group. The composite of the two equivalences above yields an equivalence between abelian categories of quantum group representations and perverse sheaves. A similar equivalence at an even root of unity can be deduced, following the Lusztig program, from earlier deep results of Kazhdan-Lusztig and Kashiwara-Tanisaki. Our approach is independent of these results and is totally different (it does not rely on the representation theory of Kac-Moody algebras). It also gives way to proving Humphreys’ conjectures on tilting U q ( g ) U_q(\mathfrak {g}) -modules, as will be explained in a separate paper.

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Section: Smooth Coinductionmentioning

confidence: 99%

Quantum groups, the loop Grassmannian, and the Springer resolution

Arkhipov

Bezrukavnikov

Ginzburg

2004

J. Amer. Math. Soc.

184

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“…Then using standard results of [APW1], [APW2], [AW], in particular, the quantum version of Kempf's theorem 7 , it follows that U k -mod is a highest weight category with weight poset Λ = X + . An argument is provided in [DS] for a somewhat weaker result-entirely adequate for our purposes here-that the modules in C form a "k-finite highest weight category.…”

Section: Quantum Groups At a Root Of Unitymentioning

confidence: 99%

On Ext-Transfer for Algebraic Groups

Cline

Parshall

Scott

2004

Transformation Groups

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This paper builds upon the work of Cline [C] and Donkin [D1] to describe explicit equivalences between some categories associated to the category of rational modules for a reductive group G and categories associated to the category of rational modules for a Levi subgroup H. As an application, we establish an Ext-transfer result from rational G-modules to rational H-modules. In case G = GLn, these results can be illustrated in terms of classical Schur algebras. In that case, we establish another category equivalence, this time between the module category for a Schur algebra and the module category for a union of blocks for a natural quotient of a larger Schur algebra. This category equivalence provides a further Ext-transfer theorem from the original Schur algebra to the larger Schur algebra. This result, announced in [PS2, (6.4b)], extends to the category level the decomposition number method of Erdmann [E2]. Finally, we indicate (largely without proof) some natural variations to situations involving quantum groups and q-Schur algebras.

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“…If we write λ = λ 0 + λ 1 with λ 0 ∈ Λ res and λ 1 ∈ Λ, one has from [APW,1.9] an isomorphism of u-modules…”

Section: Infinitesimal Verma Modulesmentioning

confidence: 99%

“…Let ε λ ∈ ∇(λ) be the element induced by the counit of u + . By the universality of ∇(λ) [APW,0.8.1] there is a homomorphism of u-modules…”

Section: Infinitesimal Verma Modulesmentioning

confidence: 99%