The Combinatorics of Category O over symmetrizable Kac-Moody Algebras

By P. Fiebig. 2003

63 citations recorded by the source corpus.

We show that the structure of a block outside the critical hyperplanes of category O over a symmetrizable Kac-Moody algebra depends only on the corresponding integral Weyl group and its action on the parameters of the Verma modules. This is done by giving a combinatorial description of the projective objects in the block. As an application, we derive the Kazhdan-Lusztig conjecture for nonintegral blocks from the integral case for finite or affine Weyl groups. We also prove the uniqueness of Verma embeddings outside the critical hyperplanes.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d193fd9dfcb38aa873f1f089bc10c3caefae20ee

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In Discrete Mathematics and Combinatorics

1. T. Arakawa
2. V. Kac
3. M. Wakimoto
4. Shi-Shyr Roan
5. E. Vasserot
6. P. Shan
7. M. Varagnolo
8. P. Fiebig

In Geometry and Topology

1. T. Arakawa
2. V. Kac
3. E. Vasserot
4. Alexander S. Kleshchev
5. M. Wakimoto
6. M. Varagnolo
7. Jonathan Brundan
8. Ben Webster

In Mathematical Physics

1. E. Vasserot
2. T. Arakawa
3. V. Kac
4. Shi-Shyr Roan
5. M. Wakimoto
6. P. Shan
7. M. Varagnolo
8. P. Fiebig

Citations in the corpus

• The combinatorics of Coxeter categories (2005, background): “Proof. The following construction of B(x) is well-known in similar situations (cf. [Soe1, Fie1 , Soe3]).”
• Sheaves on moment graphs and a localization of Verma flags (2005, methodology): “In the following we make extensive use of the base change results that are described in [Fie1], 2.3, and [ Fie2 ].”
• Rationality of admissible affine vertex algebras in the category O (2012, methodology): “lbgmodule homomorphism W(w◦λ) → W(w′◦λ), which is unique up to multiplication bya nonzeroconstant. The existence ofsucha resolutionwasconjectured in [FKW], and was proved in [A3] by applying a result [Fie] of Fiebig and a method [Ark] of Arkhipov: Fiebig’s equivalence between different blocks of cate”
• Categorifications and cyclotomic rational double affine Hecke algebras (2013, background): “Next, a theorem of Fiebig asserts that the structure of the category O of a Kac-Moody algebra only depends on the associated Coxeter system [24].”
• Quantum Reduction for Affine Superalgebras (2003, methodology): “Recall that a vertex algebra is called strongly generated by a collection of fields F if normally ordered products of fields from C[∂] F span the space of fields of this vertex algebra.”