BGG categories in prime characteristics
Henning Haahr Andersen

TL;DR
This paper explores the structure of the BGG category for quantum groups at roots of unity in positive characteristic, establishing key theorems about simple modules, Verma modules, and tilting modules, with implications for classical cases.
Contribution
It introduces a Steinberg tensor product theorem for simple modules in the quantum BGG category at roots of unity in positive characteristic.
Findings
Proved a Steinberg tensor product theorem for simple modules.
Established a finite filtration and linkage principle for Verma modules.
Identified the special Verma module as simple and projective/injective.
Abstract
Let be a simple complex Lie algebra. In this paper we study the BGG category for the quantum group with being a root of unity in a field of characteristic . We first consider the simple modules in and prove a Steinberg tensor product theorem for them. This result reduces the problem of determining the corresponding irreducible characters to the same problem for a finite subset of finite dimensional simple modules. Then we investigate more closely the Verma modules in . Except for the special Verma module, which has highest weight , they all have infinite length. Nevertheless, we show that each Verma module has a certain finite filtration with an associated strong linkage principle. The special Verma module turns out to be both simple and projective/injective. This leads to a family of…
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