Category $𝒪$ for quantum groups

Andersen, Henning Haahr, and Mazorchuk, Volodymyr. "Category $\mathcal {O}$ for quantum groups." Journal of the European Mathematical Society 017.2 (2015): 405-431. <http://eudml.org/doc/277373>.

@article{Andersen2015,
abstract = {In this paper we study the BGG-categories $\mathcal \{O\}_q$ associated to quantum groups. We prove that many properties of the ordinary BGG-category $\mathcal \{O\}$ for a semisimple complex Lie algebra carry over to the quantum case. Of particular interest is the case when $q$ is a complex root of unity. Here we prove a tensor decomposition for both simple modules, projective modules, and indecomposable tilting modules. Using the known Kazhdan-Lusztig conjectures for $\mathcal \{O\}$ and for finite dimensional $U_q$-modules we are able to determine all irreducible characters as well as the characters of all indecomposable tilting modules in $\mathcal \{O\}_q$. As a consequence of these results we are able to recover also a known result, namely that the generic quantum case behaves like the classical category $\mathcal \{O\}$.},
author = {Andersen, Henning Haahr, Mazorchuk, Volodymyr},
journal = {Journal of the European Mathematical Society},
keywords = {quantized highest weights modules; specialization at roots of unity; tensor decompositions; tilting modules; category ; quantized enveloping algebras; tilting modules},
language = {eng},
number = {2},
pages = {405-431},
publisher = {European Mathematical Society Publishing House},
title = {Category $\mathcal \{O\}$ for quantum groups},
url = {http://eudml.org/doc/277373},
volume = {017},
year = {2015},
}

TY - JOUR
AU - Andersen, Henning Haahr
AU - Mazorchuk, Volodymyr
TI - Category $\mathcal {O}$ for quantum groups
JO - Journal of the European Mathematical Society
PY - 2015
PB - European Mathematical Society Publishing House
VL - 017
IS - 2
SP - 405
EP - 431
AB - In this paper we study the BGG-categories $\mathcal {O}_q$ associated to quantum groups. We prove that many properties of the ordinary BGG-category $\mathcal {O}$ for a semisimple complex Lie algebra carry over to the quantum case. Of particular interest is the case when $q$ is a complex root of unity. Here we prove a tensor decomposition for both simple modules, projective modules, and indecomposable tilting modules. Using the known Kazhdan-Lusztig conjectures for $\mathcal {O}$ and for finite dimensional $U_q$-modules we are able to determine all irreducible characters as well as the characters of all indecomposable tilting modules in $\mathcal {O}_q$. As a consequence of these results we are able to recover also a known result, namely that the generic quantum case behaves like the classical category $\mathcal {O}$.
LA - eng
KW - quantized highest weights modules; specialization at roots of unity; tensor decompositions; tilting modules; category ; quantized enveloping algebras; tilting modules
UR - http://eudml.org/doc/277373
ER -