# Category $\mathcal{O}$ for quantum groups

Henning Haahr Andersen; Volodymyr Mazorchuk

Journal of the European Mathematical Society (2015)

- Volume: 017, Issue: 2, page 405-431
- ISSN: 1435-9855

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topAndersen, 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 -

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