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Rigidity of tilting modules

Research output: Working paper/Preprint Working paper

Standard

Rigidity of tilting modules. / Haahr Andersen, Henning; Kaneda, Masaharu.

arXiv.org, 2009.

Research output: Working paper/Preprint Working paper

Harvard

Haahr Andersen, H & Kaneda, M 2009 'Rigidity of tilting modules' arXiv.org. <https://arxiv.org/abs/0909.2935>

APA

Haahr Andersen, H., & Kaneda, M. (2009). Rigidity of tilting modules. arXiv.org. https://arxiv.org/abs/0909.2935

CBE

Haahr Andersen H, Kaneda M. 2009. Rigidity of tilting modules. arXiv.org.

MLA

Haahr Andersen, Henning and Masaharu Kaneda Rigidity of tilting modules. arXiv.org. 2009., 41 p.

Vancouver

Haahr Andersen H, Kaneda M. Rigidity of tilting modules. arXiv.org. 2009.

Author

Haahr Andersen, Henning ; Kaneda, Masaharu. / Rigidity of tilting modules. arXiv.org, 2009.

Bibtex

@techreport{55c714b0220711dfb95d000ea68e967b,
title = "Rigidity of tilting modules",
abstract = "Let $U_q$ denote the quantum group associated with a finite dimensional semisimple Lie algebra. Assume that $q$ is a complex root of unity of odd order and that $U_q$ is %the quantum group version obtained via Lusztig's $q$-divided powers construction. We prove that all regular projective (tilting) modules for $U_q$ are rigid, i.e., have identical radical and socle filtrations. Moreover, we obtain the same for a large class of Weyl modules for $U_q$. On the other hand, we give examples of non-rigid indecomposable tilting modules as well as non-rigid Weyl modules. These examples are for type $B_2$ and in this case as well as for type $A_2$ we calculate explicitly the Loewy structure for all regular Weyl modules. We also demonstrate that these results carry over to the modular case when the highest weights in question are in the so-called Jantzen region. At the same time we show by examples that as soon as we leave this region non-rigid tilting modules do occur.",
keywords = "math.RT, math.QA, 17B37; 20G05",
author = "{Haahr Andersen}, Henning and Masaharu Kaneda",
year = "2009",
language = "English",
publisher = "arXiv.org",
type = "WorkingPaper",
institution = "arXiv.org",

}

RIS

TY - UNPB

T1 - Rigidity of tilting modules

AU - Haahr Andersen, Henning

AU - Kaneda, Masaharu

PY - 2009

Y1 - 2009

N2 - Let $U_q$ denote the quantum group associated with a finite dimensional semisimple Lie algebra. Assume that $q$ is a complex root of unity of odd order and that $U_q$ is %the quantum group version obtained via Lusztig's $q$-divided powers construction. We prove that all regular projective (tilting) modules for $U_q$ are rigid, i.e., have identical radical and socle filtrations. Moreover, we obtain the same for a large class of Weyl modules for $U_q$. On the other hand, we give examples of non-rigid indecomposable tilting modules as well as non-rigid Weyl modules. These examples are for type $B_2$ and in this case as well as for type $A_2$ we calculate explicitly the Loewy structure for all regular Weyl modules. We also demonstrate that these results carry over to the modular case when the highest weights in question are in the so-called Jantzen region. At the same time we show by examples that as soon as we leave this region non-rigid tilting modules do occur.

AB - Let $U_q$ denote the quantum group associated with a finite dimensional semisimple Lie algebra. Assume that $q$ is a complex root of unity of odd order and that $U_q$ is %the quantum group version obtained via Lusztig's $q$-divided powers construction. We prove that all regular projective (tilting) modules for $U_q$ are rigid, i.e., have identical radical and socle filtrations. Moreover, we obtain the same for a large class of Weyl modules for $U_q$. On the other hand, we give examples of non-rigid indecomposable tilting modules as well as non-rigid Weyl modules. These examples are for type $B_2$ and in this case as well as for type $A_2$ we calculate explicitly the Loewy structure for all regular Weyl modules. We also demonstrate that these results carry over to the modular case when the highest weights in question are in the so-called Jantzen region. At the same time we show by examples that as soon as we leave this region non-rigid tilting modules do occur.

KW - math.RT

KW - math.QA

KW - 17B37; 20G05

M3 - Working paper

BT - Rigidity of tilting modules

PB - arXiv.org

ER -