Serre functors for Lie algebras and superalgebras

Volodymyr Mazorchuk[1]; Vanessa Miemietz[2]

  • [1] Uppsala University Department of Mathematics Box 480 751 06, Uppsala (Sweden)
  • [2] University of East Anglia School of Mathematics Norwich NR4 7TJ (United Kingdom)

Annales de l’institut Fourier (2012)

  • Volume: 62, Issue: 1, page 47-75
  • ISSN: 0373-0956

Abstract

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We propose a new realization, using Harish-Chandra bimodules, of the Serre functor for the BGG category 𝒪 associated to a semi-simple complex finite dimensional Lie algebra. We further show that our realization carries over to classical Lie superalgebras in many cases. Along the way we prove that category 𝒪 and its parabolic generalizations for classical Lie superalgebras are categories with full projective functors. As an application we prove that in many cases the endomorphism algebra of the basic projective-injective module in (parabolic) category  𝒪 for classical Lie superalgebras is symmetric. As a special case we obtain that in these cases the algebras describing blocks of the category of finite dimensional modules are symmetric. We also compute the latter algebras for the superalgebra  𝔮 ( 2 ) .

How to cite

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Mazorchuk, Volodymyr, and Miemietz, Vanessa. "Serre functors for Lie algebras and superalgebras." Annales de l’institut Fourier 62.1 (2012): 47-75. <http://eudml.org/doc/251081>.

@article{Mazorchuk2012,
abstract = {We propose a new realization, using Harish-Chandra bimodules, of the Serre functor for the BGG category $\mathcal\{O\}$ associated to a semi-simple complex finite dimensional Lie algebra. We further show that our realization carries over to classical Lie superalgebras in many cases. Along the way we prove that category $\mathcal\{O\}$ and its parabolic generalizations for classical Lie superalgebras are categories with full projective functors. As an application we prove that in many cases the endomorphism algebra of the basic projective-injective module in (parabolic) category $\mathcal\{O\}$ for classical Lie superalgebras is symmetric. As a special case we obtain that in these cases the algebras describing blocks of the category of finite dimensional modules are symmetric. We also compute the latter algebras for the superalgebra $\mathfrak\{q\}(2)$.},
affiliation = {Uppsala University Department of Mathematics Box 480 751 06, Uppsala (Sweden); University of East Anglia School of Mathematics Norwich NR4 7TJ (United Kingdom)},
author = {Mazorchuk, Volodymyr, Miemietz, Vanessa},
journal = {Annales de l’institut Fourier},
keywords = {Lie superalgebra; module; Harish-Chandra bimodule; Serre functor; quiver; category $\mathcal\{O\}$; category },
language = {eng},
number = {1},
pages = {47-75},
publisher = {Association des Annales de l’institut Fourier},
title = {Serre functors for Lie algebras and superalgebras},
url = {http://eudml.org/doc/251081},
volume = {62},
year = {2012},
}

TY - JOUR
AU - Mazorchuk, Volodymyr
AU - Miemietz, Vanessa
TI - Serre functors for Lie algebras and superalgebras
JO - Annales de l’institut Fourier
PY - 2012
PB - Association des Annales de l’institut Fourier
VL - 62
IS - 1
SP - 47
EP - 75
AB - We propose a new realization, using Harish-Chandra bimodules, of the Serre functor for the BGG category $\mathcal{O}$ associated to a semi-simple complex finite dimensional Lie algebra. We further show that our realization carries over to classical Lie superalgebras in many cases. Along the way we prove that category $\mathcal{O}$ and its parabolic generalizations for classical Lie superalgebras are categories with full projective functors. As an application we prove that in many cases the endomorphism algebra of the basic projective-injective module in (parabolic) category $\mathcal{O}$ for classical Lie superalgebras is symmetric. As a special case we obtain that in these cases the algebras describing blocks of the category of finite dimensional modules are symmetric. We also compute the latter algebras for the superalgebra $\mathfrak{q}(2)$.
LA - eng
KW - Lie superalgebra; module; Harish-Chandra bimodule; Serre functor; quiver; category $\mathcal{O}$; category
UR - http://eudml.org/doc/251081
ER -

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