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Representations of the General Linear Lie Superalgebra in the BGG Category \(\mathcal{O}\)

  • Jonathan BrundanEmail author
Chapter
Part of the Developments in Mathematics book series (DEVM, volume 38)

Abstract

This is a survey of some recent developments in the highest weight repesentation theory of the general linear Lie superalgebra \(\mathfrak{g}\mathfrak{l}_{n\vert m}(\mathbb{C})\). The main focus is on the analog of the Kazhdan–Lusztig conjecture as formulated by the author in 2002, which was finally proved in 2011 by Cheng, Lam and Wang. Recently another proof has been obtained by the author joint with Losev and Webster, by a method which leads moreover to the construction of a Koszul-graded lift of category \(\mathcal{O}\) for this Lie superalgebra.

Key words

General linear Lie superalgebra Category \(\mathcal{O}\) 

Mathematics Subject Classification (2010):

17B10 17B37. 

Notes

Acknowledgements

Special thanks go to Catharina Stroppel for several discussions which influenced this exposition. I also thank Geoff Mason, Ivan Penkov and Joe Wolf for providing me the opportunity to write a survey article of this nature. In fact I gave a talk on exactly this topic at the Seminar “Lie Groups, Lie Algebras and their Representations” at Riverside in November 2002, when the super Kazhdan–Lusztig conjecture was newborn.

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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of OregonEugeneUSA

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