Twisting functors and generalized Verma modules
HTML articles powered by AMS MathViewer
- by Ian M. Musson PDF
- Proc. Amer. Math. Soc. 147 (2019), 1013-1022 Request permission
Abstract:
Let $\mathfrak {g}$ be a reductive Lie algebra. We give a condition that ensures that the character of a generalized Verma module is well behaved under a twisting functor. We show that a similar result holds for basic classical simple Lie superalgebras. The result is used in another work of the author to obtain a Jantzen sum formula for certain highest weight modules over type A Lie superalgebras.References
- H. H. Andersen and N. Lauritzen, Twisted Verma modules, Studies in memory of Issai Schur (Chevaleret/Rehovot, 2000) Progr. Math., vol. 210, Birkhäuser Boston, Boston, MA, 2003, pp. 1–26. MR 1985191
- Henning Haahr Andersen and Catharina Stroppel, Twisting functors on $\scr O$, Represent. Theory 7 (2003), 681–699. MR 2032059, DOI 10.1090/S1088-4165-03-00189-4
- Sergey Arkhipov, Algebraic construction of contragradient quasi-Verma modules in positive characteristic, Representation theory of algebraic groups and quantum groups, Adv. Stud. Pure Math., vol. 40, Math. Soc. Japan, Tokyo, 2004, pp. 27–68. MR 2074588, DOI 10.2969/aspm/04010027
- I. N. BernšteÄn, I. M. Gel′fand, and S. I. Gel′fand, A certain category of ${\mathfrak {g}}$-modules, Funkcional. Anal. i PriloĹľen. 10 (1976), no. 2, 1–8 (Russian). MR 0407097
- Kevin Coulembier and Volodymyr Mazorchuk, Primitive ideals, twisting functors and star actions for classical Lie superalgebras, J. Reine Angew. Math. 718 (2016), 207–253. MR 3545883, DOI 10.1515/crelle-2014-0079
- James E. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, Vol. 9, Springer-Verlag, New York-Berlin, 1972. MR 0323842
- James E. Humphreys, Representations of semisimple Lie algebras in the BGG category $\scr {O}$, Graduate Studies in Mathematics, vol. 94, American Mathematical Society, Providence, RI, 2008. MR 2428237, DOI 10.1090/gsm/094
- Jens Carsten Jantzen, Moduln mit einem höchsten Gewicht, Lecture Notes in Mathematics, vol. 750, Springer, Berlin, 1979 (German). MR 552943
- Oleksandr Khomenko and Volodymyr Mazorchuk, On Arkhipov’s and Enright’s functors, Math. Z. 249 (2005), no. 2, 357–386. MR 2115448, DOI 10.1007/s00209-004-0702-8
- Volodymyr Mazorchuk, Lectures on algebraic categorification, QGM Master Class Series, European Mathematical Society (EMS), ZĂĽrich, 2012. MR 2918217, DOI 10.4171/108
- Ian M. Musson, On the center of the enveloping algebra of a classical simple Lie superalgebra, J. Algebra 193 (1997), no. 1, 75–101. MR 1456569, DOI 10.1006/jabr.1996.7000
- Ian M. Musson, Lie superalgebras and enveloping algebras, Graduate Studies in Mathematics, vol. 131, American Mathematical Society, Providence, RI, 2012. MR 2906817, DOI 10.1090/gsm/131
- Ian M. Musson, Ĺ apovalov elements and the Jantzen sum formula for contragredient Lie superalgebras, arXiv 1710.10528.
- Dennis Hasselstrøm Pedersen, Twisting functors for quantum group modules, J. Algebra 447 (2016), 580–623. MR 3427652, DOI 10.1016/j.jalgebra.2015.09.046
Additional Information
- Ian M. Musson
- Affiliation: Department of Mathematical Sciences, University of Wisconsin-Milwaukee, Milwaukee, Wisconsin 53211
- MR Author ID: 189473
- Email: musson@uwm.edu
- Received by editor(s): April 3, 2018
- Received by editor(s) in revised form: June 20, 2018
- Published electronically: December 7, 2018
- Additional Notes: This research was partly supported by Simons Foundation grant 318264.
- Communicated by: Kailash C. Misra
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 147 (2019), 1013-1022
- MSC (2010): Primary 17B10
- DOI: https://doi.org/10.1090/proc/14310
- MathSciNet review: 3896052