Abstract
We classify all simple bounded highest weight modules of a basic classical Lie superalgebra \( \mathfrak{g} \). In particular, our result leads to the classification of the simple weight modules with finite weight multiplicities over all classical Lie superalgebras. We also obtain some character formulas of strongly typical bounded highest weight modules of \( \mathfrak{g} \).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
G. Benkart, D. Britten, F. Lemire, Modules with bounded weight multiplicities for simple Lie algebras, Math. Z. 225 (1997), 333–353.
D. Britten, F. Lemire, A classification of simple Lie modules having a 1-dimensional weight space, Trans. Amer. Math. Soc. 299 (1987), 683–697.
K. Coulembier, On a class of tensor product representations for orthosymplectic superalgebras, J. Pure Appl. Algebra 217 (2013), 819–837.
V. V. Deodhar, O. Gabber, V. Kac, Structure of some categories of representations of infinite-dimensional Lie algebras, Adv. Math. 45 (1982), 92–116.
I. Dimitrov, O. Mathieu, I. Penkov, On the structure of weight modules, Trans. Amer. Math. Soc. 352 (2000), 2857–2869.
T. J. Enright, On the fundamental series of a real semisimple Lie algebra: their irreducibility, resolutions and multiplicity formulae, Ann Math. 110 (1979), 1–82.
T. Ferguson, M. Gorelik, D. Grantcharov, Bounded highest weight modules of \( \mathfrak{osp} \)(1, 2n), Proc. Symp. Pure Math., AMS, Vol. 92 (2016), 135–143.
S. Fernando, Lie algebra modules with finite dimensional weight spaces I, Trans. Amer. Math. Soc. 322 (1990), 757–781.
V. Futorny, The Weight Representations of Semisimple Finite-dimensional Lie Algebras, PhD Thesis, Kiev University, 1987.
M. Gorelik, Annihilation theorem and separation theorem for basic classical Lie superalgebras, J. Amer. Math. Soc. 15, (2002), 113–165.
M. Gorelik, Strongly typical representations of the basic classical Lie superalgebras, J. Amer. Math. Soc. 15, (2002), 167–184.
M. Gorelik, D. Grantcharov, Bounded highest weight modules of \( \mathfrak{q} \)(n), Int. Math. Res. Not. 2014(22) (2014), 6111–6154.
M. Gorelik, V. Kac, Characters of (relatively) integrable modules over affine Lie superlagebras, Japan. J. Math. 10 (2015), 135–235.
M. Gorelik, V. Serganova Snowake modules and Enright functor for Kac–Moody superalgebras, arXiv:1906.07074 (2019).
D. Grantcharov, Explicit realizations of simple weight modules of classical Lie superalgebras, Cont. Math. 499 (2009), 141–148.
D. Grantcharov, V. Serganova, Category of \( \mathfrak{sp} \)(2n)-modules with bounded weight multiplicities, Mosc. Math. J. 6 (2006), 119–134.
D. Grantcharov, V. Serganova, On weight modules of algebras of twisted differential operators on the projective space, Transform. Groups 21 (2016), 87–114.
C. Hoyt, Weight modules for D(2, 1, α), in: Advances in Lie Superalgebras, Springer INdAM Ser., Vol. 7, Springer, Cham, 2014, pp. 91–100.
K. Iohara, Y. Koga, Enright functors for Kac–Moody superalgebras, Abh. Math. Semin. Univ. Hambg. 82 (2012), no. 2, 205–226.
A. Joseph, Some ring theoretical techniques and open problems in enveloping algebras, in: Noncommutative Rings, MSRI Publications, Vol. 24, Springer, New York, NY, 1992, pp. 27–67.
V. G. Kac, Lie superalgebras, Adv. Math. 26 (1977), 8–96.
V. G. Kac, Infinite-dimensional Lie Algebras, 3rd edition, Cambridge University Press, Cambridge, 1990.
M. Kashiwara, T. Tanisaki, Kazhdan–Lusztig conjecture for symmetrizable Kac–Moody algebras III. Positive rational case, Asian J. Math. 2 (1998), no. 4, 779–832.
M. Kashiwara, T. Tanisaki, Characters of the irreducible modules with non-critical highest weights over affine Lie algebras, in: Representations and Quantizations (Shanghai, 1998), China High. Educ. Press, Beijing, 2000, pp. 275–296.
O. Mathieu Classification of irreducible weight modules, Ann. Inst. Fourier 50 (2000), 537–592.
V. Serganova, Finite-dimensional representations of algebraic supergroups, in: Proceedings of International Congress of Mathematicians|Seoul 2014, Vol. 1, Kyung Moon Sa, Seoul, 2014, pp. 603–632.
V. Serganova, Kac–Moody superalgebras and integrability, in Developments and Trends in Infinite-dimensional Lie Theory, Progress in Math., Vol. 288, Birkhäuser Boston, Boston, MA, 2011, pp. 169–218.
Н. Н. Шаповалов, Об одной билинейной форме на универсальной обертывающей алгебре комплексной полупростой алгебры Ли, Функц. анализ и его прил. 6 (1972), вып. 4, 65–70. Engl. transl.: N. N. Shapovalov, On a bilinear form on the universal enveloping algebra of a complex semisimple Lie algebra, Funct. Analysis Appl. 6 (1972), 307–312.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Maria Gorelik is supported in part by the Minerva foundation with funding from the Federal German Ministry for Education and Research.
Dimitar Grantcharov is supported in part by Simons Collaboration Grant 358245.
Rights and permissions
About this article
Cite this article
GORELIK, M., GRANTCHAROV, D. SIMPLE BOUNDED HIGHEST WEIGHT MODULES OF BASIC CLASSICAL LIE SUPERALGEBRAS. Transformation Groups 26, 893–914 (2021). https://doi.org/10.1007/s00031-020-09616-x
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00031-020-09616-x