Abstract
We construct a family of homomorphisms between Weyl modules for affine Lie algebras in characteristic p, which supports our conjecture on the strong linkage principle in this context. We also exhibit a large class of reducible Weyl modules beyond level one, for p not necessarily small.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
H. H. Andersen, The strong linkage principle, J. reine angew. Math. 315 (1980), 53–59.
J. Brundan, A. Kleshchev, Cartan determinants and Shapovalov forms, Math. Ann. 324 (2002), 431–449.
R. W. Carter, Lie Algebras of Finite and Affine Type, Cambridge Studies in Advanced Mathematics, Vol. 96, Cambridge University Press, Cambridge, 2005.
V. Chari, N. Jing, Realization of level one representations of \( {U}_q\left(\widehat{\mathfrak{g}}\right) \) at a root of unity, Duke Math. J. 108 (2001), 183–197.
R. W. Carter, G. Lusztig, On the modular representations of the general linear and symmetric groups, Math. Z. 136 (1974), 193–242.
R. W. Carter, M. T. J. Payne, On homomorphisms between Weyl modules and Specht modules, Math. Proc. Cambridge Philos. Soc. 87 (1980), 419–425.
C. De Concini, V. G. Kac, D. A. Kazhdan, Boson-Fermion correspondence over Z, in: Infinite-dimensional Lie Algebras and Groups (Luminy-Marseille, 1988), Adv. Ser. Math. Phys. 7, World Sci. Publ., Teaneck, NJ, 1989, pp. 124–137.
[Don80] S. Donkin, The blocks of a semisimple algebraic group, J. Algebra 67 (1980), 36–53.
S. Doty, The strong linkage principle, Amer. J. Math. 111 (1989), 135–141.
J. Franklin, Homomorphisms Between Verma Modules and Weyl Modules in Characteristic p, PhD thesis, University of Warwick, 1981.
J. Franklin, Homomorphisms between Verma modules in characteristic p, J. Algebra 112 (1988), 58–85.
H. Garland, The arithmetic theory of loop algebras, J. Algebra 53 (1978), 480–551.
J. E. Humphreys, Modular representations of classical Lie algebras and semi-simple groups, J. Algebra 19 (1971), 51–79.
J. E. Humphreys, Representations of Semisimple Lie Algebras in the BGG Category O, Graduate Studies in Mathematics, Vol. 94, American Mathematical Society, Providence, RI, 2008.
J. C. Jantzen, Darstellungen halbeinfacher Gruppen und kontravariante For-men, J. reine angew. Math. 290 (1977), 117–141.
J. C. Jantzen, Representations of Algebraic Groups, 2nd ed., Mathematical Surveys and Monographs, Vol. 107, American Mathematical Society, Providence, RI, 2003.
V. G. Kac, Infinite-dimensional Lie Algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. Russian transl.: В. Кац, Бесконечномерные алгебры Ли, Mиp, M., 1993.
V. G. Kac, D. A. Kazhdan, Structure of representations with highest weight of infinite-dimensional Lie algebras, Adv. in Math. 34 (1979), 97–108.
S. Kumar, Bernstein-Gelfand-Gelfand resolution for arbitrary Kac-Moody algebras, Math. Ann. 286 (1990), 709–729.
O. Mathieu, On some modular representations of affine Kac-Moody algebras at the critical level, Compos. Math. 102 (1996), 305–312.
D. Mitzman, Integral Bases for Affine Lie Algebras and Their Universal Enveloping Algebras, Contemporary Mathematics 40, American Mathematical Society, Providence, RI, 1985.
Н. Н. Шаповалов, Об одной билинейной форме на универсальной обёртывающей 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. Anal. Appl. 6 (1972), no. 4, 307–312.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
LAI, C. ON WEYL MODULES OVER AFFINE LIE ALGEBRAS IN PRIME CHARACTERISTIC. Transformation Groups 21, 1123–1153 (2016). https://doi.org/10.1007/s00031-016-9382-9
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00031-016-9382-9