Highest weight modules for algebraic groups arising from quantum groups
type: Item
instance of: scholarly article
title: Highest weight modules for algebraic groups arising from quantum groups
zbMATH Open document ID: 0917.17004
published in: Journal of Algebra
publication date: 1999-06-05T00:00:00Z
review text: Let \(L_p(\lambda)\) be the irreducible module for a quantum group \(U_p(\mathfrak g)\) at a \(p\)-th root of \(1\). Lusztig constructed a module \(\overline L_p(\lambda)\) for the corresponding simply connected semisimple algebraic group \(G\) of the same root system as the semisimple Lie algebra \(\mathfrak g\), and conjectured that \(\overline L_p(\lambda)\) is irreducible when \(\lambda\) is a \(p\)-restricted weight. This conjecture has been proved by \textit{H. H. Andersen, J. C. Jantzen} and \textit{W. Soergel} [Representations of quantum groups at a \(p\)-th root of unity and of semisimple groups in characteristic \(p\): Independence of \(p\), Astérisque 220 (1994; Zbl 0802.17009)]. In this paper, the author follows Lusztig's conjecture to get a class of highest weight modules \(\overline L_{p^r}(\lambda)\) for the algebraic group \(G\) from invariant lattices in the irreducible modules \(L_{p^r}(\lambda)\) for the quantum group \(U_\xi (\mathfrak g)\) at a \(p^r\)-th root of \(1\). Assuming the above Lusztig conjecture at \(p\)-th roots of \(1\), the main result of this paper shows that, under certain generic conditions on \(\lambda\) in the bottom \(p^3\) alcove, \[ [\overline L_p(\lambda):L(\mu)]=[V_p(\lambda):L_p(\mu)], \] where \(V_p(\lambda)\) is the quantum Weyl module at \(p\)-th root of \(1\). The right hand side can be computed using Lusztig's quantum conjecture, which has been proved by Kashiwara and Tanisaki combined with the work of Kazhdan and Lusztig. For larger \(r\), a recursive formula is also provided.
reviewed by: Ye Jiachen
Mathematics Subject Classification ID: 17B10
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 17B37
zbMATH DE Number: 1221664
zbMATH Keywords: quantum group
zbMATH Keywords: algebraic group
zbMATH Keywords: highest weight module
author: Zongzhu Lin
MaRDI profile type: Publication
full work available at URL: jabr.1998.7488
OpenAlex ID: W2071590330
cites work: On the structure of the cohomology of line bundles on G/B
cites work: On the Generic Structure of Cohomology Modules for Semisimple Algebraic Groups
cites work: The strong linkage principle.
cites work: Filtrations of cohomology modules for Chevalley groups
cites work: Extensions of Modules for Algebraic Groups
cites work: Loewy Series of Modules for the First Frobenius Kernel in a Reductive Algebraic Group
cites work: Q4296997
cites work: Representations of quantum algebras
cites work: Injective Modules for Quantum Algebras
cites work: Abstract Kazhdan-Lusztig theories
cites work: Q3922829
cites work: Filtration patterns for representations of algebraic groups and their Frobenius kernels
cites work: On the structure of Weyl modules
cites work: Q3998721
cites work: Introduction to Lie Algebras and Representation Theory
cites work: Generic Cartan invariants for Frobenius kernels and Chevalley groups
cites work: Q3797389
cites work: Kazhdan-Lusztig conjecture for affine Lie algebras with negative level
cites work: Kazhdan-Lusztig conjecture for affine Lie algebras with negative level. II: Nonintegral case
cites work: Representations of Coxeter groups and Hecke algebras
cites work: Tensor Structures Arising from Affine Lie Algebras. I
cites work: Q3699004
cites work: Induced representations of Hopf algebras: Applications to quantum groups at roots of 1
cites work: A Mackey decomposition theorem and cohomology for quantum groups at roots of 1
cites work: Structure of cohomology of line bundles on G/B for semisimple groups
cites work: Socle series of cohomology groups of line bundles on \(G/B\)
cites work: Extensions between simple modules for Frobenius kernels
cites work: Extensions of modules over Hopf algebras arising from Lie algebras of Cartan type
cites work: Q3816243
cites work: Finite Dimensional Hopf Algebras Arising From Quantized Universal Enveloping Algebras
cites work: Q4869249
cites work: Canonical Bases Arising from Quantized Enveloping Algebras. II
cites work: Quantum groups at roots of 1
cites work: Quivers, Perverse Sheaves, and Quantized Enveloping Algebras
cites work: Q3900232
cites work: Monodromic systems on affine flag manifolds
cites work: Notes on some topics in the representation theory of linear algebraic groups
cites work: Representations of Chevalley Groups in Characteristic p
cites work: Filtrations of principal indecomposable modules of Frobenius kernels of reductive groups
DOI: 10.1006/JABR.1998.7488
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description: scientific article; zbMATH DE number 1221664