Some Koszul properties of standard and irreducible modules

zbMATH Open1335.20049arXiv1305.6274MaRDI QIDQ3464528

Brian J. Parshall, Leonard L. Scott

Publication date: 27 January 2016

Abstract: Let $G$ be a simple, simply connected algebraic group over an algebraically closed field of positive characteristic $p$. In recent work, the authors have studied a graded analogue of the category of rational $G$-modules. These gradings are not natural but are "forced" on related algebras though filtrations, often obtained from appropriate quantum structures. This paper presents new results on Koszul modules for the graded algebras obtained through this forced grading process. Most of these results require that the Lusztig character formula holds for all restricted $p$-regular weights, but the paper begins to investigate how these and previous results might be established when the Lusztig character formula is only assumed to hold on a proper poset ideal in the Jantzen region. This opens up the possibility of inductive arguments.

Full work available at URL: https://arxiv.org/abs/1305.6274

zbMATH Keywords

graded algebrasfiltrationssimply connected algebraic groupsKoszul modulesLusztig character formulacategories of rational modulesrestricted \(p\)-regular weights

Mathematics Subject Classification ID

Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) (17B10) Representation theory for linear algebraic groups (20G05) Quadratic and Koszul algebras (16S37) Filtered associative rings; filtrational and graded techniques (16W70)

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