On graded representations of modular Lie algebras over commutative algebras

2021/06/09 by Westaway, Matthew
#17B10 #17B50 (Primary) 13C60 #18G05 (Secondary) #FOS: Mathematics #Representation Theory (math.RT) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2106.04994

Abstract

We develop the theory of a category \mathscr C_A which is a generalisation to non-restricted \mathfrak g-modules of a category famously studied by Andersen, Jantzen and Soergel for restricted \mathfrak g-modules, where \mathfrak g is the Lie algebra of a reductive group G over an algebraically closed field \mathbb K of characteristic p>0. Its objects are certain graded bimodules. On the left, they are graded modules over an algebra U_χ associated to \mathfrak g and to χ∈\mathfrak g^* in standard Levi form. On the right, they are modules over a commutative Noetherian S(\mathfrak h)-algebra A, where \mathfrak h is the Lie algebra of a maximal torus of G. We develop here certain important modules Z_A,χ(λ), Q_A,χ^I(λ) and Q_A,χ(λ) in \mathscr C_A which generalise familiar objects when A=\mathbb K, and we prove some key structural results regarding them.

Related

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• Restriction to Levi subalgebras and generalization of the category O
• The BGG Category for Generalised Reductive Lie Algebras
• Lie algebras constructed with Lie modules and their positively and negatively graded modules
• Filtrations in Modular Representations of Reductive Lie Algebras