On graded representations of modular Lie algebras over commutative algebras

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We develop the theory of a category CA which is a generalisation to non-restricted g-modules of a category famously studied by Andersen, Jantzen and Soergel for restricted g-modules, where g is the Lie algebra of a reductive group G over an algebraically closed field 𝕂 of characteristic p>0. Its objects are certain graded bimodules. On the left, they are graded modules over an algebra Uχ associated to g and to χ∈g∗ in standard Levi form. On the right, they are modules over a commutative Noetherian S(h)-algebra A, where h is the Lie algebra of a maximal torus of G. We develop here certain important modules ZA,χ(λ), QIA,χ(λ) and QA,χ(λ) in CA which generalise familiar objects when A=𝕂, and we prove some key structural results regarding them.
Original languageEnglish
JournalJournal of Pure and Applied Algebra
Publication statusAccepted/In press - 5 Jan 2022


  • Modular Lie algebra
  • Baby Verma module
  • Induction
  • Projective cover

ASJC Scopus subject areas

  • Algebra and Number Theory


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