On graded representations of modular Lie algebras over commutative algebras

Matthew Westaway

doi:10.1016/j.jpaa.2022.107033

On graded representations of modular Lie algebras over commutative algebras

Matthew Westaway

Mathematics

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Abstract

We develop the theory of a category C _A 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 U _χ 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 define here certain important modules Z _A,χ(λ), Q _A,χ ^I(λ) and Q _A,χ(λ) in C _A which generalise familiar objects when A=K, and we prove some key structural results regarding them.

Original language	English
Article number	107033
Number of pages	52
Journal	Journal of Pure and Applied Algebra
Volume	226
Issue number	8
Early online date	26 Jan 2022
DOIs	https://doi.org/10.1016/j.jpaa.2022.107033
Publication status	Published - Aug 2022

Bibliographical note

Funding Information:
The author was supported during this research by EPSRC grant EP/R018952/1 . He would like to thank Simon Goodwin for many useful discussions about this subject and for his opinions on an earlier version of this paper, as well as the referee for their comments. The author has no competing interests to declare.

Keywords

Modular Lie algebra
Baby Verma module
Induction
Projective cover

ASJC Scopus subject areas

Algebra and Number Theory

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10.1016/j.jpaa.2022.107033Licence: Creative Commons: Attribution (CC BY)

1-s2.0-S0022404922000299-mainFinal published version, 857 KBLicence: Creative Commons: Attribution (CC BY)

https://arxiv.org/abs/2106.04994Licence: Creative Commons: Attribution-NonCommercial-NoDerivs (CC BY-NC-ND)

Representation theory of modular Lie algebras and superalgebras
Goodwin, S.
Engineering & Physical Science Research Council
1/07/18 → 31/12/22
Project: Research Councils

Cite this

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title = "On graded representations of modular Lie algebras over commutative algebras",

abstract = "We develop the theory of a category C A 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 U χ 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 define here certain important modules Z A,χ(λ), Q A,χ I(λ) and Q A,χ(λ) in C A which generalise familiar objects when A=K, and we prove some key structural results regarding them. ",

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N2 - We develop the theory of a category C A 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 U χ 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 define here certain important modules Z A,χ(λ), Q A,χ I(λ) and Q A,χ(λ) in C A which generalise familiar objects when A=K, and we prove some key structural results regarding them.

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On graded representations of modular Lie algebras over commutative algebras

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

Access to Document

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Projects

Representation theory of modular Lie algebras and superalgebras

Cite this