Category O for Takiff Lie algebras

Matthew Chaffe

doi:10.1007/s00209-023-03262-1

Category O for Takiff Lie algebras

Matthew Chaffe^*

^*Corresponding author for this work

Mathematics

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Abstract

We study category O for Takiff Lie algebras g⊗C[ϵ]/(ϵ2) where g is the Lie algebra of a reductive algebraic group over C. We decompose this category as a direct sum of certain subcategories and use an analogue of parabolic induction functors and twisting functors for BGG category O to prove equivalences between these subcategories. We then use these equivalences to compute the composition multiplicities of the simple modules in the Verma modules in terms of composition multiplicities in the BGG category O for reductive subalgebras of g. We conclude that the composition multiplicities are given in terms of the Kazhdan–Lusztig polynomials.

Original language	English
Article number	14
Number of pages	35
Journal	Mathematische Zeitschrift
Volume	304
Issue number	1
Early online date	20 Apr 2023
DOIs	https://doi.org/10.1007/s00209-023-03262-1
Publication status	Published - May 2023

Keywords

Article
Lie algebras
Representation theory
Takiff algebras
Category O
17B10
17B45
16D90

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Category O for Takiff Lie algebras
Chaffe, M., 6 May 2022, arXiv.
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abstract = "We study category O for Takiff Lie algebras g⊗C[ϵ]/(ϵ2) where g is the Lie algebra of a reductive algebraic group over C. We decompose this category as a direct sum of certain subcategories and use an analogue of parabolic induction functors and twisting functors for BGG category O to prove equivalences between these subcategories. We then use these equivalences to compute the composition multiplicities of the simple modules in the Verma modules in terms of composition multiplicities in the BGG category O for reductive subalgebras of g. We conclude that the composition multiplicities are given in terms of the Kazhdan–Lusztig polynomials.",

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AB - We study category O for Takiff Lie algebras g⊗C[ϵ]/(ϵ2) where g is the Lie algebra of a reductive algebraic group over C. We decompose this category as a direct sum of certain subcategories and use an analogue of parabolic induction functors and twisting functors for BGG category O to prove equivalences between these subcategories. We then use these equivalences to compute the composition multiplicities of the simple modules in the Verma modules in terms of composition multiplicities in the BGG category O for reductive subalgebras of g. We conclude that the composition multiplicities are given in terms of the Kazhdan–Lusztig polynomials.

KW - Article

KW - Lie algebras

KW - Representation theory

KW - Takiff algebras

KW - Category O

KW - 17B10

KW - 17B45

KW - 16D90

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VL - 304

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

IS - 1

M1 - 14

ER -

Category O for Takiff Lie algebras

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Category O for Takiff Lie algebras

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