Whittaker modules for the planar Galilean conformal algebra and its central extension

2020/07/08 by Qiufan Chen, Yufeng Yao, Chen, Qiufan +3 · 3 citations
Mathematics · #17B10 #17B35 #17B65 #17B68 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2007.04046

openalex publication_date 2020/07/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let G be the planar Galilean conformal algebra and \widetildeG be its universal central extension. Then G (resp. \widetildeG) admits a triangular decomposition: G=G⁺\oplusG⁰\oplusG^- (resp. \widetildeG=\widetildeG⁺⊕\widetildeG⁰⊕\widetildeG^-). In this paper, we study universal and generic Whittaker G-modules (resp. \widetildeG-modules) of type ϕ, where ϕ:G⁺=\widetildeG⁺\longrightarrowℂ is a Lie algebra homomorphism. We classify the isomorphism classes of universal and generic Whittaker modules. Moreover, we show that a generic Whittaker modules of type ϕ is irreducible if and only if ϕ is nonsingular. For the nonsingular case, we completely determine the Whittaker vectors in universal and generic Whittaker modules. For the singular case, we concretely construct some proper submodules of generic Whittaker modules.

Cited by

• Simple modules of the planar Galilean conformal algebra from tensor products
• Irreducible modules over the universal central extension of the planar Galilean conformal algebra
• Tensor product modules over the planar Galilean conformal algebra from free modules of rank one

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