Whittaker modules and hyperbolic Toda lattices

2024/01/01 by Limeng Xia, Xia, Limeng
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Algebraic structures and combinatorial models #FOS: Mathematics #Nonlinear Waves and Solitons #Quantum Algebra (math.QA)

paper · pdf · doi:10.48550/arxiv.2401.00680

openalex publication_date 2024/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let \sg be a complex finite-dimensional simple Lie algebra and let \sg_l be the corresponding generalized Takiff algebra. This paper studies the affine variety \ssf+\sb_l where \ssf is similar to a principal nilpotent element of \sg and \sb_l is a subalgebra corresponding to the Borel subalgebra \sb of \sg. Inspired by Kostant's work then we deal with two questions. One of them is to construct the Whittaker model for the G_l-invariants of symmetric algebra S(\sg_l) where G_l is the adjoint group of \sg_l and G_l acts on S(\sg_l) by coadjoint action, and then to classify all nonsingular Whittaker modules over \sg_l. Another one is to describe the symplectic structure of the manifold Z⊆\ssf+\sb_l of normalized Jacobi elements. Then the Hamiltonian corresponding to a fundamental invariant provides a class of hyperbolic Toda lattices. In particular, a simplest example describes the state of a dynamical system consisting of a positive mass particle and a negative mass particle.

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• Blocks and modules for Whittaker pairs
• Whittaker modules of central extensions of Takiff superalgebras and finite supersymmetric W-algebras
• Toda systems for Takiff algebras
• Affine Lie algebras representations induced from Whittaker modules