Simple smooth modules over the Lie algebras of polynomial vector fields

2025/06/23 by Li, Zhiqiang, Cheng, Cunguang, Liu, Shiyuan +3 · 1 citation
#FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2506.18262

Abstract

Let \mathfrakg:=\rm Der(ℂ[t₁, t₂,⋯, t_n]) and L:=\rm Der(ℂ[[t₁, t₂,⋯, t_n]]) be the Witt Lie algebras. Clearly, \mathfrakg is a proper subalegbra of L. Surprisingly, we prove that simple smooth modules over \mathfrakg are exactly the simple modules over L studied by Rodakov (no need to take completion). Then we find an easy and elementary way to classify all simple smooth modules over \mathfrakg. When the height ℓ_V≥2 or n=1, any nontrivial simple smooth \mathfrakg-module V is isomorphic to an induced module from a simple smooth \mathfrakg_≥0-module V^(ℓ_V). When ℓ_V=1 and n≥2, any such module V is the unique simple quotient of the tensor module F(P₀,M) for some simple \gl_n-module M, where P₀ is a particular simple module over the Weyl algebra K⁺_n. We further show that a simple \mathfrakg-module V is a smooth module if and only if the action of each of n particular vectors in \mathfrakg is locally finite on V.

Citations

• Simple smooth modules over the Ramond algebra and applications to vertex operator superalgebras
• Simple restricted modules over the Heisenberg-Virasoro algebra as VOA modules
• Simple weight modules with finite weight multiplicities over the Lie algebra of polynomial vector fields
• Classification of simple bounded weight modules of the Lie algebra of vector fields on \Cⁿ
• Simple restricted modules for Neveu-Schwarz algebra
• Local Systems of Vertex Operators, Vertex Superalgebras and Modules
• Local systems of vertex operators, vertex superalgebras and modules

Cited by

• Quasi-Whittaker modules

Related

• Classification of Simple Cuspidal Modules over Nongraded Witt Lie Algebras
• Tensor modules over the Lie algebras of divergence zero vector fields on ℂⁿ
• Simple weight modules with finite weight multiplicities over the Lie algebra of polynomial vector fields
• Classification of simple bounded weight modules of the Lie algebra of vector fields on \Cⁿ
• Tensor modules over Witt superalgebras