2024/03/12 by Boujemaâ Agrebaoui, Agrebaoui, Boujemaa, Walid Mhiri +1
Mathematics · Physics and Astronomy · #17B10 #17B20 #17B68 #17B86 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Nonlinear Waves and Solitons #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2403.07381
openalex publication_date 2024/03/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let An=ℂ[ti±1,~1≤ i≤ n] and W(n)μ=Andμ the solenoidal Lie algebra introduced by Y.Billig and V.Futorny in \citeBiFu2, where μ=(μ1,…,μn)∈ℂn is a generic vector and dμ=∑i=1nμiti(∂)/(∂ ti). We consider the semi-direct product Lie algebra WA(n)μ:=W(n)μ\ltimes An. In the first part, We prove that WA(n)μ has a unique three-dimensional universal central extension. In fact we construct a higher rank Heisenberg-Virasoro algebra (see \citeLiuGuo, LdZ). It will be denoted by HVir(n)μ and it will be called the solenoidal Heisenberg-Virasoro algebra. Then we will study Harish-Chandra modules of HVir(n)μ following \citeLiuGuo. We will obtain two classes of Harich-Chandra modules: generalized highest weight modules(GHW modules) and intermediate series modules. Our results are particular cases of \citeLiuGuo. In the end, we will construct HVir(n)μ Verma modules using the lexicographic order on ℤn. In particular we give examples of irreducible weight modules which have infinite dimensional weight spaces.