Highest weight representations of a Lie algebra of Block type

2005/11/30 by Yuezhu Wu, Wu, Yuezhu, Yucai Su +1
Mathematics · #17B10 #17B65 #17B68 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT) #math.QA #math.RT #msc:17B10 #msc:17B65 #msc:17B68

paper · pdf · doi:10.48550/arxiv.math/0511733

LaTeX, 13 pages

arxiv created 2005/11/30 · openalex publication_date 2005/11/30 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For a field F of characteristic zero and an additive subgroup G of F, a Lie algebra B(G) of lock type is defined with basis \L_a,i,c|a ∈ G, i>-2\ and relations [L_a,i,L_b,j]=((i+1)b-(j+1)a)L_a+b,i+j+a\d_a,-b\d_i+j,-2c, [c,L_a,i]=0. Given a total order \succ on G compatible with its group structure, and any Λ∈ B(G)₀^*, a Verma B(G)-module M(Λ,\succ) is defined, and the irreducibility of M(Λ,\succ) is completely determined. Furthermore, it is proved that an irreducible highest weight B(Z)-module is quasifinite if and only if it is a proper quotient of a Verma module.

Related

• Verma modules over a Block Lie algebra
• Classification of quasifinite representations of a Lie algebra related to Block type
• Quasifinite representations of a class of Block type Lie algebras \BB
• Block type Lie algebras and their representations
• Verma modules over a class of Block type Lie algebras