Abstract
For an additive subgroup G of a field \( \mathbb{F} \) of characteristic zero, a Lie algebra
(G) of Block type is defined with basis {L α,i | α ∈ G, i ∈ ℤ+} and relations [L α,i , L β,j ] = (β − α)L α+β,i+j (αj − βi)L α+β,i+j−1. It is proved that an irreducible highest weight
(ℤ)-module is quasifinite if and only if it is a proper quotient of a Verma module. Furthermore, for a total order ≻ on G and any Λ ∈
(G) *0 (the dual space of
(G)0 = span{L 0,i | i ∈ ℤ+}), a Verma
(G)-module M(Λ, ≻) is defined, and the irreducibility of M(Λ, ≻) is completely determined.
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Supported by NSF Grant No. 10471091 of China, the Grant of “One Hundred Talents Program” from the University of Science and Technology of China
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Yue, X.Q., Su, Y.C. Highest weight representations of a family of Lie algebras of Block type. Acta. Math. Sin.-English Ser. 24, 687–696 (2008). https://doi.org/10.1007/s10114-007-0986-9
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DOI: https://doi.org/10.1007/s10114-007-0986-9