Let $A=mathbb{C}[t_1^{pm1},t_2^{pm1}]$ be the algebra of Laurentpolynomials in two variables and $B$ be the set of skew derivations of $A$. Let$L$ be the universal central extension of the derived Lie subalgebra of the Liealgebra $Atimes B$. Set $widetilde{L}=Loplusmathbb{C} d_1oplusmathbb{C}d_2$, where $d_1$, $d_2$ are two degree derivations. A Harish-Chandra module isdefined as an irreducible weight module with finite dimensional weight spaces.In this paper, we prove that a Harish-Chandra module of the Lie algebra$widetilde{L}$ is a uniformly bounded module or a generalized highest weight(GHW for short) module. Furthermore, we prove that the nonzero levelHarish-Chandra modules of $widetilde{L}$ are GHW modules. Finally, we classifyall the GHW Harish-Chandra modules of $widetilde{L}$.
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机译：让$ a = mathbb {c} [t_1 ^ { pm1}，t_2 ^ { pm1}，t_2 ^ { pm1}] $是两个变量中Laurentpolynomials的代数，$ b $ be为$ a $的偏差派生集。让$ l $是LieveBraa的派生谎言子晶晶级的通用中央扩展名为B $。设置$ widetilde {l} = l oplus mathbb {c} d_1 oplus mathbb {c} d_2 $，其中$ d_1 $，$ d_2 $是两个学位派生。一个Harish-Chandra模块作为一个不可缩小的重量模块，具有有限尺寸重量空间。本文证明了Lie代数$ Widetilde {L} $的Harish-Chandra模块是一个均匀有界模块或广义最高权重（短暂的ghw）模块。此外，我们证明了$ widetilde {l} $的非零级漂亮的Chandra模块是GHW模块。最后，我们为$ widetilde {l} $ classifyall the ghw harish-chandra模块。
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