Irreducible Modules Over Witt Algebras and Over sl(n+1) (C)
Irreducible Modules Over Witt Algebras and Over sl(n+1) (C)
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维特代数和 sl(n 1) 上的不可约模 (C)
DOI：
10.1007/s10468-017-9738-4
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发表时间：
2018
影响因子：
0.6
通讯作者：
Zhao KM
中科院分区：
文献类型：
--
作者：
Tan Haijun;Zhao KM
在本文中，通过使用“扭曲技术”，我们从模块 A 获得一类新模块，高于 Witt 代数 Wn< inline-graphic> ，对于任何 bε ℂ< inline-graphic> ，在 Weyl 代数 Kn< inline-graphic> （洛朗多项式）上。我们给出 Abto 不可约的充要条件，并确定两个这样的不可约 Wn< inline-graphic> 模块同构的充要条件。由于 픩 n+ 1 (ℂ) < inline-graphic> 是 Wn< inline-graphic> 的子代数，因此上述所有不可约的 Wn< inline-graphic> 模块 Ab 都可以被视为 픩 n+ 1 (ℂ) < inline-graphic> 模块。对于一类这样的 픩 n+ 1 (ℂ) < inline-graphic> -模块，用 Ω1− a(λ1, λ2,⋯, λn) 表示，其中 a∈ ℂ, λ1, λ2,⋯, λn∈ ℂ* < inline-graphic> ，我们确定这些 픩 n+ 1 (ℂ) < inline-graphic> -模块的充分必要条件是不可约的。如果 픩 n+ 1 (ℂ) < inline-graphic> -module Ω1− a(λ1, λ2,⋯, λn) 是可约的，我们证明它有一个唯一的非平凡子模 W1− a(λ1, λ2,... λn) 并且商模是有限维 픩 n+ 1 (ℂ) < inline-graphic> -module 具有最高权重 mΛnfor一些非负整数 m∈ ℤ+ < inline-graphic> 。我们还确定两个 픩 n+ 1 (ℂ) < inline-graphic> -形式为 Ω1− a(λ1, λ2,⋯, λn) 或形式为 W1− a(λ1, λ2,... λn) 的模块同构的充分必要条件。
In this paper, by using the “twisting technique” we obtain a class of new modules Abover the Witt algebras Wn< inline-graphic> from modules A over the Weyl algebras Kn< inline-graphic> (of Laurent polynomials) for any b∈ ℂ< inline-graphic> . We give necessary and sufficient conditions for Abto be irreducible, and determine necessary and sufficient conditions for two such irreducible Wn< inline-graphic> -modules to be isomorphic. Since 픩 n+ 1 (ℂ) < inline-graphic> is a subalgebra of Wn< inline-graphic> , all the above irreducible Wn< inline-graphic> -modules Abcan be considered as 픩 n+ 1 (ℂ) < inline-graphic> -modules. For a class of such 픩 n+ 1 (ℂ) < inline-graphic> -modules, denoted by Ω1− a(λ1, λ2,⋯, λn) where a∈ ℂ, λ1, λ2,⋯, λn∈ ℂ∗ < inline-graphic> , we determine necessary and sufficient conditions for these 픩 n+ 1 (ℂ) < inline-graphic> -modules to be irreducible. If the 픩 n+ 1 (ℂ) < inline-graphic> -module Ω1− a(λ1, λ2,⋯, λn) is reducible, we prove that it has a unique nontrivial submodule W1− a(λ1, λ2,... λn) and the quotient module is the finite dimensional 픩 n+ 1 (ℂ) < inline-graphic> -module with highest weight mΛnfor some non-negative integer m∈ ℤ+ < inline-graphic> . We also determine necessary and sufficient conditions for two 픩 n+ 1 (ℂ) < inline-graphic> -modules of the form Ω1− a(λ1, λ2,⋯, λn) or of the form W1− a(λ1, λ2,... λn) to be isomorphic.