Whittaker modules for the affine Lie algebra A(1)((1))
Whittaker modules for the affine Lie algebra A(1)((1))
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仿射李代数 A(1)((1)) 的 Whittaker 模
DOI：
10.1016/j.aim.2015.11.020
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发表时间：
2016
影响因子：
1.7
通讯作者：
Zhao Kaiming
中科院分区：
文献类型：
--
作者：
Adamovic Drazen;Lu Rencai;Zhao Kaiming
证明了A1（1）型仿射李代数sl 2 ∞在非临界水平上的泛非退化Whittaker模的不可约性.这些模可以成为环sl 2 <$= sl 2 <$+ Cd上具有相同Whittaker函数和中心荷的单Whittaker模.为了得到具有非临界水平的简单退化Whittaker sl 2-模，我们必须对sl 2的中心特征标取模.在临界水平的情况下，万有Whittaker模是可约的.证明了泛Whittaker sl 2 <$-模与顶点代数V − 2（sl 2）的中心元素的标量作用生成的子模的商是sl 2 <$-模。我们还明确地描述了在临界水平上sl 2 ∞的泛Whittaker模的简单子。令人惊奇的是，对于相同的Whittaker函数，一些简单退化的sl 2 <$Whittaker模在临界水平上具有d的半单作用，而另一些具有d的自由作用.最后，利用顶点代数技巧，给出了sl 2 ∞的一类简单广义Whittaker模在临界水平上的Wakimoto型构造.这个族包含了临界水平上的所有经典Whittaker模。在非临界水平上，我们也得到了sl 2 ∞的退化Whittaker模的Wakimoto型实现.
We prove the irreducibility of the universal non-degenerate Whittaker modules for the affine Lie algebra sl 2 ˆ of type A 1 (1) with noncritical level. These modules can become simple Whittaker modules over sl 2˜= sl 2 ˆ+ C d with the same Whittaker function and central charge. We have to modulo a central character for sl 2 to obtain simple degenerate Whittaker sl 2 ˆ-modules with noncritical level. In the case of critical level the universal Whittaker module is reducible. We prove that the quotient of universal Whittaker sl 2 ˆ-module by a submodule generated by a scalar action of central elements of the vertex algebra V− 2 (sl 2) is simple as sl 2 ˆ-module. We also explicitly describe the simple quotients of universal Whittaker modules at the critical level for sl 2˜. Quite surprisingly, with the same Whittaker function some simple degenerate sl 2˜ Whittaker modules at the critical level have semisimple action of d and others have free action of d. At last, by using vertex algebraic techniques we present a Wakimoto type construction of a family of simple generalized Whittaker modules for sl 2 ˆ at the critical level. This family includes all classical Whittaker modules at critical level. We also have Wakimoto type realization for degenerate Whittaker modules for sl 2 ˆ at noncritical level.