扭曲量子仿射代数的表示理论
Representations of twisted quantum affine algebras
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中文总结 AI 辅助
本文发展扭曲量子仿射代数的虚Verma模表示理论，构造对应Kashiwara代数，证明其PBW型基、不可约性准则等核心结果，为扭曲量子与经典理论建立相容性。
中文摘要 AI 辅助
我们发展了扭曲量子仿射代数的虚Verma模表示理论，并构造了对应的Kashiwara代数。与未扭曲情形相比，扭曲情形带来了诸多新的困难：PBW根向量具有非平凡的轨道结构，虚根空间带有重数，且单位根本质上进入了定义交换关系中。我们的第一个主要结果是，针对仿射根系统的自然虚分划，给出了扭曲量子虚Verma模的显式PBW型基，这为扭曲量子理论与经典理论提供了标准PBW理论无法直接得到的精确相容性。随后我们确定了扭曲量子虚Verma模的结构与不可约性：证明了由虚根向量生成的Heisenberg子模仅在非零中心电荷时不可约，并建立了零中心电荷时约化扭曲虚Verma模的对应不可约性准则。论文第二部分引入了扭曲情形下的Kashiwara代数，其生成元所需的电流公式与未扭曲情形有本质差异，且是构造Omega算子所必需的；我们推导了由此得到的Omega算子交换关系，包含了扭曲情形特有的单位根因子，这些关系给出了与约化扭曲虚Verma模相关的Kashiwara代数的新表示；我们证明了负电流代数是该Kashiwara代数上的单模，并构造了由Omega算子刻画的对称非退化双线性型。
英文摘要
We develop the representation theory of imaginary Verma modules for twisted quantum affine algebras and construct the corresponding Kashiwara algebras. The twisted case presents substantial new difficulties compared with the untwisted setting: the PBW root vectors have nontrivial orbit structure, the imaginary root spaces occur with multiplicities, and roots of unity enter essentially into the defining commutation relations. Our first main result is an explicit PBW-type basis for twisted quantum imaginary Verma modules associated with the natural imaginary partition of the affine root system. This provides a precise compatibility between the twisted quantum and classical theories that is not immediate from the standard PBW theory. We then determine the structure and irreducibility of the twisted quantum imaginary Verma modules. We prove that the Heisenberg submodule generated by the imaginary root vectors is irreducible precisely at nonzero central charge, and establish the corresponding irreducibility criterion for the reduced twisted imaginary Verma modules at zero central charge. The second main part of the paper introduces the Kashiwara algebra in the twisted setting. The required current formula for the generators differs essentially from the untwisted formula and is needed to construct the Omega operators. We derive the resulting Omega-operator commutation relations, including the root-of-unity factors specific to the twisted cases. These relations lead to a new presentation of the Kashiwara algebra associated with the reduced twisted imaginary Verma modules. We prove that the negative current algebra is a simple module over this Kashiwara algebra and construct a symmetric non-degenerate bilinear form characterized by the Omega operators.
