A comprehensive research of vertex algebras, especially the W-algebras

在仿射Kac-Moody代数的容许表示上，证明了Frenkel，Kac和Wakimoto关于双边BGG分解存在性的猜想，Adamovic和Milas关于相应顶点算子代数的猜想，以及Feigin和Frenkel关于其奇异支撑的猜想.在仿射Kac-Moody代数的临界水平表示上，证明了新的键原理，建立了手征Borel-Weil-Bott定理。在W-代数上，证明了Kac和Wakimoto发现的例外W-代数的C_2-上有限性，并证明了仿射Kac-Moody代数的容许表示。我们在临界水平上得到了W-代数的各种结果。

On admissible representations of affine Kac-Moody algebras, we proved the he conjecture of Frenkel, Kac and Wakimoto on the existence of two-sided BGG resolutions, the conjecture of Adamovic and Milas on the corresponding vertex operators algebra, and the conjecture of Feigin and Frenkel on their singular supports. On critical level representations of affine Kac-Moody algebras, we proved the new linkage principal and established a chiral Borel-Weil-Bott theorem. On W-algebras, we prove the C_2-cofinitness of the exceptional W-algebras discovered by Kac and Wakimoto and prove the admissible representations of affine Kac-Moody algebras. We obtained various results on the W-algebras at the critical levels.