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

关于仿射Kac-Moody代数的容许表示，我们证明了Frenkel、Kac和Wakimoto关于双边BGG解的存在性的猜想，Adamovic和Milas关于相应的顶点算子代数的猜想，以及Feigin和Frenkel关于其奇异支集的猜想。在仿射Kac-Moody代数的临界层表示上，证明了新的联结原理，建立了手征Borel-Weil-Bott定理。在W-代数上，我们证明了Kac和Wakimoto发现的例外W-代数的C2-余有限性，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关于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.