发表机构
McGill University; Universität Bonn; Université de Sherbrooke（麦吉尔大学; 波恩大学; 舍布鲁克大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究与复单链单李代数相关的移位杨代数表示范畴$\mathcal{O}$，证明其格罗滕迪克环同构于双无限Bott - Samelson簇的考克斯环，借此证明猜想等并得到一些结果。
AI 中文摘要
本文研究与$\mathbb{C}$上的单链单李代数$\mathfrak{g}$相关的移位杨代数表示的范畴$\mathcal{O}$。特别地，我们证明这个范畴的（复化）格罗滕迪克环同构于开放双无限Bott - Samelson簇的考克斯环，它是我们从交替堆的Bott - Samelson簇构造的一个射影簇。利用Francone - Leclerc的工作，通过将上述格罗滕迪克环与Geiss - Hernandez - Leclerc定义的簇代数等同，我们证明了Hernandez - Zhang的一个猜想。我们的方法还给出了朗兰兹对偶群$G^{\vee}$对这个格罗滕迪克环的作用，并表明第一作者和第五作者及其合作者工作中定义的移位余积给出了截断移位杨代数的余积。这个机制进而使我们能够证明Frenkel - Hernandez和Geiss - Hernandez - Leclerc关于扩展$QQ$ - 系统的进一步猜想，并得到Hernandez - Leclerc定义的对偶性的一个推广。
英文摘要
In this paper, we study the category $\mathcal{O}$ of representations of shifted Yangians associated to a simply-laced simple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$. In particular, we prove that the (complexified) Grothendieck ring of this category is isomorphic to the Cox ring of the open bi-infinite Bott-Samelson variety, which is a pro-variety we construct from Bott-Samelson varieties for alternating heaps. Using work of Francone-Leclerc, we prove a conjecture of Hernandez-Zhang by identifying the above Grothendieck ring with a cluster algebra defined by Geiss-Hernandez-Leclerc. Our methods also yield an action of the Langlands dual group $G^{\vee}$ on this Grothendieck ring, and show that the shifted coproducts defined in work of the first and fifth authors with collaborators give rise to coproducts for truncated shifted Yangians. This machinery then allows us to prove further conjectures of Frenkel-Hernandez and Geiss-Hernandez-Leclerc on extended $QQ$-systems, and to obtain a generalization of a duality defined by Hernandez-Leclerc.
Comments142 pages, comments are welcome. V2: added results in Section 11 and references elsewhere. V3: corrected section 12
