Title: 1-shifted Lie bialgebras and their quantizations Show Chinese title

Title: 1-平移的李双代数及其量子化

Abstract: In this paper, we define (cohomologically) 1-shifted Manin triples and 1-shifted Lie bialgebras, and study their properties. We derive many results that are parallel to those found in ordinary Lie bialgebras, including the double construction and the existence of a 1-shifted $r$-matrix satisfying the classical Yang-Baxter equation. Turning to quantization, we first construct a canonical quantization for each 1-shifted metric Lie algebra $\mathfrak{g}$, producing a deformation to the symmetric monoidal category of $\mathfrak{g}$ modules over a formal variable $\hbar$. This quantization is in terms of a curved differential graded algebra. Under a further technical assumption, we construct quantizations of transverse Lagrangian subalgebras of $\mathfrak{g}$, which is a pair of DG algebras connected by Koszul duality, and give rise to monoidal module categories of the quantized double. Finally, we apply this to Manin triples arising from Lie algebras of loop groups, and construct 1-shifted meromorphic $r$-matrices. The resulting quantizations are the cohomologically-shifted analogue of Yangians. Show Chinese abstract

Abstract: 在本文中，我们定义了（上同调意义下的）1-位移的Manin三元组和1-位移的李双代数，并研究了它们的性质。 我们得出了许多与普通李双代数中发现的结果平行的结果，包括双构造和存在一个满足经典Yang-Baxter方程的1-位移$r$-矩阵。 转向量化问题，我们首先为每个1-位移的度量李代数$\mathfrak{g}$构造了一个规范的量化，从而在形式变量$\hbar$上将对称单子范畴的$\mathfrak{g}$模块进行了变形。 这种量化是通过一个弯曲的微分分次代数来实现的。 在进一步的技术假设下，我们构造了$\mathfrak{g}$的横截拉格朗日子代数的量化，这是一对由Koszul对偶连接的DG代数，并产生了量化双代数的张量模范畴。 最后，我们将此方法应用于来自环群李代数的Manin三元组，并构造了1-位移的有理函数$r$-矩阵。 所得的量化结果是杨氏代数的上同调位移类比。