The field of quantum $GL(N,\mathbb{C})$ in the C$^*$-algebraic setting

A note on reduced and von Neumann algebraic free wreath products

In this paper, we study operator algebraic properties of the reduced and von Neumann algebraic versions of the free wreath products $\mathbb G \wr_* S_N^+$, where $\mathbb G$ is a compact matrix quantum group. Based on recent result on their corepresentation theory by Lemeux and Tarrago, we prove that $\mathbb G \wr_* S_N^+$ is of Kac type whenever $\mathbb G$ is, and that the reduced version of $\mathbb G \wr_* S_N^+$ is simple with unique trace state whenever $N \geq 8$. Moreover, we prove that the reduced von Neumann algebra of $\mathbb G \wr_* S_N^+$ does not have property $Γ$.

Borel-Weil Theorem for Algebraic Supergroups

We study the structure of an algebraic supergroup $\mathbb{G}$ and establish the Borel-Weil theorem for $\mathbb{G}$ to give a systematic construction of all simple supermodules over an arbitrary field. Especially when $\mathbb{G}$ has a distinguished parabolic super-subgroup, we show that the set of all simple supermodules of $\mathbb{G}$ is parameterized by the set of all dominant weights for the even part of $\mathbb{G}$, prove a super-analogue of the Kempf vanishing theorem, and give a description of Euler characteristics.

三角代数的代数版本及P¹×P¹的代数版本与Hochschild上同调

Algebraic versions of $\mathbb{T}^2$ and of $\mathbb{P}^1\times\mathbb{P}^1$ and Hochschild cohomology

本文研究了三角代数及其变形的Hochschild上同调，揭示了其杯积结构与维度变化的非直观特性。

Vertex algebraic intertwining operators among generalized Verma modules for $\widehat{\mathfrak{sl}(2,\mathbb{C})}$

We construct vertex algebraic intertwining operators among certain generalized Verma modules for $\widehat{\mathfrak{sl}(2,\mathbb{C})}$ and calculate the corresponding fusion rules. Additionally, we show that under some conditions these intertwining operators descend to intertwining operators among one generalized Verma module and two (generally non-standard) irreducible modules. Our construction relies on the irreducibility of the maximal proper submodules of generalized Verma modules appearing in the Garland-Lepowsky resolutions of standard $\widehat{\mathfrak{sl}(2,\mathbb{C})}$-modules. W…

代数概形导出中心上的 $\n\mathbb{E}_2$-代数结构

$\mathbb{E}_2$-algebra structures on the derived center of an algebraic scheme

该研究在形变量子化框架下建立Lurie高阶中心理论与代数概形霍赫希尔德上同调的关联，证明霍赫希尔德复形的$\n\mathbb{E}_2$-代数结构与经典Gerstenhaber括号、上积在仿射光滑情形下一致，深化了对概形导出中心代数结构的认知。

Twisted Quadrics and Algebraic Submanifolds in $\mathbb{R}^n$

We propose a general procedure to construct noncommutative deformations of an algebraic submanifold $M$ of $\mathbb{R}^n$, specializing the procedure [G. Fiore, T. Weber, Twisted submanifolds of $\mathbb{R}^n$, arXiv:2003.03854] valid for smooth submanifolds. We use the framework of twisted differential geometry of Aschieri et al. (Class. Quantum Grav. 23, 1883-1911, 2006), whereby the commutative pointwise product is replaced by the $\star$-product determined by a Drinfel'd twist. We actually simultaneously construct noncommutative deformations of all the algebraic submanifolds $M_c$ that are…