$\mathbb{R}^n$中的扭二次曲面与代数子流形
Twisted Quadrics and Algebraic Submanifolds in $\mathbb{R}^n$
摘要：
本文提出了构造$\mathbb{R}^n$的代数子流形$M$的非交换变形的一般方法，并将[G. Fiore，T. Weber，Twisted submanifold of $\mathbb{R}^n$，arXiv：2003.03854]对光滑子流形有效。我们使用的框架扭曲微分几何的Aestheri等人。（类。量子重力23，1883-1911，2006），其中交换逐点积被由Drinfel'd扭曲确定的$\star$ -积代替。我们实际上同时构造了所有代数子流形$M_c$的非交换变形，这些子流形是$f^a(x)$的水平集，其中$f^a(x)=0$是由$M$的点求解的多项式方程，使用基于与所有$M_c$相切的向量场的李代数$Ξ_t$的扭曲。扭曲Cartan演算在扭曲$Ξ_t$下是自动等变的.如果我们赋予$\mathbb{R}^n$一个度量，那么扭曲和投影到法向或切向分量是交换的，将列维-奇维塔联络投影到扭曲的$M$是一致的，特别是扭曲高斯定理成立，只要扭曲是基于基灵向量场。扭代数二次曲面可以用生成元和$\star$ -多项式关系来刻画。我们明确工作的基础上的所有二次曲面的阿贝尔或约旦扭曲的$\mathbb{R}^3$除了椭圆，特别是扭曲圆柱嵌入扭曲欧几里得$\mathbb{R}^3$和扭曲双曲面嵌入扭曲闵可夫斯基$\mathbb{R}^3$ [后者是扭曲（反）德西特空间$dS_2$，$AdS_2$ ]。
英文摘要：
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 level sets of the $f^a(x)$, where $f^a(x)=0$ are the polynomial equations solved by the points of $M$, employing twists based on the Lie algebra $Ξ_t$ of vector fields that are tangent to all the $M_c$. The twisted Cartan calculus is automatically equivariant under twisted $Ξ_t$. If we endow $\mathbb{R}^n$ with a metric, then twisting and projecting to normal or tangent components commute, projecting the Levi-Civita connection to the twisted $M$ is consistent, and in particular a twisted Gauss theorem holds, provided the twist is based on Killing vector fields. Twisted algebraic quadrics can be characterized in terms of generators and $\star$-polynomial relations. We explicitly work out deformations based on abelian or Jordanian twists of all quadrics in $\mathbb{R}^3$ except ellipsoids, in particular twisted cylinders embedded in twisted Euclidean $\mathbb{R}^3$ and twisted hyperboloids embedded in twisted Minkowski $\mathbb{R}^3$ [the latter are twisted (anti-)de Sitter spaces $dS_2$, $AdS_2$].
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
