Noncommutative Algebraic Geometry and Noncommutative Invariant Theory

这项研究项目涉及与不变理论和代数几何有关的非交换代数。代数是贯穿整个数学的一个基本对象。代数的一个用途是对有关几何结构的信息进行编码。数学和物理中出现的许多代数都是非对易的，即两个元素的乘积取决于元素相乘的顺序。PI将研究与非交换几何和非交换不变量理论相关的非交换代数，主要研究非交换代数的两个子领域：非交换射影几何(NCPAG)和非交换不变理论(NCIT)。第一个领域是在20世纪80年代发起的，目的是研究代数，特别是三维Sklyanin代数Skly3，它的环论行为不能用纯代数技巧确定。PI计划继续使用NCPAG的技术来建立关于Skly3以及物理和李理论中出现的代数的结果，例如Virasoro代数和其他相关代数。第二个研究领域NCIT的目标是将经典不变理论中的结果推广到非对易环境。为此，我们用Hopf代数(或量子群)在非对易正则代数上的作用来代替群在交换多项式环上的作用，该正则代数与其交换多项式环具有同调性质。虽然Hopf代数很难作用于一个代数，但PI已经在研究Hopf代数在交换域上的作用、它们的量子化以及由NCPAG产生的代数。PI还打算继续贡献由这些作用产生的代数的结果，例如不变子环和sash乘积代数。这将导致新的代数例子，其表示理论值得进一步研究。

This research project concerns noncommutative algebra with connections to both invariant theory and algebraic geometry. An algebra is a fundamental object throughout mathematics. One use of algebras is to encode information about geometric structures. Many algebras arising in mathematics and physics are noncommutative, i.e., the product of two elements depends on the order in which elements are multiplied. The PI will investigate noncommutative algebras that are related to noncommutative geometry and noncommutative invariant theory.The research of the principal investigator lies in two subfields of noncommutative algebra entitled Noncommutative Projective Algebraic Geometry (NCPAG) and Noncommutative Invariant Theory (NCIT). The first area was launched in the 1980s to examine algebras, especially the three-dimensional Sklyanin algebras Skly3, whose ring-theoretic behavior could not be determined using purely algebraic techniques. The PI plans to continue using techniques of NCPAG to establish results on Skly3 and on algebras arising in physics and Lie theory, such as the Virasoro algebra and other related algebras. The goal of the second area of research, NCIT, is to extend results in classical invariant theory to a noncommutative setting. To do so, we replace an action of a group on a commutative polynomial ring by an action of a Hopf algebra (or quantum group) on a noncommutative regular algebra that shares homological properties with its commutative counterpart. Although it is difficult for a Hopf algebra to act on an algebra, the PI has works in progress on actions of Hopf algebras on commutative domains, on their quantizations, and on algebras arising from NCPAG. The PI also intends to continue to contribute results on algebras arising from these actions, such as invariant subrings and smash product algebras. This will lead to new examples of algebras whose representation theory merit further investigation.