Noncommutative Algebraic Geometry and Noncommutative Invariant Theory

这个研究计画是关于非交换代数与不变量理论和代数几何的连结。代数是贯穿数学的基本对象。代数的一个用途是对几何结构的信息进行编码。在数学和物理中出现的许多代数都是非交换的，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。主要研究者的研究方向是非交换代数的两个子领域，即非交换射影代数几何（NCPAG）和非交换不变量理论（NCIT）。第一个领域是在20世纪80年代开始研究代数，特别是三维Sklyanin代数Skly3，其环论行为无法使用纯代数技术确定。PI计划继续使用NCPAG的技术来建立Skly 3以及物理学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。 第二个研究领域的目标，NCIT，是在经典不变理论的结果扩展到非交换设置。为此，我们将交换多项式环上的群的作用替换为非交换正则代数上的Hopf代数（或量子群）的作用，该非交换正则代数与交换对应物共享同调性质。虽然它是很难的一个Hopf代数的行动代数，PI的工作进展的行动的Hopf代数交换域，对他们的量化，并对代数所产生的NCPAG。PI还打算继续贡献结果代数所产生的这些行动，如不变子环和粉碎产品代数。这将导致新的例子代数的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.