Noncommutative Algebraic Geometry and Noncommutative Invariant Theory

这个研究项目涉及非交换代数，它与不变理论和代数几何都有联系。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。数学和物理中出现的许多代数是不可交换的，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。主要研究非交换代数的两个子领域：非交换射影代数几何（NCPAG）和非交换不变理论（NCIT）。第一个领域是在20世纪80年代开始研究代数，特别是三维Sklyanin代数，它的环理论行为不能用纯代数技术来确定。PI计划继续使用NCPAG的技术来建立关于Skly3和物理学和李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的结果扩展到非交换设置。为此，我们用Hopf代数（或量子群）在与其交换对应物具有同调性质的非交换正则代数上的作用来替换交换多项式环上的群的作用。虽然Hopf代数很难作用于代数上，但是PI在Hopf代数在交换域上的作用、它们的量化以及由NCPAG产生的代数上进行了研究。PI还打算继续贡献由这些作用产生的代数的结果，例如不变子代数和粉碎积代数。这将导致代数的新例子，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.