Research in Noncommutative Algebra: Hopf Algebra Actions on Noetherian Artin-Schelter Regular Algebras and Noncommutative McKay Correspondence

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Invariants such as dimensions and symmetries are useful tools in mathematics and other disciplines. Understanding links between different invariants is a demanding task in modern mathematics. This research project is to study noncommutative algebras (mathematical structures in which xy does not necessarily equal yx) by using algebraic, combinatorial, geometric, and other invariants, and to build a bridge between the subject of noncommutative algebra and other active research areas. The PI will investigate the structure of several important families of algebras and work on central questions in the subject. Since noncommutative algebras have been used extensively, this project will deepen the understanding of other mathematical areas including noncommutative algebraic geometry, commutative algebra and mathematical physics.A central theme of the proposal is the noncommutative McKay correspondence, a concept motivated by the classical McKay correspondence that has recently been extended to several new areas. Specific topics include noncommutative quotient singularities of Hopf algebra actions on Artin-Schelter regular algebras; skew Calabi-Yau property and the Nakayama automorphism of Artin-Schelter Gorenstein algebras; and the noncommutative discriminant of algebras which are module-finite over their center. The PI has introduced a number of invariants with fruitful applications in the study of automorphism groups and locally nilpotent derivations of noncommutative algebras, as well as the noncommutative Zariski cancellation problem. The PI will continue to search for distinct invariants of noncommutative algebras, to develop foundations for new research directions, and to work on central open questions in the field. The noncommutative McKay correspondence is one essential guideline for the interplay between noncommutative algebra, Hopf algebra and theory of quantum groups, noncommutative invariant theory, and noncommutative algebraic geometry.