An algebra is a fundamental mathematical concept that encodes information about objects in mathematics and other disciplines. Understanding the structure of algebras is an important task on its own as well as for other fields in mathematics. Invariants of an algebra are supposed to capture distinguished features of its representing objects, and identities or equations between the invariants reveal the insights behind these objects. This proposal focuses on the research of noncommutative algebras that arise from several subjects such as noncommutative projective geometry, noncommutative invariant theory, and the study of infinite dimensional Hopf algebras. The PI will investigate several long-term projects; one of his ultimate goals is to discover powerful invariants that characterize noncommutative algebras. The PI will involve graduate students in this research.
This research project concerns three topics in noncommutative algebra: higher dimensional Artin-Schelter regular algebras, noncommutative invariant theory and noncommutative McKay correspondence, and the Zariski cancellation problem for noncommutative algebras, all of which are intimately connected with each other. The PI continues to develop the foundations for new research directions, to search for distinct invariants of noncommutative algebras and identities between the invariants, and to work on important open questions in the field. The work involves two exciting ideas, discriminant of noncommutative algebras and homological identities, which have many surprising applications. Invariants such as the discriminant and those involved in homological identities touch the core of the structure of noncommutative algebras. This project investigates a large number of interesting questions that will stimulate research in noncommutative algebra and related areas. Results of the project have promise for significant impact in the subject and beyond.
