Topics in Noncommutative Algebra

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A noncommutative algebra is a fundamental concept that represents solutions to a system of equations of several non-commuting variables. Noncommutative algebras are frequently used to encode models for networks, communication, quantum computing, and chemical molecules, as well as many other objects in sciences and engineering. It is generally impossible to assess a given algebra by dealing with its elements, while invariants of an algebra capture many key features of the algebra. Then understanding invariants becomes extremely beneficial for understanding different aspects of these algebras. This project focuses on new invariants of noncommutative algebras that arise from several subjects such as noncommutative projective geometry, category theory, combinatorics, and the study of infinite dimensional Hopf algebras or quantum groups. The principal investigator will involve graduate students and postdoctoral fellows in this research.This project concerns several central topics in the field of noncommutative algebra with connections to noncommutative geometry, combinatorics, the representation theory of quivers and category theory. The principal investigator plans to study several invariants of noncommutative algebras and their associated categories, to develop foundations for new research directions, and to classify algebraic objects that describe noncommutative spaces. Specifically the project investigates noncommutative discriminants, Frobenius-Perron dimension, primitive cohomology, and other effective invariants and crucial structures in noncommutative algebras, quiver representations and tensor triangulated categories. The principal investigator continues to search for methods for the automorphism problem, the isomorphism problem, and different versions of the cancellation problem in noncommutative algebra. Finally one of PI's ultimate goals is to construct new Hopf algebra domains of Gelfand-Kirillov dimension two in positive characteristic and new tensor triangulated structures on the quiver representations of either finite or tame representation type. The principal investigator will train students in fields closely related to his research.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.