Noncommutative Algebras and Monoidal Triangulated Categories

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Many models in the sciences and engineering are based on mathematical settings that involve commuting variables. However, starting with quantum mechanics, a great number of models emerged that led to important mathematical problems involving variables that no longer commute. Noncommutative Algebra is one of the major areas of mathematics that studies those structures. This project addresses key problems about the properties and symmetries of noncommutative objects, as well as the representations of the corresponding noncommutative algebras. The latter area of representations theory investigates noncommutative algebras through all possible ways to present them in terms of matrices. Three major approaches are used: (1) Poisson geometry--geometry arising from the deformation of commutative objects to noncommutative ones, (2) cluster algebras--a combinatorial approach based on intricate internal transformations of the objects, called cluster mutations and (3) monoidal triangulated categories--general abstract algebraic structures arising from considering all representations of an algebra simultaneously. These research activities will be used as the foundation for the training of graduate and undergraduate students and for mentoring of mathematics postdocs. In more detail, in this project the PI will investigate the structure of quantum symmetric spaces, Nichols algebras, monoidal triangulated categories, as well as support theories for finite dimensional algebras and, more generally, finite tensor categories. The following three broad directions will be pursued: (1) Root of unity quantum cluster algebras will be investigated in two interrelated plans: description of their discriminant ideals and classification of irreducible representations. This will be based on the theory of Poisson orders and Cayley-Hamilton algebras. (2) Quantum cluster algebra structures will be constructed on quantum flag varieties, quantum Bott-Samelson varieties and quantum symmetric spaces. The irreducible representations and discriminant ideals of their root of unity quantum counterparts will be studied through the techniques developed in part 1. The representation theory of quantum symmetric pairs and quantum supergroups at roots of unity will be developed using star products and Poisson orders. (3) Methods for the classification of the noncommutative Balmer spectra of the stable categories of finite tensor categories will be developed. They will be used for the descriptions of the cohomological supports of finite dimensional Hopf algebras, and more generally, finite tensor categories.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.