Noncommutative Algebras and Related Categorical Structures

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Many models in the sciences and engineering lead to mathematical problems about solutions of equations in commuting variables. However, starting with quantum mechanics, a great number of models emerged that led to mathematical problems involving variables that no longer commute. Noncommutative Algebra is one of the areas of mathematics that studies those structures. The research projects that are funded by this award investigate the interrelations between the commutative and noncommutative settings, using a branch of geometry called Poisson geometry. Three other approaches are used to study the noncommutative setting: a combinatorial approach based on intricate internal transformations of the objects, called cluster mutations; an algebraic approach that investigates intrinsically defined structures called Calabi-Yau categories; and a noncommutative geometric approach using quantum versions of symmetric spaces. The four approaches are simultaneously used to carry out a detailed study of the properties and symmetries of noncommutative objects. Further, the noncommutative objects are shown to exhibit various forms of rigidity. This is used to settle problems in algebra, geometry, combinators, and dynamical systems that were previously posed without any reference to the noncommutative setting. These research activities will be used as the foundation for the training of graduate and undergraduate students and for mentoring mathematics postdocs.The research projects funded under this awaard investigate problems in noncommutative algebra, quantum symmetric spaces, and noncommutative projective algebraic geometry and the relations of these problems to Poisson geometry, combinatorics, triangulated categories, and integrable systems. On the one hand, the program aims at using methods from the latter areas to describe the structure and representations of quantum cluster algebras at roots of unity, the Drinfeld doubles of Nichols algebras, the algebras that appear in the theory of quantum symmetric pairs, and the algebras that describe noncommutative projective spaces. In the opposite direction, previously posed problems in the latter areas are converted to problems for noncommutative algebras and their representation categories, and are then resolved within that setting. One of the directions of this program is the construction of universal K-matrices on the symmetric subalgebras of the Drinfeld doubles of Nichols algebras, and using this to study the ring theoretic properties of Nichols algebras. Another direction aims at the classification of irreducible representations of Nichols algebras of diagonal type using Poisson orders and noncommutative discriminants. A third direction develops a general setting for the study of finite dimensional representations of quantum cluster algebras at root of unity using Poisson geometry and Cayley-Hamilton algebras. Three additional directions investigate the geometry of noncommutative projective spaces modeled by higher dimensional elliptic algebras, the structure of 2-Calabi-Yau categories via categorical C-vectors and dynamical systems, and the construction of integral quantum cluster algebra structures on the canonical forms of quantized coordinate rings of varieties in theory of Lie groups.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.