Support theories: axiomatics, realizations and calculations

点击翻译按钮获取中文摘要

Representation theory is a study of symmetries of linear spaces. Going back to its founders, Frobenius, Schur, Burnside, and Brauer, who first developed the subject over a century ago, the standard approach is to decompose a linear space with its symmetry into a sum of "simple ones," and then classify the simple ones into coherent and (relatively) comprehensible lists and families. The "Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups" is a classical and fundamental example of this strategy in action. In this project, the PI turns to the study of symmetries which do not subject themselves to such nice decompositions as in the classical case, which live in the world of "wild" representation theories. Such theories are ubiquitous in mathematics. They arise in algebra, topology, math physics, combinatorics, and, of course, representation theory itself. The PI will employ the new and rapidly developing subject of tensor triangular geometry, which combines topological, homological, and categorical techniques, to induce some structure in this wild representation territory, thus advancing our general understanding of this complicated world of symmetries. This project will provide research and training opportnities for undergraduate and graduate students.In more detail, this projects will advance knowledge in several new directions within the realm of tensor triangular geometry. It builds on PI's expertise in support theories in modular settings and branches out to different areas where the representation theories to be studied differ from the setting of finite groups in one or more significant aspects. The most interesting is when the theory is monoidal but not symmetric, such as for small quantum groups and their Borel subalgebras. Besides quantum groups, the PI will study and develop further the tensor triangular picture for complex Lie superalgebras, Nichols algebras of diagonal type, Schur algebras, Frobenius kernels, and finite supergroup schemes. The PI will also study local properties of some representation categories associated to Gorenstein algebras and the cohomology of finite dimensional Hopf and Nichols algebras.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.