Invariants of rings of differential operators: Gelfand-Kirillov rationality, categories of modules, aplications

Author(s)

João Fernando Schwarz

• Document type: Doctoral Thesis
• Press: São Paulo.
• Institution: Universidade de São Paulo (USP). Instituto de Matemática e Estatística (IME/SBI)
• Defense date: 2018-11-13

Examining board members

Hugo Luiz Mariano; Nikolai Alexandrovitch Goussevskii; Kostiantyn Iusenko; Plamen Emilov Kochloukov; Daniel Levcovitz

• Advisor: Vyacheslav Futorny

Abstract

This thesis discussess how, given the rigidity results on the Weyl Algebra An(k), its invariant subrings can nonetheless have an interesting invariant theory: from the structural point of view, a birrational equivalence study under the Gelfand-Kirillov philosophy gives us the Noncommutative Noether Problem, of which we obtain many new results (Chapter 4). From the point of view of representations, we obtain that their invariant rings, in many cases, have a natural theory of Gelfand-Tsetlin modules just like the Weyl Algebra (Chapter 5), and a natural notion of holonomic modules (Chapter 6). We discuss analogues results for algebras which are similar to the Weyl Algebra, such as the ring of differential operators on the torus and the generalized Weyl algebras (Chapters 2,4,5). As applications, we have a Gelfand-Kirillov Conjecture for spherical subalgebras of Cherednik (Chapter 4); for the Gelfand-Kirillov Conjecture of many Galois algebras (Chapter 5 and 7); and the problem to give a Galois structure to the algebra U(L), where L is a simple Lie algebra of type B,C,D -generalizing the case A (Chapter 5). A chapter about the Quantum Noether Problem and a resume of the article Quantum Linear Galois Algebras\" ends the thesis. (AU)

• FAPESP's process: 14/25612-1 - Extensions of Noether's problem and Gelfand-Kirillov's conjecture to certain classes of noncommutative algebras
• Grantee: João Fernando Schwarz
• Support Opportunities: Scholarships in Brazil - Doctorate