Extensions of Noether's problem and Gelfand-Kirillov's conjecture to certain classes of noncommutative algebras

• Grant number: 14/25612-1
• Support type: Scholarships in Brazil - Doctorate
• Effective date (Start): March 01, 2015
• Effective date (End): November 30, 2018
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Algebra
• Cooperation agreement: Coordination of Improvement of Higher Education Personnel (CAPES)
• Principal Investigator: Vyacheslav Futorny
• Grantee: João Fernando Schwarz
• Home Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil
• Associated research grant: 14/09310-5 - Algebraic structures and their representations, AP.TEM
• Associated scholarship(s): 16/14648-0 - Geometric methods in representation theory, BE.EP.DR

Abstract

Noether's Problem is a classical problem in the theory of commuative fields. J. Alev and F. Dumas introduced a noncommutative version of this problem, where the Weyl skew-fields substitute the role of the fields of rational functions. This topic was studied by the student for obtaining the Masters Degree, together with it's relation to another famous problem: Gelfand-Kirillov's Conjecture. During his studies, the student, together with his adivsor V. Futorny and Farkhod Eshmatov, obtained a new result, about invariants of the Weyl skew-fields under the action of pseudo-reflection groups. This PhD project seeks to continue the research developed in the process off obtaining the Master's degree. We search generalizations and applications of Noether's Problem and Gelfand-Kirillov's Conjecture for new classes of noncommutative algebras. We plan to study invariants under the action of Weyl groups on the ring of differential operators on the torus; to study possible applications of Noncommutative Noether's Problem to the theory of Galois Algebras, together with versions of Gelfand-Kirillov's Conjecture in this context. Applications and extensions of Noether's Problem and Gelfand-Kirillov's Conjecture will also be studied in the setting of Cherednik Algebras (in particular, the rational ones).Finally, we will search for generalizations of these results for a generalization of the concept of Galois Algebras, where the role of finite groups will be replaced by actions of Hopf algebras. (AU)

Scientific publications

(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)

FUTORNY, VYACHESLAV; SCHWARZ, JOAO. Algebras of invariant differential operators. Journal of Algebra, v. 542, p. 215-229, JAN 15 2020. Web of Science Citations: 0.

ESHMATOV, FARKHOD; FUTORNY, VYACHESLAV; OVSIENKO, SERGIY; SCHWARZ, JOAO FERNANDO. NONCOMMUTATIVE NOETHER'S PROBLEM FOR COMPLEX REFLECTION GROUPS. Proceedings of the American Mathematical Society, v. 145, n. 12, p. 5043-5052, DEC 2017. Web of Science Citations: 2.

FUTORNY, VYACHESLAV; SCHWARZ, JOAO. Galois orders of symmetric differential operators. ALGEBRA & DISCRETE MATHEMATICS, v. 23, n. 1, p. 35-46, 2017. Web of Science Citations: 0.

Academic Publications

(References retrieved automatically from State of São Paulo Research Institutions)

SCHWARZ, João Fernando. Invariants of rings of differential operators: Gelfand-Kirillov rationality, categories of modules, aplications. 2018. Doctoral Thesis - Universidade de São Paulo (USP). Instituto de Matemática e Estatística São Paulo.