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

Grant number: 14/25612-1
Support Opportunities: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
Acordo de Cooperação: Coordination of Improvement of Higher Education Personnel (CAPES)
Principal Investigator:Vyacheslav Futorny
Grantee:João Fernando Schwarz
Host 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


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)

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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, . (18/18146-5, 16/14648-0, 14/25612-1, 18/23690-6)
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, . (13/22068-6, 14/25612-1, 14/09310-5)
FUTORNY, VYACHESLAV; SCHWARZ, JOAO. Galois orders of symmetric differential operators. ALGEBRA & DISCRETE MATHEMATICS, v. 23, n. 1, p. 35-46, . (14/09310-5, 14/25612-1)
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 (IME/SBI) São Paulo.

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