School of Mathematics, Statistics and Physics

Staff Profile

Dr Martina Balagovic

Lecturer in Pure Mathematics

Research

Research Interests 

Google Scholar

algebra, representation theory, quantum groups, quantum affine algebras, Kac-Moody algebras, double affine Hecke algebras and their degenerations, quantum symmetric pairs, Lie superalgebras, diagram algebras


PhD projects

Rational Cherednik algebras in positive characteristic and their representations


Teaching

PhD topic - Rational Cherednik algebras in positive characteristic and their representations

Rational Cherednik algebras are a family of associative algebras parametrized by a reflection group and some numerical parameters. They were defined in 2002 by Etingof and Ginzburg, and can be thought of as a degeneration of double affine Hecke algebras, defined in 1993 by Cherednik. Since their introduction they have been a very active area of research, with links to integrable systems, symplectic geometry and combinatorics. Of particular interest is their representation theory, where a certain category O of ``nice" representations has been defined and studied by many authors, but is still not completely described in many cases.

Much less has been done over fields of positive characteristic, and many questions remain open here. The aim of this project is to study rational Cherednik algebras, and in particular their representations, over algebraically closed fields of positive characteristic. Typical questions would be describing category O for all rational Cherednik algebras associated to a particular reflection group, describing the block decomposition of category O, describing the centers of rational Cherednik algebras or its spherical subalgebra, and describing the polynomial representation of the algebra. We propose to work in a case where the characteristic of the field divides the order of the group, creating more involved phenomena.

Students should have basic knowledge of algebra. Knowledge of representation theory, rational Cherednik algebras or finite characteristic phenomena is not necessary.



Publications