Dr Martina Balagovic
Lecturer in Pure Mathematics
- Email: email@example.com
- Telephone: +44 (0) 191 208 5370
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
Rational Cherednik algebras in positive characteristic and their representations
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.
- Balagovic M, Kolb S. The bar involution for quantum symmetric pairs. Representation Theory 2015, 19, 186-210.
- Balagovic M. Irreducible modules for the degenerate double affine Hecke algebra of type A as submodules of Verma modules. Journal of Combinatorial Theory, Series A 2015, 133, 97-138.
- Balagovic M. Degeneration of Trigonometric Dynamical Difference Equations for Quantum Loop Algebras to Trigonometric Casimir Equations for Yangians. Communications in Mathematical Physics 2015, 334(2), 629-659.
- Balagovic M, Chen H. Category O for rational Cherednik algebras Ht,c(GL2(Fp),h) in characteristic p. Journal of Pure and Applied Algebra 2013, 217(9), 1683–1699.
- Balagovic M, Chen H. Representations of rational Cherednik algebras in positive characteristic. Journal of Pure and Applied Algebra 2013, 217(4), 716–740.
- Balagovic M, Policastro C. Category O for the rational Cherednik algebra associated to the complex reflection group G12. Journal of Pure and Applied Algebra 2012, 216(4), 857–875.
- Balagovic M. Chevalley restriction theorem for vector-valued functions on quantum groups. Representation Theory 2011, 15, 617-645.
- Balagovic M, Puranik A. Irreducible representations of the rational Cherednik algebra associated to the Coxeter group H3. Journal of Algebra 2014, 405, 259-290.
- Balagovic M, Balasubramanian A. On the lower central series quotients of a graded associative algebra. Journal of Algebra 2011, 328(1), 287-300.
- Balagovic M, Kolb S. Universal K-matrix for quantum symmetric pairs. Journal für die reine und angewandte Mathematik 2016, e-pub ahead of print.