Lie Theory Day

Thursday 25 November 2010

Institute of Mathematics, University of Oxford

Gibson Building, room RI.0.48 (map)

For information, registration for the conference dinner (by Wednesday 17 November) and application for funding, email rouquier@maths.ox.ac.uk


Abstracts
Leonard Scott, "New graded methods in representation theory"

    This talk reports on joint work with Brian Parshall. An ungraded finite dimensional algebra A can possess many of the best properties we know from Lie theory, but still fail to have the grading that would be expected if it were a Koszul algebra. It is in general a very hard problem to establish such a grading when the algebra A is not known to have one already, for some external reason. In particular, one confronts the very difficult algebraic problem of establishing an isomorphism of the algebra A and gr A, the graded algebra obtained by adding up the quotients of successive powers of the radical series of A. In this work we take a new approach, of working with gr A and transferring to it good properties of A without establishing any isomorphism. We show this works in some significant cases involving quotients of quantum groups (e.g., regular blocks of q-Schur algebras and their generalizations), and with more restrictions (including the use of primes for which the Lusztig character formulaa holds, and using only restricted and smaller weights), quotients of hyperalgebras in positive characteristic. In particular, we are able to show gr A is Koszul and transfer the other known good properties of A (quasi-heredity, parity conditions) to gr A. Additional results and applications will be discussed as time permits.

Peng Shan, "Fock spaces and cyclotomic rational double affine Hecke algebras"

    I will explain the construction of a crystal structure on the set of simple modules in the category O of cyclotomic rational double affine Hecke algebras and its relation to Fock spaces

Kobi Kremnizer, "Modular and root of unity representation theory"

    I'll give a geometric description of the category of representations of a quantum group at a root of unity. This will be in terms of the Springer resolution in characteristic zero. A similar description exists for the category of representations of a Lie algebra in positive characteristic. I'll explain how from this one can deduce that the modular and root of unity representation theories "look alike". This is joint work with Erik Backelin.

Kevin McGerty (Oxford) "Geometrization of Quantum Frobenius"

    The classical Frobenius morphism is an endomorphism of an algebraic group defined over a field of positive characteristic. Lusztig discovered a q-analogue of this endomorphism for quantum groups where the deformation parameter is specialized to a root of unity. We will discuss a realization of this map at the level of sheaves on the moduli space of quiver representations, and if time permits speculate on the relation of this construction with combinatorial observations of Lusztig and Kashiwara.

Eric Vasserot, "Affine Hecke algebras of classical type and quiver-Hecke algebras"

    Enomoto, Kashiwara and Miemietz have recently given some conjectures describing an analogue of Ariki's theorem for affine Hecke algebras of types B and D. This would give a powerful combinatorial tool to study their simple modules. In this talk we'll report on a proof of this conjecture. It relies on a new family of algebras which are analogues of quiver-Hecke algebras.