Glider Representations

Date: 6 June 2018

Venue: Campus Middelheim, G0.10 - Middelheimlaan 1 - 2020 Antwerpen (route: UAntwerpen, Campus Middelheim)

Time: 4:00 PM

Organization / co-organization: Department of Mathematics and Computer Science

PhD candidate: Frederik Caenepeel

Principal investigator: Wendy Lowen & Freddy Van Oystaeyen

Short description: PhD defence Frederik Caenepeel - Faculty of Science, Department of Mathematics and Computer Science



Abstract

Several mathematical objects such as rings, algebras, Lie algebras and groups are studied by means of their representations. Given such an object A, one associates to any element g of A a square matrix M_g in M_n(K) (with fixed n and fixed field K) and we denote this by a map from A to the algebra of square matrices M_n(K). If this map satisfies some particular conditions, we call this an A-representation or an A-module. This idea of representations originates at the end of the 19th century with the work of Frobenius and has ever since known a boost in popularity.

The theory of glider representations combines the ideas from classical representation theory explained above and the theory of filtered rings. If one fixes a filtered ring FR with subring F_0R = S, then a glider representation is given by a descending chain of S-modules  … <  M_i < … < M_2 < M_1 < M  such that there exist F_iR-actions on M_j for every i smaller or equal than j with the additional property that F_iRM_j < M_(j-i). The underlying thought is that by descending deeper into the M-chain, the more action we get from the ring R.

In the first part of the dissertation we develop the general theory from scratch. Inspired by classical representation theory, we introduce the notions of a subglider, irreducible gliders and finitely generated gliders and we prove several properties. After we developed enough machinery, we want to get our hands dirty and try to apply this new theory to various existing theories. We chose to make a distinction based on the type of filtration we are working with. In the second part of the thesis we study so called ring or algebra filtrations. These are filtrations for which all the F_iR are in fact rings or algebras and they appear naturally in the theory of groups and Lie algebras. In Chapter 4 we treat the ingredients to perform a Clifford theory and in Chapter 5 we introduce a generalized character theory for glider representations over finite algebra filtrations of group algebras. Chapter 6 contains a study of chains of semisimple Lie algebras by introducing so called Verma gliders. In the third part of the thesis we discuss glider representations for standard filtrations, that is, filtrations generated by their degree 1 part F_1R (F_nR = F_1R^n for all n). In Chapter 7 we start from the standard filtrations on coordinate rings of affine varieties and develop filtered localization theory to arrive at sheaves of glider representations for various topological spaces.



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