Infinitely Generated Group Representations and Large Lattices
| Thesis title in Czech: | Nekonečně generované reprezentace grup a zobecněné mříže |
|---|---|
| Thesis title in English: | Infinitely Generated Group Representations and Large Lattices |
| Academic year of topic announcement: | 2021/2022 |
| Thesis type: | dissertation |
| Thesis language: | angličtina |
| Department: | Department of Algebra (32-KA) |
| Supervisor: | doc. Mgr. Pavel Příhoda, Ph.D. |
| Author: | hidden - assigned and confirmed by the Study Dept. |
| Date of registration: | 01.03.2022 |
| Date of assignment: | 01.03.2022 |
| Confirmed by Study dept. on: | 07.03.2022 |
| Guidelines |
| The goal of the thesis is to study and improve existing methods for understanding generalized lattices over orders
in separable algebras. The case of representation of groups over Dedekind domains (eventually also over Gorenstein rings) will be of particular interest. The work is supposed to be done with the cooperation of prof. Dolors Herbera from Universitat Autonoma de Barcelona under the cotutelle agreement between Charles University and Universitat Autonoma. |
| References |
| C. Curtis, I. Reiner Methods of representation theory. Vol. I. With applications to finite groups and orders. John Wiley and Sons, New York 1981.
D. Herbera, P. Příhoda: Big projective modules over semilocal noetherian rings, J. Reine Angew. Math 648 (2010), 111 - 148. D. Herbera, P. Příhoda and Roger Wiegand: Big Pure Projective Modules over Commutative Noetherian Domains: Comparsion With the Completion, preprint. P. Příhoda: Fair-sized projective modules, Rend. Semin. Mat. Univ. Padova, 123 (2010), 141 - 167. W. Rump, Large lattices over orders, Proc. London Math. Soc 91 (2005), 105 - 128. |
| Preliminary scope of work in English |
| The study of idempotent ideals in integral group rings was initiated by Akasaki who proved that an integral group ring of a finite group possesses non-free large projectives if and only if the group is not solvable. Later, Puninski realized that idempotent ideals can be used to study the problem of full decomposability of generalized lattices over orders (for example determine whether an action of a group on Z^{(\omega)} is a direct sum of actions on finitely generated subgroups of Z^{(\omega)}).
Existing techniques anable to classify generalized lattices over some orders. One of the main goal of the thesis will be to study and improve such techniques to make them more practical. |