In this article we study the properties of preprojective algebras of representation finite species. To understand the structure of a preprojective algebra, one often studies its Nakayama automorphism. A complete description of the Nakayama automorphism is given by Brenner, Butler and King when the algebra is given by a path algebra. We partially generalize this result to the species case, i.e. we manage to describe the Nakayama automorphism up to an unknown constant. We show that the preprojective algebra of a representation finite species is an almost Koszul algebra. With this we know that almost Koszul complexes exist. It turns out that the almost Koszul complex for a representation finite species is given by a mapping cone of a certain chain map. We also study a higher dimensional analogue of representation finite hereditary algebras called d-representation finite algebras. One source of $d$-representation finite algebras comes from taking tensor products. By introducing a functor called the Segre product, we manage to give a complete description of the almost Koszul complex of the preprojective algebra of a tensor product of two species with relations with certain properties, in terms of the knowledge of the given species with relations. This allows us to compute the almost Koszul complex explicitly for certain species with relations more easily.
| Identifer | oai:union.ndltd.org:UPSALLA1/oai:DiVA.org:uu-471668 |
| Date | January 2022 |
| Creators | Söderberg, Christoffer |
| Publisher | Uppsala universitet, Algebra, logik och representationsteori, Uppsala |
| Source Sets | DiVA Archive at Upsalla University |
| Language | English |
| Detected Language | English |
| Type | Licentiate thesis, monograph, info:eu-repo/semantics/masterThesis, text |
| Format | application/pdf |
| Rights | info:eu-repo/semantics/openAccess |
| Relation | U.U.D.M. report / Uppsala University, Department of Mathematics, 1101-3591 ; 2022:1 |
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