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Sub-bimodules of the identity bimodule for cyclic quivers
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We describe the combinatorics of the multisemigroup with multiplicities for the tensor category of subbimodules of the identity bimodule, foran arbitrary non-uniform orientation of a finite cyclic quiver

Keywords [en]
representation theory, quiver, bimodule, multisemigroup
National Category
Algebra and Logic
Identifiers
URN: urn:nbn:se:uu:diva-327259OAI: oai:DiVA.org:uu-327259DiVA, id: diva2:1129912
Available from: 2017-08-07 Created: 2017-08-07 Last updated: 2017-08-09
In thesis
1. Semigroups, multisemigroups and representations
Open this publication in new window or tab >>Semigroups, multisemigroups and representations
2017 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of four papers about the intersection between semigroup theory, category theory and representation theory. We say that a representation of a semigroup by a matrix semigroup is effective if it is injective and define the effective dimension of a semigroup S as the minimal n such that S has an effective representation by square matrices of size n.

A multisemigroup is a generalization of a semigroup where the multiplication is set-valued, but still associative.

A 2-category consists of objects, 1-morphisms and 2-morphisms. A finitary 2-category has finite dimensional vector spaces as objects and linear maps as morphisms. This setting permits the notion of indecomposable 1-morphisms, which turn out to form a multisemigroup.

Paper I computes the effective dimension Hecke-Kiselman monoids of type A. Hecke-Kiselman monoids are defined by generators and relations, where the generators are vertices and the relations depend on arrows in a given quiver.

Paper II computes the effective dimension of path semigroups and truncated path semigroups. A path semigroup is defined as the set of all paths in a quiver, with concatenation as multiplication. It is said to be truncated if we introduce the relation that all paths of length N are zero.

Paper III defines the notion of a multisemigroup with multiplicities and discusses how it better captures the structure of a 2-category, compared to a multisemigroup (without multiplicities).

Paper IV gives an example of a family of 2-categories in which the multisemigroup with multiplicities is not a semigroup, but where the multiplicities are either 0 or 1. We describe these multisemigroups combinatorially.

Place, publisher, year, edition, pages
Uppsala: Department of Mathematics, 2017. p. 29
Series
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 102
Keywords
representation theory, semigroups, multisemigroups, category theory, 2-categories
National Category
Algebra and Logic
Identifiers
urn:nbn:se:uu:diva-327270 (URN)978-91-506-2647-6 (ISBN)
Public defence
2017-09-25, Polhemssalen, Lägerhyddsvägen 1, Uppsala, 13:15 (English)
Opponent
Supervisors
Available from: 2017-08-31 Created: 2017-08-07 Last updated: 2017-08-31

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