Thesis supervisor: Gabriella Eszter Böhm
belső konzulens: Attila Nagy
Location of studies (in Hungarian): Wigner Research Centre of the Hungarian Academy of Sciences Abbreviation of location of studies: WFK
Description of the research topic:
Crossed module of groups is a classical notion, it was introduced by Whitehead in 1949. His original motivation came from algebraic topology: he meant to describe the higher versions of the fundamental group. Since then crossed modules found applications in many branches of mathematics and even in physics: in homological algebra, non-abelian cohomology, combinatorial group theory, modelling homotopy n-types (and double loop spaces for n=3), and in topological quantum field theory. Crossed modules admit several equivalent descriptions, each of them of conceptual or practical use in some of the above applications. A fully categorical treatment, based on the use of semi-abelian categories, was given by Janelidze in 2003.
Crossed modules of Hopf algebras were introduced recently and they already found interesting applications. They share many features of the group case. This raises the demand of a unifying treatment which should include some further relevant examples as well. While groups constitute a semi-abelian category, Hopf algebras do not. Thus the new theory should go beyond Janelidze’s study. Since groups are Hopf monoids in the category of sets, and Hopf algebras are Hopf monoids in the category of vector spaces, the unifying framework is expected to be provided by Hopf monoids in symmetric monoidal categories.
In the proposed PhD project the following groups of questions should be attacked.
• Which ones of the equivalent descriptions of crossed modules of groups can be translated to the new, more general setting, first of all to Hopf algebras over fields?
• How to define higher crossed modules of Hopf algebras, and next of Hopf monoids in symmetric monoidal categories? To what other algebraic structures are they equivalent to?
• Which occurrences of crossed modules of groups in various applications do have counterparts for the new kinds of crossed modules? For which instances of Hopf monoids?
• Is it possible to develop the theory beyond Hopf monoids in symmetric monoidal categories: in braided monoidal, or even in duoidal categories? Can one include – in this, or another way – in the study also weak Hopf algebras?
Required language skills: English Further requirements: An adequate level of English for a smooth scientific communication, and some experience in algebra with a basic knowledge of category theory.
Number of students who can be accepted: 1
Deadline for application: 2017-05-31
2024. IV. 17. ODT ülés Az ODT következő ülésére 2024. június 14-én, pénteken 10.00 órakor kerül sor a Semmelweis Egyetem Szenátusi termében (Bp. Üllői út 26. I. emelet).
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