# Techniques from monoidal category theory applied to generalized Lie algebras and various types of dualities

Onderzoeksoutput: PhD Thesis

## Samenvatting

Lie algebras play a prominent role in a large number of different branches of mathematics. In the past decades, various generalizations -in different directions- of this algebraic structure have appeared in literature. Motivated by the way the field of Hopf algebra theory benefited from the interaction with the theory of monoidal categories on the one hand, and the strong relationship between Hopf algebras and Lie algebras on the other hand, the natural question arose whether it is possible to study Lie algebras within the framework of monoidal categories, and whether Lie theory could also benefit from this point of view. Some of these connections between the above-mentioned theories can be described by means of certain duality theorems. The concept of duality -in a broad sense- and certain types of autodual objects in particular, are of considerable interest in the monoidal categorical context. The main goal of this work is, roughly speaking, twofold. On the one hand, we wish to establish different types of dualities for generalized Lie (and Hopf) algebras; as Lie (and Hopf) algebras allow for a study in general additive, symmetric monoidal categories, one can expect the afore-mentioned duality results to be extended to this more abstract setting, allowing to formulate duality theorems for those generalized structures which are not covered by the classical approach. On the other hand, and of a slightly different nature, we propose a study of a particular type of autodual objects in certain braided monoidal categories, which are in close connection to quadratic modules. The chief aim here is to provide a tool which can be used to understand some aspects of the structure of categories of representations of finite groups.
Originele taal-2 English Vrije Universiteit Brussel Caenepeel, Stefaan, PromotorVercruysse, Joost, Promotor, Externe Persoon Brussels Published - 2013

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