Novikov, LR- en post-Lie algebra structuren, en hun relatie met NIL-affiene kristallografische acties

摘要

The study of simply transitive and crystallographic NIL-affine actions on the Lie algebra level leads to different concepts, including Novikov, LR- and post-Lie algebra structures, which are studied in this thesis. In our research we can distinguish three aspects: construction, existence and structure. In the construction aspect, we search for examples by using different techniques as the lifting of such structures, using theoretical considerations and using computer experiments. In the existence aspect, we try to answer the question which Lie algebras admit such structures. In the structure aspect, we study the algebraic structure of Lie algebras admitting a Novikov, LR- or post-Lie algebra structure. The results obtained here, are of great importance in the construction and existence aspect. For Novikov and LR-structures we try to find out what we can say about ideals, quotients, subalgebras,... For post-Lie algebra structures we study how the existence of such a structure imposes certain algebraic conditions on the Lie algebras involved and how the algebraic structures of these Lie algebras depend on each other.