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Project executive summary 

The classification of finite objects (for example, associative/Lie/Poisson/Hopf algebras of a given finite dimension) is an old and notoriously difficult problem. Having the extension problem and the factorization problem as sources of inspiration we have introduced and begun an extensive study program focusing on two problems, namely the Extending Structures (ES) problem and the Global Extension (GE) problem, which are the specific objectives of the present research project. The ES problem can be stated as follows: Let O be a mathematical object in a given category C (e.g. the category of Lie/associative/Hopf/Poisson algebras) and consider c to be a fixed cardinal. Describe and classify all objects of the category C which contain and stabilize O as a subobject of codimension c. Beyond its elementary statement, the problem is very difficult but also very tempting, given the numerous subsequent problems it raises. The GE problem can be stated as follows: Let A be a fixed object in a certain category C (for instance, the category of Lie/associative/Hopf/Poisson algebras etc.) and c a given cardinal. Describe and classify, up to an isomorphism that co-stabilizes A, all objects X in the category C for which there exists an epimorphism f: X→A whose kernel has dimension c. Like the ES-problem, the GE-problem is also very general and has a large number of subsequent problems and applications, such as the classification of  co-flag objects of a given finite dimension.

Overview of the results obtained 

Both the Extending Structures (ES) problem and the Global Extension (GE) problem have been successfully investigated for various classes of algebras (such as associative/Lie/Poisson/Hopf algebras). Classification techniques for the aforementioned algebras were obtained; more precisely, given a certain (type of) algebra A, we developed a recursive method of describing and classifying all algebras (of the same type as A) which contain A as a subalgebra. The approach considered for the classification problem allowed us to explicitly construct various non-abelian cohomological objects.

Furthermore, many properties of the classes of algebras considered have been investigated through the lense of certain symmetry groups which in this very abstract context are in fact (Poisson) Hopf algebras. For example, we proved that the automorphism groups as well as the group gradings of a given (type of) algebra can be expressed in terms of these symmetry groups.

As another notable application, we mention that a rich class of new solutions of the celebrated Yang-Baxter equation on sets was obtained. Furthermore, relying on combinatorial and graph theory techniques, the number of  isomorphism types of certain classes of solutions on a finite set was explicitly determined by using classical combinatorial numbers (e.g., by Euler's partition number, Davis' number, Landau's number or Harary's number).

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