Data di Pubblicazione:
2020
Citazione:
AN INTRINSIC APPROACH TO THE NON-ABELIAN TENSOR PRODUCT / D. Di Micco ; tutor: S. Mantovani, T. Van der Linden ; coordinatore: V. Mastropietro. Università degli Studi di Milano, 2020 Jan 28. 32. ciclo, Anno Accademico 2019. [10.13130/di-micco-davide_phd2020-01-28].
Abstract:
The notion of a non-abelian tensor product of groups first appeared in a paper where Brown and Loday generalised a theorem on CW-complexes by using the new notion of non-abelian tensor product of two groups acting on each other, instead of the usual tensor product of abelian groups. In particular, they took two groups acting on each other and they defined their non-abelian tensor product via an explicit presentation. This led to the development of an algebraic theory based on this construction. Many results were obtained treating the properties which are satisfied by this non-abelian tensor product as well as some explicit calculations in particular classes of groups. In order to state many of their results regarding this tensor product, Brown and Loday needed to require, as an additional condition, that the two groups M and N acted on each other compatibly: these amount to the existence of a group L and of two crossed modules structures of M and N on L such that the original actions are induced from these crossed module structures. Furthermore, they proved that the non-abelian tensor product is part of a so-called crossed square of groups: this particular crossed square is the pushout of a specific diagram in the category of crossed squares of groups. Note that crossed squares are a 2-dimensional version of crossed modules of groups. Following the idea of generalising the algebraic theory arising from the study of the non-abelian tensor product of groups, Ellis gave a definition of non-abelian tensor product of Lie algebras, and obtained similar results. Further generalisations have been studied in the contexts of Leibniz algebras, restricted Lie algebras, Lie-Rinehart algebras, Hom-Lie algebras, Hom-Leibniz algebras, Hom-Lie-Rinehart algebras, Lie superalgebras and restricted Lie superalgebras. The aim of our work is to build a general version of non-abelian tensor product, having the specific definitions in the categories of groups and Lie algebras as particular instances. In order to do so we first extend the concept of a pair of compatible actions (introduced in the case of groups by Brown and Loday and in the case of Lie algebras by Ellis) to semi-abelian categories. This is indeed the most general environment in which we are able to talk about actions, due to the concept of internal actions. In this general context, we give a diagrammatic definition of the compatibility conditions for internal actions, which specialises to the particular definitions known for groups and Lie algebras. We then give a new construction of the Peiffer product in this setting and we use these tools to show that in any semi-abelian category satisfying the "Smith-is-Huq" condition, asking that two actions are compatible is the same as requiring that these actions are induced from a pair of internal crossed modules over a common base object. Thanks to this equivalence, in order to deal with the generalisation to the semi-abelian context of the non-abelian tensor product, we are able to use a pair of internal crossed modules over a common base object instead of a pair of compatible internal actions, whose formalism is far more intricate. Now we fix a semi-abelian category A satisfying "Smith-is-Huq" and we show that, for each pair of internal L-crossed modules, it is possible to construct an internal crossed square which is the pushout (in the category of crossed squares) of the general version of the diagram used by Brown and Loday in the groups case. The non-abelian tensor product is then defined as a piece of this internal crossed square. We show that if A is the category of groups or the category of Lie algebras, this general construction coincides with the specific notions of non-abelian tensor products already known in these settings. We constr
Tipologia IRIS:
Tesi di dottorato
Keywords:
semi-abelian categories; commutator theory; crossed modules; crossed squares; non-abelian tensor product; groupoids; double groupoids
Elenco autori:
D. DI MICCO
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