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Category of G-Groups and Its Spectral Category

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Category of G-Groups and Its Spectral Category Communications in Algebra ISSN: 0092-7872 (Print) 1532-4125 (Online) Journal homepage: http://www.tandfonline.com/loi/lagb20 Category of G-Groups and its Spectral Category María José Arroyo Paniagua & Alberto Facchini To cite this article: María José Arroyo Paniagua & Alberto Facchini (2017) Category of G-Groups and its Spectral Category, Communications in Algebra, 45:4, 1696-1710, DOI: 10.1080/00927872.2016.1222409 To link to this article: http://dx.doi.org/10.1080/00927872.2016.1222409 Accepted author version posted online: 07 Oct 2016. Published online: 07 Oct 2016. Submit your article to this journal Article views: 12 View related articles View Crossmark data Full Terms & Conditions of access and use can be found at http://www.tandfonline.com/action/journalInformation?journalCode=lagb20 Download by: [UNAM Ciudad Universitaria] Date: 29 November 2016, At: 17:29 COMMUNICATIONS IN ALGEBRA® 2017, VOL. 45, NO. 4, 1696–1710 http://dx.doi.org/10.1080/00927872.2016.1222409 Category of G-Groups and its Spectral Category María José Arroyo Paniaguaa and Alberto Facchinib aDepartamento de Matemáticas, División de Ciencias Básicas e Ingeniería, Universidad Autónoma Metropolitana, Unidad Iztapalapa, Mexico, D. F., México; bDipartimento di Matematica, Università di Padova, Padova, Italy ABSTRACT ARTICLE HISTORY Let G be a group. We analyse some aspects of the category G-Grp of G-groups. Received 15 April 2016 In particular, we show that a construction similar to the construction of the Revised 22 July 2016 spectral category, due to Gabriel and Oberst, and its dual, due to the second Communicated by T. Albu. author, is possible for the category G-Grp. KEYWORDS Group actions; Semi-abelian categories; Spectral categories 2000 MATHEMATICS SUBJECT CLASSIFICATION 20J15 1. Introduction The original motivation of this paper was to give an adequate categorical setting to the techniques employed in [4]. In that paper, Andrea Lucchini and the second author obtained results concerning the direct product decompositions of a group G as a direct product of indecomposable normal subgroups employing techniques essentially based on the notion of normal homomorphism [12, p. 81]. Here, if N and M are normal subgroups of G, a homomorphism f : N → G/M is normal if it commutes with all inner automorphisms of G. In particular, a construction reminiscent of the construction of the spectral category [5] was applied to the set of all normal endomorphisms of the group G. In order to place these techniques into a proper setting, we study groups, where now is the group G, and G acts on its normal subgroups N and its quotient groups G/M via inner automorphisms. To this end, we consider the category G-Grp of all pairs (H, ϕ), where H is any group and ϕ : G → Aut(H) is a group homomorphism. In other words, G-Grp is the category whose objects are the socalled Ggroups. The notion of Ggroup is a classical one, and sometimes G is called the operator group acting on H ([1, Denition 5.5] and [13, Denition 8.1]). In this context, normal subgroups N of G and quotient groups G/M become objects of G-Grp, and normal homomorphisms become morphisms in G-Grp. Thus we focus on the study of the category G-Grp, which is a semiabelian category in the sense of [9]; cf. [1] and [6]. We determine the free G-groups and show that the injective objects in the category G-Grp are only the trivial ones, like in the case of the category Grp of groups. There are two more categories in strict relation with the category G-Grp. One is the category of all Gsemidirect products (i.e., the category of pointed objects over the group G [1, Section 5]) and the other is the category of all le kGmodule Hopf algebras. We adapt the GabrielOberst construction of spectral category and its ′ dual to the category G-Grp, getting two categories Spec(G-Grp) and CG. This setting works particularly well on the full subcategory CG of G-Grp consisting of all objects (H, ϕ) of G-Grp for which the image of the group homomorphism ϕ : G → Aut(H) contains the group Inn(H) of all inner automorphisms C C′ of H. Furthermore, there is a local canonical functor of G to the product category Spec(G-Grp) × G (Theorem 6.7). Also, we obtain that the endomorphism monoid of an object H of the category Spec(G- Grp) is a regular monoid whenever H is a Ggroup in CG. CONTACT Alberto Facchini [email protected] Dipartimento di Matematica, Università di Padova, Via Trieste 63, 35121 Padova, Italy. © 2017 Taylor & Francis COMMUNICATIONS IN ALGEBRA® 1697 The reader will notice the analogy between the category G-Grp of Ggroups and the category RMod of all le modules over a xed ring R. The category RMod has as objects the pairs (M, λ), where M is an additive abelian group and λ: R → End(M) is a ring morphism. In the category G-Grp the morphisms f : (H, ϕ) → (H′, ϕ′) are the group homomorphisms f : H → H′ such that f (gh) = gf (h) for every g ∈ G, h ∈ H, exactly as in the category RMod. We write mappings on the le (we stress this, because in group theory mappings are oen written on the right, sometimes as exponents. Moreover, writing mappings on the le has as a consequence that le actions correspond to homomorphisms, and right actions correspond to antihomomorphisms. We will come back to this point in Remark 2.3). Let G be a group. As we have already said above, a (le) Ggroup is a pair (H, ϕ), where H is a group and ϕ : G → Aut(H) is a group homomorphism. Since we write mappings on the le, a Ggroup can be equivalently dened as a group H endowed with a mapping ·: G × H → H, (g, h) → gh, called le scalar multiplication, such that (a) g(hh′) = (gh)(gh′) (b) (gg′)h = g(g′h) (c) 1Gh = h for every g, g′ ∈ G and every h, h′ ∈ H. As far as terminology is concerned, it is convenient to be very precise. The notion of Ggroup is a particular case of the notion of group as treated in [12, p. 28] (for any set , an group is a pair (H, ϕ), where H is a group and ϕ : → End(H) is a mapping). At the same time, this notion of - group is a particular case of the more general notion of a group with multiple operators studied in [7]. Also, recall that a (le) Gset is a pair (X, σ), where X is a set and σ : G → SX is a group homomorphism into the symmetric group SX on X. Equivalently, a Gset is a set X with a mapping ·: G × X → X, ′ ′ ′ (g, x) → gx, such that (gg )x = g(g x) and 1Gx = x for every g, g ∈ G and x ∈ X. 2. G-groups and their morphisms The rst part of this section is devoted to establishing the elementary terminology about Ggroups. A Ggroup morphism f : (H, ϕ) → (H′, ϕ′) is a group homomorphism f : H → H′ such that f (gh) = gf (h) ′ ′ for every g ∈ G, h ∈ H. We will denote by HomG(H, H ) the set of all Ggroup morphisms of H into H . Thus Ggroups form a category, which we will denote by G-Grp. Notice that the category Grp of groups and the category 1Grp, where 1 is the trivial group (with one element), are isomorphic categories. This is the analog of the fact that the category Ab of abelian groups and the category Z−Mod of modules over the ring Z of integers are isomorphic categories, because 1 and Z are the initial objects in the category of groups and the category of rings, respectively. It is clearly possible to generalize our study of the category G-Grp to a more general setting, though this will not be our primary aim in this paper. The generalization is the following. Let C be a category and M a monoid. We can obviously dene the category CM of all Mobjects in C as the category of all pairs (C, ϕ), where C is an object in C and ϕ : M → End(C) is a monoid homomorphism. If we view M as a category with one object, CM is the category of all functors M → C. When M is a group, every homomorphism M → End(C) factors uniquely though Aut(C), and so we can write ϕ : M → Aut(C) instead of ϕ : M → End(C). When C is the category Grp of groups and G is a group, CG turns out to be the category G-Grp of Ggroups. Now for any category C and any monoid M, there is a forgetful functor U : CM → C, which associates to every object (C, ϕ) of CM (the category of Mobjects in C) the object C of C. Suppose now that C has Mindexed coproducts. This is the case, for instance, for M = G and C = Grp, where, for any family of groups Hg indexed by the elements g of G, the Mindexed coproduct is the free product of the groups Hg. The forgetful functor U : CM → C has a le adjoint F : C → CM. Here F is dened on objects by F(C) = (M ·C, ϕ), where M ·C is the Mindexed coproduct of |M| copies of C, and ϕ : M → End(M ·C) 1698 M. J. A. PANIAGUA AND A. FACCHINI is constructed as follows. For every m ∈ M, let ιm : C → M·C be the coproduct injection.
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