Connected gradings and fundamental group

📝 Abstract

The main purpose of this paper is to provide explicit computations of the fundamental group of several algebras. For this purpose, given a $k $-algebra $A $, we consider the category of all connected gradings of $A$ by a group $G$ and we study the relation between gradings and Galois coverings. This theoretical tool gives information about the fundamental group of $A $, which allows its computation using complete lists of gradings.

💡 Analysis

The main purpose of this paper is to provide explicit computations of the fundamental group of several algebras. For this purpose, given a $k $-algebra $A $, we consider the category of all connected gradings of $A$ by a group $G$ and we study the relation between gradings and Galois coverings. This theoretical tool gives information about the fundamental group of $A $, which allows its computation using complete lists of gradings.

📄 Content

arXiv:0906.3069v4 [math.RA] 31 May 2010 Connected gradings and fundamental group Claude Cibils, Mar´ıa Julia Redondo and Andrea Solotar ∗ Abstract The main purpose of this paper is to provide explicit computations of the fundamental group of several algebras. For this purpose, given a k-algebra A, we consider the category of all connected gradings of A by a group G and we study the relation between gradings and Galois coverings. This theoretical tool gives information about the fundamental group of A, which allows its computation using complete lists of gradings. 2000 Mathematics Subject Classification : 16W50, 16S50. Keywords : grading, Galois covering, fundamental group. 1 Introduction The main goal of this article is to provide explicit computations of the intrinsic fun- damental group of some algebras. For this, we study in detail the relation between gradings and Galois coverings of the algebra considered as a k-linear category with one object. Particular attention is paid to matrix algebras, since the problem of clas- sifying gradings of these algebras has been extensively treated in the literature (see [1, 2, 4, 6, 7, 8, 9, 11, 14, 17, 21], and also [3]). We recall that the intrinsic fundamental group of an algebra has been defined in [13] using Galois coverings. We make use of an equivalence between the category of Galois coverings and its full subcategory with objects obtained from the smash product ∗This work has been supported by the projects UBACYTX212, PIP-CONICET 5099, and CONICET-CNRS. The second and third authors are research members of CONICET (Argentina). The third author is a Regular Associate of ICTP Associate Scheme. 1 construction, which is deeply attached to connected gradings. We replace algebras by linear categories over a base ring: a category over a ring k is considered as an algebra with several objects, see [23], and a k-algebra A can be viewed as a k-category with a single object and endomorphism ring equal to A. Note that in [19, 20] a relation between gradings and coverings is established for quivers with relations. In this paper we consider an intrinsic context, i.e., the categories are not given by a presentation. When performing the computation of the fundamental group of an algebra, one faces the problem of classifying and organizing their connected gradings. The meth- ods we introduce allow the computation of the fundamental groups of matrix algebras, triangular matrix algebras, group algebras and diagonal algebras. We restrict to con- nected gradings and we prove that the matrix algebras do not admit a universal grading. Indeed, there exist at least two non-isomorphic Galois coverings or, equivalently, two non-isomorphic connected gradings which are simply connected, in the sense that they have no nontrivial Galois coverings. In particular this provides a confirmation of the fact that the fundamental group of an algebra takes into account the matrix structure, in other words it is not a Morita invariant. In Section 2 we show that the connectedness of gradings is the right notion which corresponds to the connectedness of the smash product associated. We recall the concept of Galois covering and we observe that the smash product construction gives examples of Galois coverings. We describe in detail the morphisms between smash coverings. In Section 3 we make an explicit comparison between Galois coverings and smash coverings of a k-category B. More precisely, we provide an equivalence between the category Gal(B, b0) of Galois coverings of B and its full subcategory Gal#(B, b0), whose objects are the smash product coverings. We consider the fundamental group that has been defined in [13] using Galois coverings and show that we can restrict to smash coverings when computing the fundamental group π1(B, b0). In the following sections we focus on the description of connected gradings of certain algebras in order to compute their fundamental group. As a rule, we wonder about the existence of a universal grading, since when such a grading exists the grading group is isomorphic to the fundamental group of the algebra. In Section 4 we consider matrix algebras: we prove that there is no universal covering by providing two non-isomorphic simply connected gradings. Despite the fact that they appear to be very different in nature, we show that they have a unique largest common nontrivial quotient. Using the classification of gradings of M2(k) given by C. Boboc, S. D˘asc˘alescu and R. Khazal [21] and of M3(k) given by C. Boboc, S. D˘asc˘alescu [9], we compute the fundamental group of these algebras in case the field is algebraically closed of characteristic different from 2 and 3, respectively. Using 2 analogous methods and the classification of Yu. A. Bahturin and M.V. Zaicev [4], we compute the fundamental group of Mp(k), where p is prime and k an algebraically closed field of characteristic zero, which is the direct product of the free group on p−1 generators with the cyclic group of order p

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