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Maria Julia Redondo

Publications and source records attributed to Maria Julia Redondo.

At least 19 recordsLinked to original sources
arXiv2018

The Gerstenhaber structure on the Hochschild cohomology of a class of special biserial algebras

We determine the Gerstenhaber structure on the Hochschild cohomology ring of a class of self-injective special biserial algebras. Each of these algebras is presented as a quotient of the path algebra of a certain quiver. In degree one, we show that the cohomology is isomorphic, as a Lie algebra, to a direct sum of copies of a subquotient of the Virasoro algebra. These copies share Virasoro degree 0 and commute otherwise. Finally, we describe the cohomology in degree $n$ as a module over this Lie algebra by providing its decomposition as a direct sum of indecomposable modules.

math.RA↗
arXiv2017

Comparison morphisms between two projective resolutions of monomial algebras

We construct comparison morphisms between two well-known projective resolutions of a monomial algebra $A$: the bar resolution and Bardzell's resolution; the first one is used to define the cup product and the Lie bracket on the Hochschild cohomology $HH^*(A)$ and the second one has been shown to be an efficient tool for computations of these cohomology groups. The constructed comparison morphisms allow us to show that the cup product restricted to even degrees of the Hochschild cohomology has a very simple description. Moreover, for $A=k Q/I$ a monomial algebra such that $dim_k \ e_i A e_j = 1$ whenever there exists an arrow $α: i \to j \in Q_1$, we describe the Lie action of the Lie algebra $HH^1(A)$ on $HH^{\ast}(A)$.

math.RA↗
arXiv2017

Cohomology of partial smash products

We define the partial group cohomology as the right derived functor of the functor of partial invariants, we relate this cohomology with partial derivations and with the partial augmentation ideal and we show that there exists a Grothendieck spectral sequence relating cohomology of partial smash algebras with partial group cohomology and algebra cohomology.

math.RA↗
arXiv2017

Gerstenhaber algebra structure on the Hochschild cohomology of quadratic string algebras

We describe the Gerstenhaber algebra structure on the Hochschild cohomology HH*$(A)$ when $A$ is a quadratic string algebra. First we compute the Hochschild cohomology groups using Barzdell's resolution and we describe generators of these groups. Then we construct comparison morphisms between the bar resolution and Bardzell's resolution in order to get formulae for the cup product and the Lie bracket. We find conditions on the bound quiver associated to string algebras in order to get non-trivial structures.

math.RA↗
arXiv2013

On universal gradings, versal gradings and Schurian generated categories

Categories over a field $k$ can be graded by different groups in a connected way; we consider morphisms between these gradings in order to define the fundamental grading group. We prove that this group is isomorphic to the fundamental group à la Grothendieck as considered in previous papers. In case the $k$-category is Schurian generated we prove that a universal grading exists. Examples of non Schurian generated categories with universal grading, versal grading or none of them are considered.

math.CT↗
arXiv2011

Full and convex linear subcategories are incompressible

Consider the intrinsic fundamental group à la Grothendieck of a linear category using connected gradings. In this article we prove that any full convex subcategory is incompressible, in the sense that the group map between the corresponding fundamental groups is injective. We start by proving the functoriality of the intrinsic fundamental group with respect to full subcategories, based on the study of the restriction of connected gradings.

math.RA↗
arXiv2011

Fundamental group of Schurian categories and the Hurewicz isomorphism

Let k be a field. We attach a CW-complex to any Schurian k-category and we prove that the fundamental group of this CW-complex is isomorphic to the intrinsic fundamental group of the k-category. This extends previous results by J.C. Bustamante. We also prove that the Hurewicz morphism from the vector space of abelian characters of the fundamental group to the first Hochschild-Mitchell cohomology vector space of the category is an isomorphism.

math.RA↗
arXiv2010

The Intrinsic Fundamental Group of a Linear Category

We provide an intrinsic definition of the fundamental group of a linear category over a ring as the automorphism group of the fibre functor on Galois coverings. If the universal covering exists, we prove that this group is isomorphic to the Galois group of the universal covering. The grading deduced from a Galois covering enables us to describe the canonical monomorphism from its automorphism group to the first Hochschild-Mitchell cohomology vector space.

math.RA↗
arXiv2010

Connected gradings and fundamental group

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.

math.RA↗
arXiv2009

Covering functors without groups

Coverings in the representation theory of algebras were introduced for the Auslander-Reiten quiver of a representation finite algebra by Riedtmann and later for finite dimensional algebras by Bongartz and Gabriel, R. Martinez-Villa and de la Peña. The best understood class covering functors is that of Galois covering functors F: A -> B determined by the action of a group of automorphisms of A. In this work we introduce the balanced covering functors which include the Galois class and for which classical Galois covering-type results still hold. For instance, if F:A -> B is a balanced covering functor, where A and B are linear categories over an algebraically closed field, and B is tame, then A is tame.

math.RT↗
arXiv2009

Combinatorial classification of piecewise hereditary algebras

We use the characteristic polynomial of the Coxeter matrix of an algebra to complete the combinatorial classification of piecewise hereditary algebras which Happel gave in terms of the trace of the Coxeter matrix. We also give a cohomological interpretation of the coefficients (other than the trace) of the characteristic polynomial of the Coxeter matrix of any finite dimensional algebra with finite global dimension.

math.RT↗
arXiv2006

Universal coefficient theorem in triangulated categories

Let T be a triangulated category, A a graded abelian category and h: T -> A a homology theory on T with values in A. If the functor h reflects isomorphisms, is full and is such that for any object x in A there is an object X in T with an isomorphism between h(X) and x, we prove that A is a hereditary abelian category and the ideal Ker(h) is a square zero ideal which as a bifunctor on T is isomorphic to Ext^1_A(h(-)[1], h(-)).

math.CT↗
arXiv2005

Cohomology of the Grothendieck construction

We consider cohomology of small categories with coefficients in a natural system in the sense of Baues and Wirsching. For any funtor L: K -> CAT, we construct a spectral sequence abutting to the cohomology of the Grothendieck construction of L in terms of the cohomology of K and of L(k), for k an object in K.

math.CT↗