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Andrea Solotar

Publications and source records attributed to Andrea Solotar.

52 records · Page 3Linked to original sources
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

Hochschild and cyclic homology of Yang-Mills algebras

The aim of this article is to compute the Hochschild and cyclic homology groups of Yang-Mills algebras, that have been defined by A. Connes and M. Dubois-Violette. We proceed here the study of these algebras that we have initiated in a previous article. The computation involves the use of a spectral sequence associated to the natural filtration on the enveloping algebra of the Lie Yang-Mills algebra. This filtration in provided by a Lie ideal which is free as Lie algebra.

math.KT↗
arXiv2010

(a,b)-Koszul algebras

Let $a$ and $b$ be two integers such that $2\le a<b$. In this article we define the notion of $(a,b)$-Koszul algebra as a generalization of $N$-Koszul algebras. We also exhibit examples and we provide a minimal graded projective resolution of the algebra $A$ considered as an $A$-bimodule, which allows us to compute the Hochschild homology groups for some examples of $(a,b)$-Koszul algebras.

math.KT↗
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↗
arXiv2008

Representation theory of Yang-Mills algebras

The aim of this article is to describe families or representations of the Yang-Mills algebras YM(n) (where n>1) defined by Connes and Dubois-Violette. We first describe irreducible finite dimensional representations. Next, we provide families of infinite dimensional representations of YM(n), big enough to separate points of the algebra. In order to prove this result, we use that all Weyl algebras Ar(k) are epimorphic images of YM(n).

math.RT↗
arXiv2008

On Morita equivalence for simple Generalized Weyl algebras

We give a necessary condition for Morita equivalence of simple Generalized Weyl algebras of classical type. We propose a reformulation of Hodges' result, which describes Morita equivalences in case the polynomial defining the Generalized Weyl algebra has degree 2, in terms of isomorphisms of quantum tori, inspired by similar considerations in noncommutative differential geometry. We study how far this link can be generalized for $n\ge 3$.

math.KT↗
arXiv2006

N-complexes as functors, amplitude cohomology and fusion rules

We consider N-complexes as functors over an appropriate linear category in order to show first that the Krull-Schmidt Theorem holds, then to prove that amplitude cohomology only vanishes on injective functors providing a well defined functor on the stable category. For left truncated N-complexes, we show that amplitude cohomology discriminates the isomorphism class up to a projective functor summand. Moreover amplitude cohomology of positive N-complexes is proved to be isomorphic to an Ext functor of an indecomposable N-complex inside the abelian functor category. Finally we show that for the monoidal structure of N-complexes a Clebsch-Gordan formula holds, in other words the fusion rules for N-complexes can be determined.

math.QA↗
arXiv2006

Derived invariance of Hochschild-Mitchell (co)homology and one-point extensions

In this article we prove derived invariance of Hochschild-Mitchell homology and cohomology and we extend to $k$-linear categories a result by Barot and Lenzing concerning derived equivalences and one-point extensions. We also prove the existence of a Happel long exact sequence and we give a generalization of this result which provides an alternative approach.

math.KT↗
arXiv2005

Hochschild-Mitchell cohomology and Galois extensions

We define $H$-Galois extensions for $k$-linear categories and a Hopf algebra $H$ and prove the existence of a Grothendieck spectral sequence for Hochschild-Mitchell cohomology, related to this situation. This spectral sequence is multiplicative and for a $H$ a group algebra decomposes as a direct sum indexed by the set of conjugacy classes of the group. We also compute some Hochschild-Mitchell cohomology groups of categories with infinite associated quiver.

math.KT↗
arXiv2005

Galois coverings, Morita equivalence and smash extensions of categories over a field

We consider categories over a field $k$ in order to prove that smash extensions and Galois coverings with respect to a finite group coincide up to Morita equivalence of $k$-categories. For this purpose we describe processes providing Morita equivalences called contraction and expansion. We prove that composition of these processes provides any Morita equivalence, a result which is related with the karoubianisation (or idempotent completion) and additivisation of a $k$-category.

math.RA↗
arXiv2002

Cohomology of split algebras and of trivial extensions

We consider associative algebras L over a field provided with a direct sum decomposition of a two-sided ideal M and a sub-algebra A - examples are provided by trivial extensions or triangular type matrix algebras. In this relative and split setting we describe a long exact sequence computing the Hochschild cohomology of L. We study the connecting homomorphism using the cup-product and we infer several results, in particular the first Hochschild cohomology group of a trivial extension never vanishes.

math.KT↗