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.
We construct comparison morphisms between two well-known projective resolutions of a monomial algebra $A$: the bar resolution $\operatorname{\mathbb{Bar}} A$ and Bardzell's resolution ...$\operatorname{\mathbb{Ap}} A$; the first one is used to define the cup product and the Lie bracket on the Hochschild cohomology $\operatorname{HH} ^*(A)$ and the second one has been shown to be an efficient tool for computation 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 $A= \mathbb{k} Q/I$ a monomial algebra such that $\dim_ \mathbb{k} e_i A e_j = 1$ whenever there exists an arrow $\alpha: i \to j \in Q_1$, we describe the Lie action of the Lie algebra $\operatorname{HH}^1(A)$ on $\operatorname{HH}^{\ast} (A)$.
The Ext-algebra for infinitesimal deformations Redondo, María Julia; Román, Lucrecia; Rossi Bertone, Fiorela
Journal of pure and applied algebra,
October 2024, 2024-10-00, Letnik:
228, Številka:
10
Journal Article
Recenzirano
Let f be a Hochschild 2-cocycle and let Af be an infinitesimal deformation of an associative finite dimensional algebra A over an algebraically closed field k. We investigate the algebra structure of ...the Ext-algebra of Af and, under some conditions on f, we describe it in terms of the Ext-algebra of A. We achieve this description by getting an explicit construction of minimal projective resolutions in modAf.
>à Grothendieck of a linear category, introduced in our earlier papers 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.>
Morita Invariance for Infinitesimal Deformations Redondo, María Julia; Román, Lucrecia; Rossi Bertone, Fiorela ...
Algebras and representation theory,
08/2022, Letnik:
25, Številka:
4
Journal Article
Recenzirano
Let
A
and
B
be two Morita equivalent finite dimensional associative algebras over a field 𝕜. It is well known that Hochschild cohomology is invariant under Morita equivalence. Since infinitesimal ...deformations are connected with the second Hochschild cohomology group, we explicitly describe the transfer map connecting
H
H
2
(
A
) with
H
H
2
(
B
). This allows us to transfer Morita equivalence between
A
and
B
to that between infinitesimal deformations of them. As an application, when 𝕜 is algebraically closed, we consider the quotient path algebra associated to
A
and describe the presentation by quiver and relations of the infinitesimal deformations of
A
.
Cohomology of partial smash products Ribeiro Alvares, Edson; Muniz Alves, Marcelo; Redondo, María Julia
Journal of algebra,
07/2017, Letnik:
482
Journal Article
Recenzirano
Odprti dostop
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 products with partial group cohomology and algebra cohomology.
Given a cluster-tilted algebra
B
, we study its first Hochschild cohomology group HH
1
(
B
) with coefficients in the
B
-
B
-bimodule
B
. If
C
is a tilted algebra such that
B
is the ...relation-extension of
C
, then we show that if
B
is tame, then HH
1
(
B
) is isomorphic, as a
k
-vector space, to the direct sum of
HH
1
(
C
)
with
k
n
B
,
C
, where
n
B
,
C
is an invariant linking the bound quivers of
B
and
C
. In the representation-finite case, HH
1
(
B
) can be read off simply by looking at the quiver of
B
.
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.
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.
Categories over a field $k$ can be graded by di erent 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.
PDF
