The Gerstenhaber Bracket and Cycles in
the Module Category of a Monomial Quadratic Algebra (2019) We establish a link between the Gerstenhaber
bracket in the Hochshild cohomology and the behaviour of cycles in
the module category of a monomial quadratic algebra A.
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Published articles in international journals with referee

On the Morita Reduced Versions of
Skew-Group Algebras of Path Algebras (2018) The Quarterly Journal of Mathematics (to appear) Let R be the skew group algebra of a finite
group acting on the path algebra of a quiver. This article develops
both theoretical and practical methods to do computations in the
Morita reduced algebra associated to R. Reiten and Riedtmann proved
that there exists an idempotent e of R such that the algebra eRe is
both Morita equivalent to R and isomorphic to the path algebra of
some quiver which was described by Demonet. This article gives
explicit formulas for the decomposition of any element of eRe as a
linear combination of paths in the quiver described by Demonet. This
is done by expressing appropriate compositions and pairings in a
suitable monoidal category which takes into account the
representation theory of the finite group. pdfarXivHAL

Crossed-Products of Calabi-Yau Algebra
by Finite Groups Journal of Pure and Applied Algebra 224 (10) (2020) 106394 Let a finite group G act on a differential
graded algebra A. This article presents necessary conditions and
sufficient conditions for the skew group algebra A*G to be
Calabi-Yau. In particular, when A is the Ginzburg dg algebra of a
quiver with an invariant potential, then A*G is Calabi-Yau and
Morita equivalent to a Ginzburg dg algebra. Some applications of
these results are derived to compare the generalised cluster
categories of A and A*G when they are defined and to compare the
higher Auslander-Reiten theories of A and A*G when A is a finite
dimensional algebra. pdfarXivHALdoi

(m,n)-Quasitilted and (m,n)-Almost
Hereditary Algebras (2017)
with Edson Ribeiro Alvares, Diane Castonguay and Tanise Carnieri
Pierin Colloquium Mathematicum (to appear) Motivated by the study of (m,n)-quasitilted
algebras, which are the piecewise hereditary algebras obtained from
quasitilted algebras of global dimension two by a sequence of
(co)tiltings involving n-1 tilting modules and m-1 cotilting
modules, we introduce (m,n)-almost hereditary algebras. These are
the algebras with global dimension m+n and such that any
indecomposable module has projective dimension at most m, or else
injective dimension at most n. We relate these two classes of
algebras, among which (m,1)-almost hereditary ones play a special
role. For these, we prove that any indecomposable module lies in the
right part of the module category, or else in an m-analog of the
left part. This is based on the more general study of algebras the
module categories of which admit a torsion-free subcategory such
that any indecomposable module lies in that subcategory, or else has
injective dimension at most n. pdfarXiv

Smash-Products of Calabi-Yau Algebras
by Hopf Algebras Journal of Noncommutative Geometry 13 (3) (2019) 887-961 Let H be a Hopf algebra and A be
an H-module algebra. This article studies the smash product A#H in
terms of Calabi-Yau duality. When the antipode of H is invertible
and A and H are homologically smooth, it is proved that A#H has Van
den Bergh duality if and only if so have A and H. Under the same
condition, it is proved that if A and H are skew-Calabi-Yau, then so
is A#H; A description of a Nakayama automorphism of A#H is given as
well as a partial converse to the implication. This leads to
necessary and/or sufficient conditions for A#H to be Calabi-Yau. As
a byproduct, it is proved that for connected graded Artin-Schelter
regular algebras, graded Nakayama automorphisms have trivial
homological determinant. The results are illustrated on the smash
products of polynomial algebras (or of the quantum plane) by
enveloping algebras of finite dimensional Lie algebras (or, by the
quantum enveloping algebra of sl(2), respectively). The results in
this article are based on the study of the inverse dualising complex
of A#H in the more general situation where A is a dg algebra. In
that setting, the article studies the interaction between taking the
smash product with H of a dg algebra and taking the (deformed)
Calabi-Yau completion of a dg algebra. The final part of the article
discusses necessary and sufficient conditions for A#H to be
Artin-Schelter regular or Gorenstein. pdfarXivHALdoi

Degrees of Irreducible Morphisms over
Perfect Fields
with Claudia Chaio and Sonia Trepode Algebras and Representation Theory 22 (2) (2019) 495-515 The module category of any artin algebra is
filtered by the powers of its radical, thus defining an associated
graded category. As an extension of the degree of irreducible
morphisms, this text introduces the degree of morphisms in the
module category in terms of the induced natural transformations
between representable functors on this graded category. When the
ground ring is a perfect field, and the given morphism behaves
nicely with respect to covering theory (as do irreducible morphisms
with indecomposable domain or indecomposable codomain), it is shown
that the degree of the morphism is finite if and only if its
associated natural transformation has a representable kernel. As a
corollary, generalisations of known results on the degrees of
irreducible morphisms over perfect fields are given. Finally, this
study is applied to the composition of paths of irreducible
morphisms in relationship to the powers of the radical.
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Covering Techniques for
Auslander-Reiten Theory
with Claudia Chaio and Sonia Trepode Journal of Pure and Applied Algebra 223 (2) (2019) 641-659 Given a finite dimensional algebra over a
perfect field the text introduces covering functors over the mesh
category of any modulated Auslander-Reiten component of the
algebra. This is applied to study the composition of irreducible
morphisms between indecomposable modules in relation with the powers
of the radical of the module category. pdfarXivHALdoi

Representation Theory of Partial
Relation Extensions
with Ibrahim Assem, Juan Carlos Bustamante, Julie Dionne and David
Smith Colloquium Mathematicum 155 (2) (2019) 157-186 Let C be a finite dimensional algebra of
global dimension at most two. A partial relation extension is any
trivial extension of C by a direct summand of its relation
C-C-bimodule. When C is a tilted algebra, this construction provides
an intermediate class of algebras between tilted and cluster tilted
algebras. The text investigates the representation theory of partial
relation extensions. When C is tilted, any complete slice in the
Auslander-Reiten quiver of C embeds as a local slice in the
Auslander-Reiten quiver of the partial relation extension; Moreover,
a systematic way of producing partial relation extensions is
introduced by considering direct sum decompositions of the potential
arising from a minimal system of relations of C. pdfarXivHALdoi

Duality for Differential Operators of
Lie-Rinehart Algebras
with Thierry Lambre Pacific Journal of Mathematics 297 (2) (2018) 405-454 Let (S,L) be a Lie-Rinehart algebra over a
commutative ring R. This article proves that, if S is flat as an
R-module and has Van den Bergh duality in dimension n, and if L is
finitely generated and projective with constant rank d as an
S-module, then the enveloping algebra of (S,L) has Van den Bergh
duality in dimension n+d. When, moreover, S is Calabi-Yau and the
d-th exterior power of L is free over S, the article proves that the
enveloping algebra is skew-Calabi-Yau, and it describes a Nakayama
automorphism of it. These considerations are specialised to Poisson
enveloping algebras. They are also illustrated on Poisson structures
over two and three dimensional polynomial algebras and on
Nambu-Poisson structures on certain two dimensional hypersurfaces.
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The strong global dimension of
piecewise hereditary algebras
with Edson Ribeiro Alvares et Eduardo N. Marcos Journal of Algebra 481 (2017) 36-67 Let T be a tilting object in a triangulated
category equivalent to the bounded derived category of a hereditary
abelian category with finite dimensional homomorphism spaces and
split idempotents. This text investigates the strong global
dimension, in the sense of Ringel, of the endomorphism algebra of
T. This invariant is expressed using the infimum of the lengths of
the sequences of tilting objects successively related by tilting
mutations and where the last term is T and the endomorphism algebra
of the first term is quasi-tilted. It is also expressed in terms of
the hereditary abelian generating subcategories of the triangulated
category. pdfarXivHALdoi

Special biserial algebras with no outer
derivations
with Ibrahim Assem et Juan Carlos Bustamante Colloquium Mathematicum 125 (1) (2011), 83-98 Let A be a special biserial algebra over an
algebraically closed field. We show that the first Hohchshild
cohomology group of A with coefficients in the bimodule A vanishes
if and only if A is representation finite and simply connected (in
the sense of Bongartz and Gabriel), if and only if the Euler
characteristic of Q equals the number of indecomposable non
uniserial projective injective A-modules (up to
isomorphism). Moreover, if this is the case, then all the higher
Hochschild cohomology groups of A vanish. pdfarXivHALdoi

Degrees of irreducible morphisms and
finite representation type
with Claudia Chaio et Sonia Trepode Journal of the London Mathematical Society sér. 2 84 (1) (2011),
35-57 We study the degree of irreducible morphisms
in any Auslander-Reiten component of a finite dimensional algebra
over an algebraically closed field. We give a characterization for
an irreducible morphism to have finite left (or right) degree. This
is used to prove our main theorem: An algebra is of finite
representation type if and only if for every indecomposable
projective the inclusion of the radical in the projective has finite
right degree, which is equivalent to require that for every
indecomposable injective the epimorphism from the injective to its
quotient by its socle has finite left degree. We also apply the
techniques that we develop: We study when the non-zero composite of
a path of n irreducible morphisms between indecomposable modules
lies in the n+1-th power of the radical; and we study the same
problem for sums of such paths when they are sectional, thus proving
a generalisation of a pioneer result of Igusa and Todorov on the
composite of a sectional path. pdfarXivHALdoi

Topological invariants of piecewise
hereditary algebras Transactions of the American Mathematical Society 363 (4) (2011),
2143-2170 We investigate the Galois coverings of
piecewise algebras and more particularly their behaviour under
derived equivalences. Under a technical assumption which is
satisfied if the algebra is derived equivalent to a hereditary
algebra, we prove that there exists a universal Galois covering
whose group of automorphisms is free and depends only on the derived
category of the algebra. As a corollary, we prove that the algebra
is simply connected if and only if its first Hochschild cohomology
vanishes. pdfarXivHALdoi

Coverings of Laura algebras: the
standard case
with Ibrahim Assem and Juan Carlos Bustamante Journal of Algebra 323 (1) (2010), 83-120 In this paper, we study the covering theory of
laura algebras. We prove that if a connected laura algebra is
standard (that is, it is not quasi-tilted of canonical type and its
connecting components are standard), then this algebra has nice
Galois coverings associated to the coverings of the connecting
component. As a consequence, we show that the first Hochschild
cohomology group of a standard laura algebra vanishes if and only if
it has no proper Galois coverings. pdfarXivHALdoi

Galois coverings of weakly shod
algebras Communications in Algebra 38 (4) (2010), 1291-1318 We investigate the Galois coverings of weakly
shod algebras. For a weakly shod algebra not quasi-tilted of
canonical type, we establish a correspondence between its Galois
coverings and the Galois coverings of its connecting component. As a
consequence, we show that a weakly shod algebra is simply connected
if and only if its first Hochschild cohomology group vanishes.
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On maximal diagonalizable Lie
subalgebras of the first Hochschild cohomology Communications in Algebra 38 (4) (2010), 1325-1340 Let A be a basic connected finite dimensional
algebra over an algebraically closed field k and with ordinary
quiver Q without oriented cycle. To any presentation of A by quiver
and admissible relations, Martinez-Villa and de La Pena have
associated the fundamental group of the presentation. Assem and de
La Pena have constructed an injective mapping from the additive
characters of this fundamental group (with values in the ground
field) to the first Hochschild cohomology group HH^{1}(A). We study
the image of these mappings associated to the different
presentations of A in terms of diagonalizable Lie subalgebras of
HH^{1}(A). Then we characterise the maximal diagonalisable subalgebras
of HH^{1}(A) when A is monomial and Q has no multiple arrows and also
when car(k)=0 and Q has no double bypass. pdfarXivHALdoi

On Galois coverings and tilting
modules Journal of Algebra 319 (12) (2008), 4961-4999 Let A be a basic connected finite dimensional
algebra over an algebraically closed field, let G be a group, let T
be a basic tilting A-module and let B the endomorphism algebra of
T. Under a hypothesis on T, we establish a correspondence between
the Galois coverings with group G of A and the Galois coverings with
group G of B. The hypothesis on T is expressed using the Hasse
diagram of basic tilting A-modules and is always verified if A is of
finite representation type. Then, we use the above correspondence to
prove that A is simply connected if and only if B is simply
connected, under the same hypothesis on T. Finally, we prove that if
a tilted algebra B of type Q is simply connected, then Q is a tree
and the first Hochschild cohomology group of B vanishes.
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The universal cover of a monomial
triangular algebra without multiple arrows Journal of Algebra and its Applications 7 (4) (2008), 443-469 Let A be a basic connected finite dimensional
algebra over an algebraically closed field k. Assuming that A is
monomial and that the ordinary quiver Q of A has no oriented cycle
and no multiple arrows, we prove that A admits a universal cover
with group the fundamental group of the underlying space of Q.
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The universal cover of an algebra
without double bypass Journal of Algebra 312 (1) (2007), 330-353 Let A be a basic finite dimensional and
connected algebra over an algebraically closed field k with zero
characteristic. If the ordinary quiver of A has no double bypasses,
we show that A admits a Galois covering which satisfies a universal
property with respect to the Galois coverings of A. This universal
property is similar to the one of the universal cover of a connected
topological space. pdfarXivHALdoi

The fundamental group of a triangular
algebra without double bypasses Comptes Rendus de l'académie des sciences de Paris, série 1 341
(2005) 211-216 Let A be a basic connected finite dimensional
algebra over a field k and let Q be the ordinary quiver of A. To any
presentation of A with Q and admissible relations, R. Martinez-Villa
and J. A. de La Pena have associated a group called the fundamental
group of this presentation. There may exist different presentations
of A with non isomorphic fundamental groups. In this note, we show
that if the field k has characteristic zero, if Q has no oriented
cycles and if Q has no double bypasses then there exists a
privileged presentation of A such that the fundamental group of any
other presentation is the quotient of the fundamental group of this
privileged presentation. pdfarXivHALdoi