Patrick Le Meur

Publications

Preprints

  1. (m,n)-Quasitilted and (m,n)-Almost Hereditary Algebras (2017)
    with Edson Ribeiro Alvares, Diane Castonguay and Tanise Carnieri Pierin
    Link to arXiv 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.
  2. Duality for Differential Operators of Lie-Rinehart Algebras (2017)
    with Thierry Lambre
    Link to arXiv 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.
  3. Degrees of Irreducible Morphisms over Perfect Fields (2017)
    with Claudia Chaio and Sonia Trepode
    Link to arXiv 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.
  4. Representation Theory of Partial Relation Extensions (2016)
    with Ibrahim Assem, Juan Carlos Bustamante, Julie Dionne and David Smith
    Link to arXiv 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.
  5. Crossed-Products of Calabi-Yau Algebras by Hopf Algebras (2015)
    Link to arXiv 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.
  6. Covering Techniques for Auslander-Reiten Theory (2015)
    with Claudia Chaio and Sonia Trepode
    Link to arXiv 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.
  7. Crossed-Products of Calabi-Yau Algebra by Finite Groups (2010)
    Link to arXiv Let a finite group G act on a differential graded Calabi-Yau algebra A over a field whose characteristic does not divide the order of G. This note studies when the crossed-product of A by G is still Calabi-Yau. Under a compatibility condition between the action of G on A and the Calabi-Yau structure of A, it is proved that this is indeed the case. Similar results are proved for general constructions of Calabi-Yau algebras such as deformed Calabi-Yau completions and Ginzburg algebras.

Published articles in international journals with referee

  1. The strong global dimension of piecewise hereditary algebras (2017)
    with Edson Ribeiro Alvares et Eduardo N. Marcos
    J. Algebra 481 (2017) 36-67
    Link to arXiv 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.
  2. Special biserial algebras with no outer derivations
    with Ibrahim Assem et Juan Carlos Bustamante
    Colloq. Math. 125 (1) (2011), 83-98
    Link to arXiv 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.
  3. Degrees of irreducible morphisms and finite representation type
    with Claudia Chaio et Sonia Trepode
    J. London Math. Soc. sér. 2 84 (1) (2011), 35-57
    Link to arXiv 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.
  4. Topological invariants of piecewise hereditary algebras
    Trans. Amer. Math. Soc. 363 (4) (2011), 2143-2170
    Link to arXiv
    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.
  5. Coverings of Laura algebras: the standard case with Ibrahim Assem and Juan Carlos Bustamante
    J. Algebra 323 (1) (2010), 83-120
    Link to arXiv
    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.
  6. Galois coverings of weakly shod algebras
    Comm. Algebra 38 (4) (2010), 1291-1318
    Link to arXiv
    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.
  7. On maximal diagonalizable Lie subalgebras of the first Hochschild cohomology
    Comm. Algebra 38 (4) (2010), 1325-1340
    Link to arXiv
    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 HH1(A). We study the image of these mappings associated to the different presentations of A in terms of diagonalizable Lie subalgebras of HH1(A). Then we characterise the maximal diagonalisable subalgebras of HH1(A) when A is monomial and Q has no multiple arrows and also when car(k)=0 and Q has no double bypass.
  8. On Galois coverings and tilting modules
    J. Algebra 319 (12) (2008), 4961-4999
    Link to arXiv
    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.
  9. /The universal cover of a monomial triangular algebra without multiple arrows
    J. Algebra Appl. 7 (4) (2008), 443-469
    Link to arXiv
    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.
  10. The universal cover of an algebra without double bypass J. Algebra 312 (1) (2007), 330-353
    Link to arXiv
    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.
  11. The fundamental group of a triangular algebra without double bypasses
    C.R. Acad. Sci. Paris, Série 1, 341 (2005) 211-216
    Link to arXiv
    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.