Sur une opérade ternaire liée aux treillis de Tamari

Frédéric Chapoton[1]

  • [1] Institut Camille Jordan, Université Claude Bernard Lyon 1, Bâtiment Braconnier, 21 Avenue Claude Bernard, F-69622 Villeurbanne Cedex

Annales de la faculté des sciences de Toulouse Mathématiques (2011)

  • Volume: 20, Issue: 4, page 843-869
  • ISSN: 0240-2963

Abstract

top
We introduce an anticyclic operad V given by a ternary generator and a quadratic relation. We show that it admits a natural basis indexed by planar binary trees. We then relate this construction to the familly of Tamari lattices ( Y n ) n ≥ 0 by defining an isomorphism between V ( 2 n + 1 ) and the Grothendieck group of the category mod Y n . This isomorphism maps the basis of V ( 2 n + 1 ) to the classes of projective modules and sends the anticyclic map of the operad V ( 2 n + 1 ) to the Coxeter transformation of the derived category of mod Y n . The Koszul duality theory for operads then allows us to compute the characteristic polynomial of the Coxeter transformation by a Legendre transform.

How to cite

top

Chapoton, Frédéric. "Sur une opérade ternaire liée aux treillis de Tamari." Annales de la faculté des sciences de Toulouse Mathématiques 20.4 (2011): 843-869. <http://eudml.org/doc/219722>.

@article{Chapoton2011,
abstract = {On introduit une opérade anticyclique $\{\bf V\}$ définie par une présentation ternaire quadratique. On montre qu’elle admet une base indexée par les arbres binaires planaires. On relie cette construction à la famille des treillis de Tamari $(\mathsf \{Y\}_n)_\{n \ge 0\}$ en construisant un isomorphisme entre $\{\bf V\}(2n+1)$ et le groupe de Grothendieck de la catégorie $\mathsf \{\}\mod \{\mathsf \{Y\}\}_n$ qui envoie la base de $\{\bf V\}(2n+1)$ sur les classes des modules projectifs et qui transforme la structure anticyclique de $\{\bf V\}$ en la transformation de Coxeter de la catégorie dérivée de $\mathsf \{\}\mod \{\mathsf \{Y\}\}_n$. La dualité de Koszul des opérades permet alors de calculer le polynôme caractéristique de cette transformation de Coxeter en utilisant une transformation de Legendre.},
affiliation = {Institut Camille Jordan, Université Claude Bernard Lyon 1, Bâtiment Braconnier, 21 Avenue Claude Bernard, F-69622 Villeurbanne Cedex},
author = {Chapoton, Frédéric},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {partial associative operad; cyclic operad; Tamari order},
language = {fre},
month = {7},
number = {4},
pages = {843-869},
publisher = {Université Paul Sabatier, Toulouse},
title = {Sur une opérade ternaire liée aux treillis de Tamari},
url = {http://eudml.org/doc/219722},
volume = {20},
year = {2011},
}

TY - JOUR
AU - Chapoton, Frédéric
TI - Sur une opérade ternaire liée aux treillis de Tamari
JO - Annales de la faculté des sciences de Toulouse Mathématiques
DA - 2011/7//
PB - Université Paul Sabatier, Toulouse
VL - 20
IS - 4
SP - 843
EP - 869
AB - On introduit une opérade anticyclique ${\bf V}$ définie par une présentation ternaire quadratique. On montre qu’elle admet une base indexée par les arbres binaires planaires. On relie cette construction à la famille des treillis de Tamari $(\mathsf {Y}_n)_{n \ge 0}$ en construisant un isomorphisme entre ${\bf V}(2n+1)$ et le groupe de Grothendieck de la catégorie $\mathsf {}\mod {\mathsf {Y}}_n$ qui envoie la base de ${\bf V}(2n+1)$ sur les classes des modules projectifs et qui transforme la structure anticyclique de ${\bf V}$ en la transformation de Coxeter de la catégorie dérivée de $\mathsf {}\mod {\mathsf {Y}}_n$. La dualité de Koszul des opérades permet alors de calculer le polynôme caractéristique de cette transformation de Coxeter en utilisant une transformation de Legendre.
LA - fre
KW - partial associative operad; cyclic operad; Tamari order
UR - http://eudml.org/doc/219722
ER -

References

top
  1. Aguiar (M.) and Sottile (F.).— Structure of the Loday-Ronco Hopf algebra of trees. J. Algebra, 295(2) p. 473-511 (2006). Zbl1099.16015MR2194965
  2. Chapoton (F.).— On the Coxeter transformations for Tamari posets. Canad. Math. Bull., 50(2) p. 182-190 (2007). Zbl1147.18007MR2317440
  3. Chapoton (F.).— Le module dendriforme sur le groupe cyclique. Ann. Inst. Fourier (Grenoble), 58(7) p. 2333-2350 (2008). Zbl1163.18004MR2498353
  4. Chapoton (F.).— Categorification of the dendriform operad. In Jean-Louis Loday and Bruno Vallette, editors, Proceedings of Operads 2009, Séminaire et Congrès. SMF, 2012. oai :arXiv.org :0909.2751. Zbl1277.18010
  5. Curtis (C. W.) and Irving Reiner (I.).— Representation theory of finite groups and associative algebras. Pure and Applied Mathematics, Vol. XI. Interscience Publishers, a division of John Wiley & Sons, New York-London (1962). Zbl0131.25601MR144979
  6. Dotsenko (V.) and Khoroshkin (A.).— Gröbner bases for operads. Duke Math. J., 153(2) p. 363-396 (2010). Zbl1208.18007MR2667136
  7. Ebrahimi-Fard (K.) and Manchon (D.).— Dendriform equations. J. Algebra, 322(11) p. 4053-4079 (2009). Zbl1229.17001MR2556138
  8. Ebrahimi-Fard (K.), Manchon (D.), and Patras (F.).— New identities in dendriform algebras. J. Algebra, 320(2) p. 708-727 (2008). Zbl1153.17003MR2422313
  9. Fomin (S.) and Zelevinsky (A.).— Cluster algebras. I. Foundations. J. Amer. Math. Soc., 15(2) p. 497-529 (electronic) (2002). Zbl1021.16017MR1887642
  10. Friedman (H.) and Tamari (D.).— Problèmes d’associativité : Une structure de treillis finis induite par une loi demi-associative. J. Combinatorial Theory, 2 p. 215-242 (1967). Zbl0158.01904MR238984
  11. Getzler (E.).— Operads and moduli spaces of genus 0 Riemann surfaces. In The moduli space of curves (Texel Island, 1994), volume 129 of Progr. Math., pages 199-230. Birkhäuser Boston, Boston, MA (1995). Zbl0851.18005MR1363058
  12. Getzler (E.) and Kapranov (M. M.).— Cyclic operads and cyclic homology. In Geometry, topology, & physics, Conf. Proc. Lecture Notes Geom. Topology, IV, p. 167-201. Int. Press, Cambridge, MA (1995). Zbl0883.18013MR1358617
  13. Getzler (E.) and Kapranov (M. M.).— Modular operads. Compositio Math., 110(1) p. 65-126 (1998). Zbl0894.18005MR1601666
  14. Gnedbaye (A. V.).— Opérades des algèbres (k + 1)-aires. In Operads : Proceedings of Renaissance Conferences (Hartford, CT/Luminy, 1995), volume 202 of Contemp. Math., pages 83-113. Amer. Math. Soc., Providence, RI (1997). Zbl0880.17003MR1436918
  15. Happel (D.).— Triangulated categories in the representation theory of finite-dimensional algebras, volume 119 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge (1988). Zbl0635.16017MR935124
  16. Happel (D.) and Unger (L.).— On a partial order of tilting modules. Algebr. Represent. Theory, 8(2) p. 147-156 (2005). Zbl1110.16011MR2162278
  17. Hivert (F.), Novelli (J.-C.), and Thibon (J.-Y.).— The algebra of binary search trees. Theoret. Comput. Sci., 339(1) p. 129-165 (2005). Zbl1072.05052MR2142078
  18. Hoffbeck (E.).— A Poincaré-Birkhoff-Witt criterion for Koszul operads. Manuscripta Math., 131(1-2) p. 87-110 (2010). Zbl1207.18009MR2574993
  19. Huang (S.) and Tamari (D.).— Problems of associativity : A simple proof for the lattice property of systems ordered by a semi-associative law. J. Combinatorial Theory Ser. A, 13 p. 7-13 (1972). Zbl0248.06003MR306064
  20. Ladkani (S.).— Universal derived equivalences of posets of cluster tilting objects (2007). 
  21. Ladkani (S.).— Universal derived equivalences of posets of tilting modules (2007). 
  22. Ladkani (S.).— On derived equivalences of categories of sheaves over finite posets. J. Pure Appl. Algebra, 212(2) p. 435-451 (2008). Zbl1127.18005MR2357344
  23. Lenzing (H.).— Coxeter transformations associated with finite-dimensional algebras. In Computational methods for representations of groups and algebras (Essen, 1997), volume 173 of Progr. Math., pages 287-308. Birkhäuser, Basel (1999). Zbl0941.16007MR1714618
  24. Loday (J.-L.).— Dialgebras. In Dialgebras and related operads, volume 1763 of Lecture Notes in Math., pages 7-66. Springer, Berlin (2001). Zbl0999.17002MR1860994
  25. Loday (J.-L.).— Arithmetree. J. Algebra, 258(1) p. 275-309 (2002). Special issue in celebration of Claudio Procesis 60th birthday. Zbl1063.16044MR1958907
  26. Loday (J.-L.) and Ronco (M. O.).— Order structure on the algebra of permutations and of planar binary trees. J. Algebraic Combin., 15(3) p. 253-270 (2002). Zbl0998.05013MR1900627
  27. Loday (J.-L.) and Vallette (B.).— Algebraic Operads. à paraître, 2010. xviii+512 pp. 
  28. Macdonald (I. G.).— Symmetric functions and Hall polynomials. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York, second edition, 1995. With contributions by A. Zelevinsky, Oxford Science Publications. Zbl0487.20007MR1354144
  29. Markl (M.) and Remm (E.).— (Non-)Koszulity of operads for n-ary algebras, cohomology and deformations (2009). 
  30. Markl (M.), Shnider (S.), and Stasheff (J.).— Operads in algebra, topology and physics, volume 96 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI (2002). Zbl1017.18001MR1898414
  31. Reading (N.).— Cambrian lattices. Adv. Math., 205(2) p. 313-353 (2006). Zbl1106.20033MR2258260
  32. Riedtmann (C.) and Schofield (A.).— On a simplicial complex associated with tilting modules. Comment. Math. Helv., 66(1) p. 70-78 (1991). Zbl0790.16013MR1090165
  33. Ronco (M.).— Primitive elements in a free dendriform algebra. In New trends in Hopf algebra theory (La Falda, 1999), volume 267 of Contemp. Math., pages 245-263. Amer. Math. Soc., Providence, RI (2000). Zbl0974.16035MR1800716
  34. Tamari (D.).— The algebra of bracketings and their enumeration. Nieuw Arch. Wisk. (3), 10 p. 131-146 (1962). Zbl0109.24502MR146227

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.