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Frédéric Chapoton
Sur une opérade ternaire liée aux treillis de Tamari
Annales de la faculté des sciences de Toulouse Sér. 6, 20 no. 4 (2011), p. 843-869, doi: 10.5802/afst.1326
Article PDF | Reviews MR 2918216 | Zbl 1248.18009

Résumé - Abstract

We introduce an anticyclic operad ${\bf V}$ 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 $(\mathsf {Y}_n)_{n \ge 0}$ by defining an isomorphism between ${\bf V}(2n+1)$ and the Grothendieck group of the category $\mathsf {}\mod {\mathsf {Y}}_n$. This isomorphism maps the basis of ${\bf V}(2n+1)$ to the classes of projective modules and sends the anticyclic map of the operad ${\bf V}(2n+1)$ to the Coxeter transformation of the derived category of $\mathsf {}\mod {\mathsf {Y}}_n$. The Koszul duality theory for operads then allows us to compute the characteristic polynomial of the Coxeter transformation by a Legendre transform.

Bibliography

[1] Aguiar (M.) and Sottile (F.).— Structure of the Loday-Ronco Hopf algebra of trees. J. Algebra, 295(2) p. 473-511 (2006).  MR 2194965 |  Zbl 1099.16015
[2] Chapoton (F.).— On the Coxeter transformations for Tamari posets. Canad. Math. Bull., 50(2) p. 182-190 (2007).  MR 2317440 |  Zbl 1147.18007
[3] Chapoton (F.).— Le module dendriforme sur le groupe cyclique. Ann. Inst. Fourier (Grenoble), 58(7) p. 2333-2350 (2008). Cedram |  MR 2498353 |  Zbl 1163.18004
[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.
[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).  MR 144979 |  Zbl 0131.25601
[6] Dotsenko (V.) and Khoroshkin (A.).— Gröbner bases for operads. Duke Math. J., 153(2) p. 363-396 (2010).  MR 2667136 |  Zbl 1208.18007
[7] Ebrahimi-Fard (K.) and Manchon (D.).— Dendriform equations. J. Algebra, 322(11) p. 4053-4079 (2009).  MR 2556138 |  Zbl pre05676107
[8] Ebrahimi-Fard (K.), Manchon (D.), and Patras (F.).— New identities in dendriform algebras. J. Algebra, 320(2) p. 708-727 (2008).  MR 2422313 |  Zbl 1153.17003
[9] Fomin (S.) and Zelevinsky (A.).— Cluster algebras. I. Foundations. J. Amer. Math. Soc., 15(2) p. 497-529 (electronic) (2002).  MR 1887642 |  Zbl 1021.16017
[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).  MR 238984 |  Zbl 0158.01904
[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).  MR 1363058 |  Zbl 0851.18005
[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).  MR 1358617 |  Zbl 0883.18013
[13] Getzler (E.) and Kapranov (M. M.).— Modular operads. Compositio Math., 110(1) p. 65-126 (1998).  MR 1601666 |  Zbl 0894.18005
[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).  MR 1436918 |  Zbl 0880.17003
[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).  MR 935124 |  Zbl 0635.16017
[16] Happel (D.) and Unger (L.).— On a partial order of tilting modules. Algebr. Represent. Theory, 8(2) p. 147-156 (2005).  MR 2162278 |  Zbl 1110.16011
[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).  MR 2142078 |  Zbl 1072.05052
[18] Hoffbeck (E.).— A Poincaré-Birkhoff-Witt criterion for Koszul operads. Manuscripta Math., 131(1-2) p. 87-110 (2010).  MR 2574993 |  Zbl 1207.18009
[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).  MR 306064 |  Zbl 0248.06003
[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).  MR 2357344 |  Zbl 1127.18005
[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).  MR 1714618 |  Zbl 0941.16007
[24] Loday (J.-L.).— Dialgebras. In Dialgebras and related operads, volume 1763 of Lecture Notes in Math., pages 7-66. Springer, Berlin (2001).  MR 1860994 |  Zbl 0999.17002
[25] Loday (J.-L.).— Arithmetree. J. Algebra, 258(1) p. 275-309 (2002). Special issue in celebration of Claudio Procesis 60th birthday.  MR 1958907 |  Zbl 1063.16044
[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).  MR 1900627 |  Zbl 0998.05013
[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.  MR 1354144 |  Zbl 0487.20007
[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).  MR 1898414 |  Zbl 1017.18001
[31] Reading (N.).— Cambrian lattices. Adv. Math., 205(2) p. 313-353 (2006).  MR 2258260 |  Zbl 1106.20033
[32] Riedtmann (C.) and Schofield (A.).— On a simplicial complex associated with tilting modules. Comment. Math. Helv., 66(1) p. 70-78 (1991).  MR 1090165 |  Zbl 0790.16013
[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).  MR 1800716 |  Zbl 0974.16035
[34] Tamari (D.).— The algebra of bracketings and their enumeration. Nieuw Arch. Wisk. (3), 10 p. 131-146 (1962).  MR 146227 |  Zbl 0109.24502
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