[Approche coalgébrique de la catégorie des algèbres de Loday infinies, différentielle souche pour les algèbres graduées -aires ou à homotopie]
Nous définissons un coproduit gradué et coassociatif tordu sur l’algèbre tensorielle d’un espace vectoriel gradué . Les codérivations (resp. codifférentielles quadratiques de “degré 1”, codifférentielles impaires quelconques) de cette co-algèbre sont en correspondance biunivoque avec les suites d’applications multilinéaires sur (resp. structures graduées de Loday sur , suites que nous appelons structures de Loday infinies sur ). Nous prouvons un théorème du modèle minimal pour les algèbres infinies de Loday et observons que la catégorie contient la catégorie comme sous-catégorie. En plus, le crochet de Lie gradué des codérivations conduit à un crochet de Lie gradué “souche” sur les espaces des cochaînes des algèbres de Loday graduées, de Loday infinies et de Loday graduées -aires. Le crochet souche se restreint aux crochets gradués de Nijenhuis-Richardson et de Grabowski-Marmo, et il encode, au-delà des cohomologies déjà mentionnées, celles des algèbres de Lie graduées, de Poisson graduées, de Jacobi graduées, Lie infinies, ainsi que celle des algèbres de Lie graduées -aires.
We define a graded twisted-coassociative coproduct on the tensor algebra the desuspension space of a graded vector space . The coderivations (resp. quadratic “degree 1” codifferentials, arbitrary odd codifferentials) of this coalgebra are 1-to-1 with sequences of multilinear maps on (resp. graded Loday structures on , sequences that we call Loday infinity structures on ). We prove a minimal model theorem for Loday infinity algebras and observe that the category contains the category as a subcategory. Moreover, the graded Lie bracket of coderivations gives rise to a graded Lie “stem” bracket on the cochain spaces of graded Loday, Loday infinity, and -ary graded Loday algebras. This stem bracket restricts to the graded Nijenhuis-Richardson and Grabowski-Marmo brackets, and it encodes, beyond the already mentioned cohomologies, those of graded Lie, graded Poisson, graded Jacobi, Lie infinity, as well as that of -ary graded Lie algebras.
Keywords: Zinbiel coalgebra, graded Loday, Lie, Poisson, Jacobi structure, strongly homotopy algebra, square-zero element method, graded cohomology, Schouten-Nijenhuis, Nijenhuis-Richardson, Grabowski-Marmo bracket, deformation theory
Mot clés : co-algèbre de Zinbiel, suites graduées de Loday, Lie, Poisson, structure de Jacobi, algèbre fortement homotopique, cohomologie, Schouten-Nijenhuis, Nijenhuis-Richardson, crochets gradués de Grabowski-Marmo, théorie de déformation
@article{AIF_2010__60_1_355_0, author = {Ammar, Mourad and Poncin, Norbert}, title = {Coalgebraic {Approach} to the {Loday} {Infinity} {Category,} {Stem} {Differential} for $2n$-ary {Graded} and {Homotopy} {Algebras}}, journal = {Annales de l'Institut Fourier}, pages = {355--387}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {60}, number = {1}, year = {2010}, doi = {10.5802/aif.2525}, zbl = {1208.53084}, mrnumber = {2664318}, language = {en}, url = {http://www.numdam.org/articles/10.5802/aif.2525/} }
TY - JOUR AU - Ammar, Mourad AU - Poncin, Norbert TI - Coalgebraic Approach to the Loday Infinity Category, Stem Differential for $2n$-ary Graded and Homotopy Algebras JO - Annales de l'Institut Fourier PY - 2010 SP - 355 EP - 387 VL - 60 IS - 1 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.2525/ DO - 10.5802/aif.2525 LA - en ID - AIF_2010__60_1_355_0 ER -
%0 Journal Article %A Ammar, Mourad %A Poncin, Norbert %T Coalgebraic Approach to the Loday Infinity Category, Stem Differential for $2n$-ary Graded and Homotopy Algebras %J Annales de l'Institut Fourier %D 2010 %P 355-387 %V 60 %N 1 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.2525/ %R 10.5802/aif.2525 %G en %F AIF_2010__60_1_355_0
Ammar, Mourad; Poncin, Norbert. Coalgebraic Approach to the Loday Infinity Category, Stem Differential for $2n$-ary Graded and Homotopy Algebras. Annales de l'Institut Fourier, Tome 60 (2010) no. 1, pp. 355-387. doi : 10.5802/aif.2525. http://www.numdam.org/articles/10.5802/aif.2525/
[1] On some generalizations of Batalin-Vilkovisky algebras, J. Pure Appl. Algebra, Volume 120 (1997) no. 2, pp. 105-141 | DOI | MR | Zbl
[2] Deformation Quantization and Cohomologies of Poisson, Graded, and Homotopy Algebras, University of Luxembourg, Paul Verlaine University of Metz (2008) (Ph. D. Thesis)
[3] Formal Poisson cohomology of twisted -matrix induced structures, Israel J. Math., Volume 165 (2008), pp. 381-411 | DOI | MR | Zbl
[4] Choix des signes pour la formalité de M. Kontsevich, Pacific J. Math., Volume 203 (2002) no. 1, pp. 23-66 | DOI | MR | Zbl
[5] Déformations et rigidité géométrique des algèbres de Leibniz, Comm. Algebra, Volume 24 (1996) no. 3, pp. 1017-1034 | DOI | MR | Zbl
[6] Deformations of algebras over a quadratic operad, Operads: Proceedings of Renaissance Conferences (Hartford, CT/Luminy, 1995) (Contemp. Math.), Volume 202, Amer. Math. Soc., Providence, RI, 1997, pp. 207-234 | MR | Zbl
[7] Leibniz and Lie algebra structures for Nambu algebra, Lett. Math. Phys., Volume 39 (1997) no. 2, pp. 127-141 | DOI | MR | Zbl
[8] Formal deformations of the Poisson Lie algebra of a symplectic manifold and star-products. Existence, equivalence, derivations, Deformation theory of algebras and structures and applications (Il Ciocco, 1986) (NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci.), Volume 247, Kluwer Acad. Publ., Dordrecht, 1988, pp. 897-960 | MR | Zbl
[9] Deformation theory of infinity algebras, J. Algebra, Volume 255 (2002) no. 1, pp. 59-88 | DOI | MR | Zbl
[10] -L,ie algebras, Sibirsk. Mat. Zh., Volume 26 (1985) no. 6, p. 126-140, 191 | MR | Zbl
[11] On the deformation of rings and algebras, Ann. of Math. (2), Volume 79 (1964), pp. 59-103 | DOI | MR | Zbl
[12] Koszul duality for operads, Duke Math. J., Volume 76 (1994) no. 1, pp. 203-272 | DOI | MR | Zbl
[13] Jacobi structures revisited, J. Phys. A, Volume 34 (2001) no. 49, pp. 10975-10990 | DOI | MR | Zbl
[14] The graded Jacobi algebras and (co)homology, J. Phys. A, Volume 36 (2003) no. 1, pp. 161-181 | DOI | MR | Zbl
[15] Duality and modular class of a Nambu-Poisson structure, J. Phys. A, Volume 34 (2001) no. 17, pp. 3623-3650 | DOI | MR | Zbl
[16] The algebraic structure in the homology of an -algebra, Soobshch. Akad. Nauk Gruzin. SSR, Volume 108 (1982) no. 2, p. 249-252 (1983) | MR | Zbl
[17] Deformation quantization of Poisson manifolds, Lett. Math. Phys., Volume 66 (2003) no. 3, pp. 157-216 | DOI | MR | Zbl
[18] From Poisson algebras to Gerstenhaber algebras, Ann. Inst. Fourier (Grenoble), Volume 46 (1996) no. 5, pp. 1243-1274 | DOI | EuDML | Numdam | MR | Zbl
[19] Supercanonical algebras and Schouten brackets, Mat. Zametki, Volume 49 (1991) no. 1, p. 70-76, 160 | MR | Zbl
[20] The multigraded Nijenhuis-Richardson algebra, its universal property and applications, J. Pure Appl. Algebra, Volume 77 (1992) no. 1, pp. 87-102 | DOI | MR | Zbl
[21] Lichnerowicz-Jacobi cohomology and homology of Jacobi manifolds: modular class and duality (1999) (arXiv: math/9910079)
[22] On the computation of the Lichnerowicz-Jacobi cohomology, J. Geom. Phys., Volume 44 (2003) no. 4, pp. 507-522 | DOI | MR | Zbl
[23] Lichnerowicz-Jacobi cohomology, J. Phys. A, Volume 30 (1997) no. 17, pp. 6029-6055 | DOI | MR | Zbl
[24] Les variétés de Poisson et leurs algèbres de Lie associées, J. Differential Geometry, Volume 12 (1977) no. 2, pp. 253-300 | MR | Zbl
[25] Les variétés de Jacobi et leurs algèbres de Lie associées, J. Math. Pures Appl. (9), Volume 57 (1978) no. 4, pp. 453-488 | MR | Zbl
[26] Une version non commutative des algèbres de Lie: les algèbres de Leibniz, Enseign. Math. (2), Volume 39 (1993) no. 3-4, pp. 269-293 | MR | Zbl
[27] Cup-product for Leibniz cohomology and dual Leibniz algebras, Math. Scand., Volume 77 (1995) no. 2, pp. 189-196 | EuDML | MR | Zbl
[28] Operads in algebra, topology and physics, Mathematical Surveys and Monographs, 96, American Mathematical Society, Providence, RI, 2002 | MR | Zbl
[29] On a general approach to the formal cohomology of quadratic Poisson structures, J. Pure Appl. Algebra, Volume 208 (2007) no. 3, pp. 887-904 | DOI | MR | Zbl
[30] -ary Lie and associative algebras, Rend. Sem. Mat. Univ. Politec. Torino, Volume 54 (1996) no. 4, pp. 373-392 Geometrical structures for physical theories, II (Vietri, 1996) | MR | Zbl
[31] Generalized Hamiltonian dynamics, Phys. Rev. D (3), Volume 7 (1973), pp. 2405-2412 | DOI | MR | Zbl
[32] Deformations of Lie algebra structures, J. Math. Mech., Volume 17 (1967), pp. 89-105 | MR | Zbl
[33] Coproduct and cogroups in the category of graded dual Leibniz algebras, Operads: Proceedings of Renaissance Conferences (Hartford, CT/Luminy, 1995) (Contemp. Math.), Volume 202, Amer. Math. Soc., Providence, RI, 1997, pp. 115-135 | MR | Zbl
[34] Infinity Algebras, Cohomology and Cyclic Cohomology, and Infinitesimal Deformations (2001) (arXiv: math/0111088)
[35] Poisson (co)homology and isolated singularities, J. Algebra, Volume 299 (2006) no. 2, pp. 747-777 | DOI | MR | Zbl
[36] Premier et deuxième espaces de cohomologie de l’algèbre de Lie des opérateurs différentiels sur une variété, à coefficients dans les fonctions, Bull. Soc. Roy. Sci. Liège, Volume 67 (1998) no. 6, pp. 291-337 | MR | Zbl
[37] On the cohomology of the Nijenhuis-Richardson graded Lie algebra of the space of functions of a manifold, J. Algebra, Volume 243 (2001) no. 1, pp. 16-40 | DOI | MR | Zbl
[38] Cohomology ring of -Lie algebras, Extracta Math., Volume 20 (2005) no. 3, pp. 219-232 | EuDML | MR | Zbl
[39] The intrinsic bracket on the deformation complex of an associative algebra, J. Pure Appl. Algebra, Volume 89 (1993) no. 1-2, pp. 231-235 | DOI | MR | Zbl
[40] On multiple generalizations of Lie algebras and Poisson manifolds, Secondary calculus and cohomological physics (Moscow, 1997) (Contemp. Math.), Volume 219, Amer. Math. Soc., Providence, RI, 1998, pp. 273-287 | MR | Zbl
[41] Graded multiple analogs of Lie algebras, Acta Appl. Math., Volume 72 (2002) no. 1-2, pp. 183-197 (Symmetries of differential equations and related topics) | DOI | MR | Zbl
Cité par Sources :