Table des matières de ce fascicule | Article précédent
Mourad Ammar; Norbert Poncin
Coalgebraic Approach to the Loday Infinity Category, Stem Differential for $2n$-ary Graded and Homotopy Algebras
(Approche coalgébrique de la catégorie des algèbres de Loday infinies, différentielle souche pour les algèbres graduées $2n$-aires ou à homotopie)
Annales de l'institut Fourier, 60 no. 1 (2010), p. 355-387
Article: sur abonnement
Class. Math.: 16W30, 16E45, 17B56, 17B70
Mots 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
[1] Füsun Akman, “On some generalizations of Batalin-Vilkovisky algebras”, J. Pure Appl. Algebra 120 (1997), p. 105-141 MR 1466615 | Zbl 0885.17020
[2] Mourad Ammar, Deformation Quantization and Cohomologies of Poisson, Graded, and Homotopy Algebras, thesis, University of Luxembourg, Paul Verlaine University of Metz, 2008
[3] Mourad Ammar & Norbert Poncin, “Formal Poisson cohomology of twisted $r$-matrix induced structures”, Israel J. Math. 165 (2008), p. 381-411 MR 2403627 | Zbl 1146.53054
[4] D. Arnal, D. Manchon & M. Masmoudi, “Choix des signes pour la formalité de M. Kontsevich”, Pacific J. Math. 203 (2002), p. 23-66 MR 1895924 | Zbl 1055.53066
[5] David Balavoine, “Déformations et rigidité géométrique des algèbres de Leibniz”, Comm. Algebra 24 (1996), p. 1017-1034 MR 1374652 | Zbl 0855.17021
[6] David Balavoine, Deformations of algebras over a quadratic operad, Operads: Proceedings of Renaissance Conferences (Hartford, CT/Luminy, 1995), Amer. Math. Soc., 1997, p. 207–234 MR 1436922 | Zbl 0883.17004
[7] Yuri L. Daletskii & Leon A. Takhtajan, “Leibniz and Lie algebra structures for Nambu algebra”, Lett. Math. Phys. 39 (1997), p. 127-141 MR 1437747 | Zbl 0869.58024
[8] M. De Wilde & P. Lecomte, 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), Kluwer Acad. Publ., 1988, p. 897–960 MR 981635 | Zbl 0685.58039
[9] Alice Fialowski & Michael Penkava, “Deformation theory of infinity algebras”, J. Algebra 255 (2002), p. 59-88 MR 1935035 | Zbl 1038.17012
[10] V. T. Filippov, “$n$-L$Q\in \mathbb{Z}^n,p\in \mathbb{N}^*$,ie algebras”, Sibirsk. Mat. Zh. 26 (1985), p. 126-140, 191 MR 816511 | Zbl 0585.17002
[11] Murray Gerstenhaber, “On the deformation of rings and algebras”, Ann. of Math. (2) 79 (1964), p. 59-103 MR 171807 | Zbl 0123.03101
[12] Victor Ginzburg & Mikhail Kapranov, “Koszul duality for operads”, Duke Math. J. 76 (1994), p. 203-272
Article | MR 1301191 | Zbl 0855.18006
[13] Janusz Grabowski & Giuseppe Marmo, “Jacobi structures revisited”, J. Phys. A 34 (2001), p. 10975-10990 MR 1872975 | Zbl 0998.53054
[14] Janusz Grabowski & Giuseppe Marmo, “The graded Jacobi algebras and (co)homology”, J. Phys. A 36 (2003), p. 161-181 MR 1959019 | Zbl 1039.53090
[15] R. Ibáñez, M. de León, B. López, J. C. Marrero & E. Padrón, “Duality and modular class of a Nambu-Poisson structure”, J. Phys. A 34 (2001), p. 3623-3650 MR 1841496 | Zbl 1021.53060
[16] T. V. Kadeishvili, “The algebraic structure in the homology of an $A(\infty )$-algebra”, Soobshch. Akad. Nauk Gruzin. SSR 108 (1982), p. 249-252 (1983) MR 720689 | Zbl 0535.55005
[17] Maxim Kontsevich, “Deformation quantization of Poisson manifolds”, Lett. Math. Phys. 66 (2003), p. 157-216
arXiv | MR 2062626 | Zbl 1058.53065
[18] Yvette Kosmann-Schwarzbach, “From Poisson algebras to Gerstenhaber algebras”, Ann. Inst. Fourier (Grenoble) 46 (1996), p. 1243-1274
Cedram | MR 1427124 | Zbl 0858.17027
[19] I. S. Krasilʼshchik, “Supercanonical algebras and Schouten brackets”, Mat. Zametki 49 (1991), p. 70-76, 160 MR 1101552 | Zbl 0723.58020
[20] P. A. B. Lecomte, P. W. Michor & H. Schicketanz, “The multigraded Nijenhuis-Richardson algebra, its universal property and applications”, J. Pure Appl. Algebra 77 (1992), p. 87-102 MR 1148273 | Zbl 0752.17019
[21] Manuel de León, Belén López, Juan C. Marrero & Edith Padrón, “Lichnerowicz-Jacobi cohomology and homology of Jacobi manifolds: modular class and duality”, arXiv: math/9910079, 1999
arXiv
[22] Manuel de León, Belén López, Juan C. Marrero & Edith Padrón, “On the computation of the Lichnerowicz-Jacobi cohomology”, J. Geom. Phys. 44 (2003), p. 507-522 MR 1943175 | Zbl 1092.53060
[23] Manuel de León, Juan C. Marrero & Edith Padrón, “Lichnerowicz-Jacobi cohomology”, J. Phys. A 30 (1997), p. 6029-6055 MR 1482695 | Zbl 0951.37033
[24] André Lichnerowicz, “Les variétés de Poisson et leurs algèbres de Lie associées”, J. Differential Geometry 12 (1977), p. 253-300
Article | MR 501133 | Zbl 0405.53024
[25] André Lichnerowicz, “Les variétés de Jacobi et leurs algèbres de Lie associées”, J. Math. Pures Appl. (9) 57 (1978), p. 453-488 MR 524629 | Zbl 0407.53025
[26] Jean-Louis Loday, “Une version non commutative des algèbres de Lie: les algèbres de Leibniz”, Enseign. Math. (2) 39 (1993), p. 269-293 MR 1252069 | Zbl 0806.55009
[27] Jean-Louis Loday, “Cup-product for Leibniz cohomology and dual Leibniz algebras”, Math. Scand. 77 (1995), p. 189-196 MR 1379265 | Zbl 0859.17015
[28] Martin Markl, Steve Shnider & Jim Stasheff, Operads in algebra, topology and physics, Mathematical Surveys and Monographs 96, American Mathematical Society, 2002 MR 1898414 | Zbl 1017.18001
[29] Mohsen Masmoudi & Norbert Poncin, “On a general approach to the formal cohomology of quadratic Poisson structures”, J. Pure Appl. Algebra 208 (2007), p. 887-904 MR 2283433 | Zbl 1174.17018
[30] P. W. Michor & A. M. Vinogradov, “$n$-ary Lie and associative algebras”, Rend. Sem. Mat. Univ. Politec. Torino 54 (1996), p. 373-392, Geometrical structures for physical theories, II (Vietri, 1996) MR 1618134 | Zbl 0928.17029
[31] Yoichiro Nambu, “Generalized Hamiltonian dynamics”, Phys. Rev. D (3) 7 (1973), p. 2405-2412 MR 455611 | Zbl 1027.70503
[32] Albert Nijenhuis & R. W. Richardson, “Deformations of Lie algebra structures”, J. Math. Mech. 17 (1967), p. 89-105 MR 214636 | Zbl 0166.30202
[33] Jean-Michel Oudom, Coproduct and cogroups in the category of graded dual Leibniz algebras, Operads: Proceedings of Renaissance Conferences (Hartford, CT/Luminy, 1995), Amer. Math. Soc., 1997, p. 115–135 MR 1436919 | Zbl 0880.17002
[34] M. Penkava, “Infinity Algebras, Cohomology and Cyclic Cohomology, and Infinitesimal Deformations”, arXiv: math/0111088, 2001
arXiv
[35] Anne Pichereau, “Poisson (co)homology and isolated singularities”, J. Algebra 299 (2006), p. 747-777 MR 2228339 | Zbl 1113.17009
[36] Norbert Poncin, “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 67 (1998), p. 291-337 MR 1677564 | Zbl 0971.17011
[37] Norbert Poncin, “On the cohomology of the Nijenhuis-Richardson graded Lie algebra of the space of functions of a manifold”, J. Algebra 243 (2001), p. 16-40 MR 1851652 | Zbl 1012.17014
[38] Mikołaj Rotkiewicz, “Cohomology ring of $n$-Lie algebras”, Extracta Math. 20 (2005), p. 219-232 MR 2243339 | Zbl 1163.17300
[39] Jim Stasheff, “The intrinsic bracket on the deformation complex of an associative algebra”, J. Pure Appl. Algebra 89 (1993), p. 231-235 MR 1239562 | Zbl 0786.57017
[40] Alexandre Vinogradov & Michael Vinogradov, On multiple generalizations of Lie algebras and Poisson manifolds, Secondary calculus and cohomological physics (Moscow, 1997), Amer. Math. Soc., 1998, p. 273–287 MR 1640439 | Zbl 1074.17501
[41] Alexandre Vinogradov & Michael Vinogradov, “Graded multiple analogs of Lie algebras”, Acta Appl. Math. 72 (2002), p. 183-197, Symmetries of differential equations and related topics MR 1907945 | Zbl 1020.17003
Mourad Ammar; Norbert Poncin
Coalgebraic Approach to the Loday Infinity Category, Stem Differential for $2n$-ary Graded and Homotopy Algebras
(Approche coalgébrique de la catégorie des algèbres de Loday infinies, différentielle souche pour les algèbres graduées $2n$-aires ou à homotopie)
Annales de l'institut Fourier, 60 no. 1 (2010), p. 355-387
Article: sur abonnement
Class. Math.: 16W30, 16E45, 17B56, 17B70
Mots 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
Résumé - Abstract
Nous définissons un coproduit gradué et coassociatif tordu sur l’algèbre tensorielle d’un espace vectoriel gradué $V$. 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 $V$ (resp. structures graduées de Loday sur $V$, suites que nous appelons structures de Loday infinies sur $V$). Nous prouvons un théorème du modèle minimal pour les algèbres infinies de Loday et observons que la catégorie $\text{Lod}_{\!\infty }$ contient la catégorie $\text{L}_{\!\infty }$ 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 $2n$-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 $2n$-aires.
Bibliographie
[2] Mourad Ammar, Deformation Quantization and Cohomologies of Poisson, Graded, and Homotopy Algebras, thesis, University of Luxembourg, Paul Verlaine University of Metz, 2008
[3] Mourad Ammar & Norbert Poncin, “Formal Poisson cohomology of twisted $r$-matrix induced structures”, Israel J. Math. 165 (2008), p. 381-411 MR 2403627 | Zbl 1146.53054
[4] D. Arnal, D. Manchon & M. Masmoudi, “Choix des signes pour la formalité de M. Kontsevich”, Pacific J. Math. 203 (2002), p. 23-66 MR 1895924 | Zbl 1055.53066
[5] David Balavoine, “Déformations et rigidité géométrique des algèbres de Leibniz”, Comm. Algebra 24 (1996), p. 1017-1034 MR 1374652 | Zbl 0855.17021
[6] David Balavoine, Deformations of algebras over a quadratic operad, Operads: Proceedings of Renaissance Conferences (Hartford, CT/Luminy, 1995), Amer. Math. Soc., 1997, p. 207–234 MR 1436922 | Zbl 0883.17004
[7] Yuri L. Daletskii & Leon A. Takhtajan, “Leibniz and Lie algebra structures for Nambu algebra”, Lett. Math. Phys. 39 (1997), p. 127-141 MR 1437747 | Zbl 0869.58024
[8] M. De Wilde & P. Lecomte, 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), Kluwer Acad. Publ., 1988, p. 897–960 MR 981635 | Zbl 0685.58039
[9] Alice Fialowski & Michael Penkava, “Deformation theory of infinity algebras”, J. Algebra 255 (2002), p. 59-88 MR 1935035 | Zbl 1038.17012
[10] V. T. Filippov, “$n$-L$Q\in \mathbb{Z}^n,p\in \mathbb{N}^*$,ie algebras”, Sibirsk. Mat. Zh. 26 (1985), p. 126-140, 191 MR 816511 | Zbl 0585.17002
[11] Murray Gerstenhaber, “On the deformation of rings and algebras”, Ann. of Math. (2) 79 (1964), p. 59-103 MR 171807 | Zbl 0123.03101
[12] Victor Ginzburg & Mikhail Kapranov, “Koszul duality for operads”, Duke Math. J. 76 (1994), p. 203-272
Article | MR 1301191 | Zbl 0855.18006
[13] Janusz Grabowski & Giuseppe Marmo, “Jacobi structures revisited”, J. Phys. A 34 (2001), p. 10975-10990 MR 1872975 | Zbl 0998.53054
[14] Janusz Grabowski & Giuseppe Marmo, “The graded Jacobi algebras and (co)homology”, J. Phys. A 36 (2003), p. 161-181 MR 1959019 | Zbl 1039.53090
[15] R. Ibáñez, M. de León, B. López, J. C. Marrero & E. Padrón, “Duality and modular class of a Nambu-Poisson structure”, J. Phys. A 34 (2001), p. 3623-3650 MR 1841496 | Zbl 1021.53060
[16] T. V. Kadeishvili, “The algebraic structure in the homology of an $A(\infty )$-algebra”, Soobshch. Akad. Nauk Gruzin. SSR 108 (1982), p. 249-252 (1983) MR 720689 | Zbl 0535.55005
[17] Maxim Kontsevich, “Deformation quantization of Poisson manifolds”, Lett. Math. Phys. 66 (2003), p. 157-216
arXiv | MR 2062626 | Zbl 1058.53065
[18] Yvette Kosmann-Schwarzbach, “From Poisson algebras to Gerstenhaber algebras”, Ann. Inst. Fourier (Grenoble) 46 (1996), p. 1243-1274
Cedram | MR 1427124 | Zbl 0858.17027
[19] I. S. Krasilʼshchik, “Supercanonical algebras and Schouten brackets”, Mat. Zametki 49 (1991), p. 70-76, 160 MR 1101552 | Zbl 0723.58020
[20] P. A. B. Lecomte, P. W. Michor & H. Schicketanz, “The multigraded Nijenhuis-Richardson algebra, its universal property and applications”, J. Pure Appl. Algebra 77 (1992), p. 87-102 MR 1148273 | Zbl 0752.17019
[21] Manuel de León, Belén López, Juan C. Marrero & Edith Padrón, “Lichnerowicz-Jacobi cohomology and homology of Jacobi manifolds: modular class and duality”, arXiv: math/9910079, 1999
arXiv
[22] Manuel de León, Belén López, Juan C. Marrero & Edith Padrón, “On the computation of the Lichnerowicz-Jacobi cohomology”, J. Geom. Phys. 44 (2003), p. 507-522 MR 1943175 | Zbl 1092.53060
[23] Manuel de León, Juan C. Marrero & Edith Padrón, “Lichnerowicz-Jacobi cohomology”, J. Phys. A 30 (1997), p. 6029-6055 MR 1482695 | Zbl 0951.37033
[24] André Lichnerowicz, “Les variétés de Poisson et leurs algèbres de Lie associées”, J. Differential Geometry 12 (1977), p. 253-300
Article | MR 501133 | Zbl 0405.53024
[25] André Lichnerowicz, “Les variétés de Jacobi et leurs algèbres de Lie associées”, J. Math. Pures Appl. (9) 57 (1978), p. 453-488 MR 524629 | Zbl 0407.53025
[26] Jean-Louis Loday, “Une version non commutative des algèbres de Lie: les algèbres de Leibniz”, Enseign. Math. (2) 39 (1993), p. 269-293 MR 1252069 | Zbl 0806.55009
[27] Jean-Louis Loday, “Cup-product for Leibniz cohomology and dual Leibniz algebras”, Math. Scand. 77 (1995), p. 189-196 MR 1379265 | Zbl 0859.17015
[28] Martin Markl, Steve Shnider & Jim Stasheff, Operads in algebra, topology and physics, Mathematical Surveys and Monographs 96, American Mathematical Society, 2002 MR 1898414 | Zbl 1017.18001
[29] Mohsen Masmoudi & Norbert Poncin, “On a general approach to the formal cohomology of quadratic Poisson structures”, J. Pure Appl. Algebra 208 (2007), p. 887-904 MR 2283433 | Zbl 1174.17018
[30] P. W. Michor & A. M. Vinogradov, “$n$-ary Lie and associative algebras”, Rend. Sem. Mat. Univ. Politec. Torino 54 (1996), p. 373-392, Geometrical structures for physical theories, II (Vietri, 1996) MR 1618134 | Zbl 0928.17029
[31] Yoichiro Nambu, “Generalized Hamiltonian dynamics”, Phys. Rev. D (3) 7 (1973), p. 2405-2412 MR 455611 | Zbl 1027.70503
[32] Albert Nijenhuis & R. W. Richardson, “Deformations of Lie algebra structures”, J. Math. Mech. 17 (1967), p. 89-105 MR 214636 | Zbl 0166.30202
[33] Jean-Michel Oudom, Coproduct and cogroups in the category of graded dual Leibniz algebras, Operads: Proceedings of Renaissance Conferences (Hartford, CT/Luminy, 1995), Amer. Math. Soc., 1997, p. 115–135 MR 1436919 | Zbl 0880.17002
[34] M. Penkava, “Infinity Algebras, Cohomology and Cyclic Cohomology, and Infinitesimal Deformations”, arXiv: math/0111088, 2001
arXiv
[35] Anne Pichereau, “Poisson (co)homology and isolated singularities”, J. Algebra 299 (2006), p. 747-777 MR 2228339 | Zbl 1113.17009
[36] Norbert Poncin, “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 67 (1998), p. 291-337 MR 1677564 | Zbl 0971.17011
[37] Norbert Poncin, “On the cohomology of the Nijenhuis-Richardson graded Lie algebra of the space of functions of a manifold”, J. Algebra 243 (2001), p. 16-40 MR 1851652 | Zbl 1012.17014
[38] Mikołaj Rotkiewicz, “Cohomology ring of $n$-Lie algebras”, Extracta Math. 20 (2005), p. 219-232 MR 2243339 | Zbl 1163.17300
[39] Jim Stasheff, “The intrinsic bracket on the deformation complex of an associative algebra”, J. Pure Appl. Algebra 89 (1993), p. 231-235 MR 1239562 | Zbl 0786.57017
[40] Alexandre Vinogradov & Michael Vinogradov, On multiple generalizations of Lie algebras and Poisson manifolds, Secondary calculus and cohomological physics (Moscow, 1997), Amer. Math. Soc., 1998, p. 273–287 MR 1640439 | Zbl 1074.17501
[41] Alexandre Vinogradov & Michael Vinogradov, “Graded multiple analogs of Lie algebras”, Acta Appl. Math. 72 (2002), p. 183-197, Symmetries of differential equations and related topics MR 1907945 | Zbl 1020.17003
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