Following a recent work [Bai C., Bellier O., Guo L., Ni X., Splitting of operations, Manin products, and Rota-Baxter operators, Int. Math. Res. Not. IMRN (in press), DOI: 10.1093/imrn/rnr266] we define what is a dendriform dior trialgebra corresponding to an arbitrary variety Var of binary algebras (associative, commutative, Poisson, etc.). We call such algebras di- or tri-Var-dendriform algebras, respectively. We prove in general that the operad governing the variety of di- or tri-Var-dendriform algebras is Koszul dual to the operad governing di- or trialgebras corresponding to Var!. We also prove that every di-Var-dendriform algebra can be embedded into a Rota-Baxter algebra of weight zero in the variety Var, and every tri-Var-dendriform algebra can be embedded into a Rota-Baxter algebra of nonzero weight in Var.
@article{bwmeta1.element.doi-10_2478_s11533-012-0138-z,
author = {Vsevolod Gubarev and Pavel Kolesnikov},
title = {Embedding of dendriform algebras into Rota-Baxter algebras},
journal = {Open Mathematics},
volume = {11},
year = {2013},
pages = {226-245},
zbl = {1262.18009},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-012-0138-z}
}
Vsevolod Gubarev; Pavel Kolesnikov. Embedding of dendriform algebras into Rota-Baxter algebras. Open Mathematics, Tome 11 (2013) pp. 226-245. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-012-0138-z/
[1] Aguiar M., Pre-Poisson algebras, Lett. Math. Phys., 2000, 54(4), 263–277 http://dx.doi.org/10.1023/A:1010818119040
[2] Andrews G.E., Guo L., Keigher W., Ono K., Baxter algebras and Hopf algebras, Trans. Amer. Math. Soc., 2003, 355(11), 4639–4656 http://dx.doi.org/10.1090/S0002-9947-03-03326-9 | Zbl 1056.16025
[3] Bai C., Bellier O., Guo L., Ni X., Splitting of operations, Manin products, and Rota-Baxter operators, Int. Math. Res. Not. IMRN (in press), DOI: 10.1093/imrn/rnr266 | Zbl 1314.18010
[4] Bai C., Guo L., Ni X., O-operators on associative algebras and dendriform algebras, preprint available at http://arxiv.org/abs/1003.2432 | Zbl 1300.16036
[5] Bai C., Liu L., Ni X., Some results on L-dendriform algebras, J. Geom. Phys., 2010, 60(6–8), 940–950 http://dx.doi.org/10.1016/j.geomphys.2010.02.007 | Zbl 1207.17002
[6] Bakalov B., D’Andrea A., Kac V.G., Theory of finite pseudoalgebras, Adv. Math., 2001, 162(1), 1–140 http://dx.doi.org/10.1006/aima.2001.1993 | Zbl 1001.16021
[7] Baxter G., An analytic problem whose solution follows from a simple algebraic identity, Pacific J. Math., 1960, 10, 731–742 | Zbl 0095.12705
[8] Belavin A.A., Drinfel’d V.G., Solutions of the classical Yang-Baxter equation for simple Lie algebras, Funct. Anal. Appl., 1982, 16(3), 159–180 http://dx.doi.org/10.1007/BF01081585
[9] Bokut L.A., Chen Yu., Deng X., Gröbner-Shirshov bases for Rota-Baxter algebras, Sib. Math. J., 2010, 51(6), 978–988 http://dx.doi.org/10.1007/s11202-010-0097-1 | Zbl 1235.16021
[10] Cartier P., On the structure of free Baxter algebras, Adv. in Math., 1972, 9, 253–265 http://dx.doi.org/10.1016/0001-8708(72)90018-7
[11] Chapoton F., Un endofoncteur de la catégorie des opérades, In: Dialgebras and Related Operads, Lecture Notes in Math., 1763, Springer, Berlin, 2001, 105–110 http://dx.doi.org/10.1007/3-540-45328-8_4
[12] Chen Y., Mo Q., Embedding dendriform algebra into its universal enveloping Rota-Baxter algebra, Proc. Amer. Math. Soc., 2011, 139(12), 4207–4216 http://dx.doi.org/10.1090/S0002-9939-2011-10889-X | Zbl 1254.17001
[13] Connes A., Kreimer D., Renormalization in quantum field theory and the Riemann-Hilbert problem. I: The Hopf algebra structure of graphs and the main theorem, Comm. Math. Phys., 2000, 210(1), 249–273 http://dx.doi.org/10.1007/s002200050779 | Zbl 1032.81026
[14] Connes A., Kreimer D., Renormalization in quantum field theory and the Riemann-Hilbert problem. II: The β-function, diffeomorphisms and the renormalization group, Comm. Math. Phys., 2001, 216(1), 215–241 http://dx.doi.org/10.1007/PL00005547 | Zbl 1042.81059
[15] Ebrahimi-Fard K., Loday-type algebras and the Rota-Baxter relation, Lett. Math. Phys., 2002, 61(2), 139–147 http://dx.doi.org/10.1023/A:1020712215075 | Zbl 1035.17001
[16] Ebrahimi-Fard K., Guo L., On products and duality of binary, quadratic, regular operads, J. Pure Appl. Algebra, 2005, 200(3), 293–317 http://dx.doi.org/10.1016/j.jpaa.2004.12.020 | Zbl 1082.18007
[17] Ebrahimi-Fard K., Guo L., Mixable shuffles, quasi-shuffles and Hopf algebras, J. Algebraic Combin., 2006, 24(1), 83–101 http://dx.doi.org/10.1007/s10801-006-9103-x | Zbl 1103.16025
[18] Ebrahimi-Fard K., Guo L., Rota-Baxter algebras and dendriform algebras, J. Pure Appl. Algebra, 2008, 212(2), 320–339 http://dx.doi.org/10.1016/j.jpaa.2007.05.025 | Zbl 1132.16032
[19] Ginzburg V., Kapranov M., Koszul duality for operads, Duke Math. J., 1994, 76(1), 203–272 http://dx.doi.org/10.1215/S0012-7094-94-07608-4
[20] Golenishcheva-Kutuzova M.I., Kac V.G., Γ-conformal algebras, J. Math. Phys., 1998, 39(4), 2290–2305 http://dx.doi.org/10.1063/1.532289 | Zbl 1031.81527
[21] Gubarev V.Yu., Kolesnikov P.S., The Tits-Kantor-Koecher construction for Jordan dialgebras, Comm. Algebra, 2011, 39(2), 497–520 http://dx.doi.org/10.1080/00927871003591967 | Zbl 1272.17032
[22] Guo L., An Introduction to Rota-Baxter Algebra, available at http://math.newark.rutgers.edu/_liguo/rbabook.pdf | Zbl 1271.16001
[23] Kolesnikov P., Identities of conformal algebras and pseudoalgebras, Comm. Algebra, 2006, 34(6), 1965–1979 http://dx.doi.org/10.1080/00927870500542945 | Zbl 1144.17020
[24] Kolesnikov P.S., Varieties of dialgebras and conformal algebras, Sib. Math. J., 2008, 49(2), 257–272 http://dx.doi.org/10.1007/s11202-008-0026-8 | Zbl 1164.17002
[25] Leinster T., Higher Operads, Higher Categories, London Math. Soc. Lecture Note Ser., 298, Cambridge University Press, Cambridge, 2004 | Zbl 1160.18001
[26] Leroux P., Construction of Nijenhuis operators and dendriform trialgebras, Int. J. Math. Math. Sci., 2004, 49–52, 2595–2615 http://dx.doi.org/10.1155/S0161171204402117 | Zbl 1116.17002
[27] Loday J.-L., Une version non commutative des algèbres de Lie: les algèbres de Leibniz, Enseign. Math., 1993, 39(3–4), 269–293
[28] Loday J.-L., Dialgebras, In: Dialgebras and Related Operads, Lecture Notes in Math., 1763, Springer, Berlin, 2001, 7–66
[29] Loday J.-L., Pirashvili T., Universal enveloping algebras of Leibniz algebras and (co)homology, Math. Ann., 1993, 296(1), 139–158 http://dx.doi.org/10.1007/BF01445099 | Zbl 0821.17022
[30] Loday J.-L., Ronco M., Trialgebras and families of polytopes, In: Homotopy Theory: Relations with Algebraic Geometry, Group Cohomology, and Algebraic K-Theory, Contemp. Math., 346, American Mathematical Society, Providence, 2004, 369–398 http://dx.doi.org/10.1090/conm/346/06296
[31] Loday J.-L., Vallette B., Algebraic Operads, Grundlehren Math. Wiss., 346, Springer, Heidelberg, 2012 | Zbl 1260.18001
[32] May J.P., Geometry of Iterated Loop Spaces, Lecture Notes in Math., 271, Springer, Berlin-New York, 1972 | Zbl 0244.55009
[33] Pozhidaev A.P., 0-dialgebras with bar-unity, Rota-Baxter and 3-Leibniz algebras, In: Groups, Rings and Group Rings, Contemp. Math., 499, American Mathematical Society, Providence, 2009, 245–256 | Zbl 1323.17007
[34] Rota G.-C., Baxter algebras and combinatorial identities I, II, Bull. Amer. Math. Soc., 1969, 75(2), 325–329, 330–334 http://dx.doi.org/10.1090/S0002-9904-1969-12156-7
[35] Semenov-Tyan-Shanskii M.A., What is a classical r-matrix?, Funct. Anal. Appl., 1983, 17(4), 259–272 http://dx.doi.org/10.1007/BF01076717 | Zbl 0535.58031
[36] Spitzer F., A combinatorial lemma and its application to probability theory, Trans. Amer. Math. Soc., 1956, 82, 323–339 http://dx.doi.org/10.1090/S0002-9947-1956-0079851-X | Zbl 0071.13003
[37] Stasheff J., What is … an operad? Notices Amer. Math. Soc., 2004, 51(6), 630–631 | Zbl 1151.18301
[38] Uchino K., Derived bracket construction and Manin products, Lett. Math. Phys., 2010, 90(1), 37–53 http://dx.doi.org/10.1007/s11005-010-0400-x | Zbl 1198.18001
[39] Uchino K., On distributive laws in derived bracket construction, preprint available at http://arxiv.org/abs/1110.4188v1
[40] Vallette B., Homology of generalized partition posets, J. Pure Appl. Algebra, 2007, 208(2), 699–725 http://dx.doi.org/10.1016/j.jpaa.2006.03.012 | Zbl 1109.18002
[41] Vallette B., Manin products, Koszul duality, Loday algebras and Deligne conjecture, J. Reine Angew. Math., 2008, 620, 105–164 | Zbl 1159.18001
[42] Voronin V., Special and exceptional Jordan dialgebras, J. Algebra Appl., 2012, 11(2), #1250029 | Zbl 1300.17022
[43] Zinbiel G.W., Encyclopedia of types of algebras 2010, preprint available at http://arxiv.org/abs/1101.0267