Anti-dendriform algebras, new splitting of operations and Novikov-type algebras
We introduce the notion of an anti-dendriform algebra as a new approach of splitting the associativity. It is characterized as the algebra with two multiplications giving their left and right multiplication operators, respectively, such that the opposites of these operators define a bimodule structu...
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| Published in | Journal of algebraic combinatorics Vol. 59; no. 3; pp. 661 - 696 |
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| Main Authors | , , |
| Format | Journal Article |
| Language | English |
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| ISSN | 0925-9899 1572-9192 |
| DOI | 10.1007/s10801-024-01303-4 |
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| Abstract | We introduce the notion of an anti-dendriform algebra as a new approach of splitting the associativity. It is characterized as the algebra with two multiplications giving their left and right multiplication operators, respectively, such that the opposites of these operators define a bimodule structure on the sum of these two multiplications, which is associative. This justifies the terminology due to a closely analogous characterization of a dendriform algebra. The notions of anti-
O
-operators and anti-Rota–Baxter operators on associative algebras are introduced to interpret anti-dendriform algebras. In particular, there are compatible anti-dendriform algebra structures on associative algebras with nondegenerate commutative Connes cocycles. There is an important observation that there are correspondences between certain subclasses of dendriform and anti-dendriform algebras in terms of
q
-algebras. As a direct consequence, we give the notion of Novikov-type dendriform algebras as an analogue of Novikov algebras for dendriform algebras, whose relationship with Novikov algebras is consistent with the one between dendriform and pre-Lie algebras. Finally, we extend to provide a general framework of introducing the notions of analogues of anti-dendriform algebras, which interprets a new splitting of operations. |
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| AbstractList | We introduce the notion of an anti-dendriform algebra as a new approach of splitting the associativity. It is characterized as the algebra with two multiplications giving their left and right multiplication operators, respectively, such that the opposites of these operators define a bimodule structure on the sum of these two multiplications, which is associative. This justifies the terminology due to a closely analogous characterization of a dendriform algebra. The notions of anti-
O
-operators and anti-Rota–Baxter operators on associative algebras are introduced to interpret anti-dendriform algebras. In particular, there are compatible anti-dendriform algebra structures on associative algebras with nondegenerate commutative Connes cocycles. There is an important observation that there are correspondences between certain subclasses of dendriform and anti-dendriform algebras in terms of
q
-algebras. As a direct consequence, we give the notion of Novikov-type dendriform algebras as an analogue of Novikov algebras for dendriform algebras, whose relationship with Novikov algebras is consistent with the one between dendriform and pre-Lie algebras. Finally, we extend to provide a general framework of introducing the notions of analogues of anti-dendriform algebras, which interprets a new splitting of operations. We introduce the notion of an anti-dendriform algebra as a new approach of splitting the associativity. It is characterized as the algebra with two multiplications giving their left and right multiplication operators, respectively, such that the opposites of these operators define a bimodule structure on the sum of these two multiplications, which is associative. This justifies the terminology due to a closely analogous characterization of a dendriform algebra. The notions of anti-O-operators and anti-Rota–Baxter operators on associative algebras are introduced to interpret anti-dendriform algebras. In particular, there are compatible anti-dendriform algebra structures on associative algebras with nondegenerate commutative Connes cocycles. There is an important observation that there are correspondences between certain subclasses of dendriform and anti-dendriform algebras in terms of q-algebras. As a direct consequence, we give the notion of Novikov-type dendriform algebras as an analogue of Novikov algebras for dendriform algebras, whose relationship with Novikov algebras is consistent with the one between dendriform and pre-Lie algebras. Finally, we extend to provide a general framework of introducing the notions of analogues of anti-dendriform algebras, which interprets a new splitting of operations. |
| Author | Gao, Dongfang Liu, Guilai Bai, Chengming |
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| Cites_doi | 10.2478/s11533-006-0014-9 10.1007/s00013-003-0796-y 10.1016/S0764-4442(97)84587-9 10.2140/pjm.1960.10.731 10.1007/s10958-009-9532-x 10.1093/imrn/rnr266 10.2140/pjm.2012.256.257 10.1016/j.aim.2005.12.004 10.1016/j.jpaa.2004.01.002 10.1007/3-540-45328-8_4 10.1016/j.jalgebra.2010.04.030 10.1007/BF02698807 10.1016/j.jalgebra.2022.07.004 10.1016/S0021-8693(02)00097-2 10.4171/jncg/64 10.1016/j.jalgebra.2004.06.022 10.1016/S0007-4497(02)01113-2 10.1016/S0022-4049(01)00052-4 10.1007/BF01078363 10.1007/3-540-45328-8_2 10.1023/A:1015064508594 10.1007/s11005-008-0259-2 10.1142/S0219498813500278 10.1002/9781119818175.ch7 10.1006/aima.1998.1759 10.1090/conm/346/06296 10.1016/S0021-8693(02)00510-0 10.1016/S0022-4049(97)00066-2 |
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| Keywords | Dendriform algebra Commutative Connes cocycle 17A36 17B40 17B63 Associative algebra 17B10 17B60 17A40 Anti-dendriform algebra 17D25 Novikov algebra |
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| References_xml | – reference: Bai, C.: An introduction to pre-Lie algebras. In: Algebra and Applications 1: Nonssociative Algebras and Categories, Wiley Online Library 245–273 (2021) – reference: FrabettiALeibniz homology of dialgebras of matricesJ. Pure Appl. Algebra1998129123141162444610.1016/S0022-4049(97)00066-2 – reference: HoltkampRComparison of Hopf algebras on treesArch. Math. (Basel)200380368383198283710.1007/s00013-003-0796-y – reference: Loday, J.-L., Ronco, M.: Trialgebras and families of polytopes. In: Homotopy Theory: Relations with Algebraic Geometry, Group Cohomology, and Algebraic K-theory, Comtep. Math. 346, pp. 369–398 (2004) – reference: Loday, J.-L.: Scindement d’associativité et algèbres de Hopf, Actes des Journées Mathématiques à la Mémoire de Jean Leray, Sémin. Congr. 9, Soc. Math. France, Paris 155–172 (2004) – reference: Aguiar, M., Loday, J.-L.: Quadri-algebras. J. Pure Appl. Algebra 191, 205–221 (2004) – reference: BurdeDLeft-symmetric algebras and pre-Lie algebras in geometry and physicsCent. Eur. J. Math.20064323357223385410.2478/s11533-006-0014-9 – reference: LerouxPEnnea-algebrasJ. Algebra2004281287302209197210.1016/j.jalgebra.2004.06.022 – reference: ChapotonFUn théorème de Cartier-Milnor-Moore-Quillen pour les bigèbres dendriformes et les algèbres bracesJ. Pure Appl. Algebra2002168118187992710.1016/S0022-4049(01)00052-4 – reference: BaiCGuoLNiXO\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\cal{O} }$$\end{document}-operators on associative algebras and associative Yang-Baxter equationsPac. J. Math.2012256257289294497610.2140/pjm.2012.256.257 – reference: LodayJ-LArithmetreeJ. Algebra2002258275309195890710.1016/S0021-8693(02)00510-0 – reference: Gel’fandIMDorfmanI. 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