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Weighted infinitesimal unitary bialgebras of rooted forests, symmetric cocycles and pre-Lie algebras
The concept of weighted infinitesimal unitary bialgebra is an algebraic meaning of the nonhomogenous associative Yang–Baxter equation. In this paper, we equip the space of decorated planar rooted forests with a coproduct which makes it a weighted infinitesimal unitary bialgebra. Further, we construc...
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Published in: | Journal of algebraic combinatorics 2021-05, Vol.53 (3), p.771-803 |
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description | The concept of weighted infinitesimal unitary bialgebra is an algebraic meaning of the nonhomogenous associative Yang–Baxter equation. In this paper, we equip the space of decorated planar rooted forests with a coproduct which makes it a weighted infinitesimal unitary bialgebra. Further, we construct an infinitesimal unitary Hopf algebra on decorated planar rooted forests in the sense of Loday and Ronco. We then introduce the concept of symmetric 1-cocycle condition, which is derived from the dual of the Hochschild cohomology. We study the universal properties of the space of decorated planar rooted forests with the symmetric 1-cocycle, leading to the notation of a weighted
Ω
-cocycle infinitesimal unitary bialgebra. As an application, we obtain the initial object in the category of free cocycle infinitesimal unitary bialgebras on the undecorated planar rooted forests, which is the object studied in the well-known noncommutative Connes–Kreimer Hopf algebra. Finally, we construct a pre-Lie algebra on decorated planar rooted forests. |
doi_str_mv | 10.1007/s10801-020-00942-7 |
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Ω
-cocycle infinitesimal unitary bialgebra. As an application, we obtain the initial object in the category of free cocycle infinitesimal unitary bialgebras on the undecorated planar rooted forests, which is the object studied in the well-known noncommutative Connes–Kreimer Hopf algebra. Finally, we construct a pre-Lie algebra on decorated planar rooted forests.</description><identifier>ISSN: 0925-9899</identifier><identifier>EISSN: 1572-9192</identifier><identifier>DOI: 10.1007/s10801-020-00942-7</identifier><language>eng</language><publisher>New York: Springer US</publisher><subject>Algebra ; Combinatorics ; Computer Science ; Convex and Discrete Geometry ; Decoration ; Forests ; Group Theory and Generalizations ; Homology ; Lattices ; Lie groups ; Mathematics ; Mathematics and Statistics ; Order ; Ordered Algebraic Structures</subject><ispartof>Journal of algebraic combinatorics, 2021-05, Vol.53 (3), p.771-803</ispartof><rights>Springer Science+Business Media, LLC, part of Springer Nature 2020</rights><rights>Springer Science+Business Media, LLC, part of Springer Nature 2020.</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-c314t-798f9075a2c788448f3a09ccce7275c83e3b2c1f428482f2cec4b7d36e347f063</cites><orcidid>0000-0003-0513-334X ; 0000-0002-2367-2497</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>314,780,784,27924,27925</link.rule.ids></links><search><creatorcontrib>Zhang, Yi</creatorcontrib><creatorcontrib>Gao, Xing</creatorcontrib><creatorcontrib>Luo, Yanfeng</creatorcontrib><title>Weighted infinitesimal unitary bialgebras of rooted forests, symmetric cocycles and pre-Lie algebras</title><title>Journal of algebraic combinatorics</title><addtitle>J Algebr Comb</addtitle><description>The concept of weighted infinitesimal unitary bialgebra is an algebraic meaning of the nonhomogenous associative Yang–Baxter equation. In this paper, we equip the space of decorated planar rooted forests with a coproduct which makes it a weighted infinitesimal unitary bialgebra. Further, we construct an infinitesimal unitary Hopf algebra on decorated planar rooted forests in the sense of Loday and Ronco. We then introduce the concept of symmetric 1-cocycle condition, which is derived from the dual of the Hochschild cohomology. We study the universal properties of the space of decorated planar rooted forests with the symmetric 1-cocycle, leading to the notation of a weighted
Ω
-cocycle infinitesimal unitary bialgebra. As an application, we obtain the initial object in the category of free cocycle infinitesimal unitary bialgebras on the undecorated planar rooted forests, which is the object studied in the well-known noncommutative Connes–Kreimer Hopf algebra. Finally, we construct a pre-Lie algebra on decorated planar rooted forests.</description><subject>Algebra</subject><subject>Combinatorics</subject><subject>Computer Science</subject><subject>Convex and Discrete Geometry</subject><subject>Decoration</subject><subject>Forests</subject><subject>Group Theory and Generalizations</subject><subject>Homology</subject><subject>Lattices</subject><subject>Lie groups</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Order</subject><subject>Ordered Algebraic Structures</subject><issn>0925-9899</issn><issn>1572-9192</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2021</creationdate><recordtype>article</recordtype><recordid>eNp9kMtOAyEUhonRxFp9AVckbkW5FliaxlvSxI3GJWEYqDTTocJ0MW8vdTTuXJ2z-L5z-QG4JPiGYCxvC8EKE4QpRhhrTpE8AjMiJEWaaHoMZlhTgbTS-hSclbLBlVJEzED77uP6Y_AtjH2IfRx8iVvbwX1tbR5hE2239k22BaYAc0oHNKTsy1CuYRm3Wz_k6KBLbnSdL9D2Ldxlj1bRw1_1HJwE2xV_8VPn4O3h_nX5hFYvj8_LuxVyjPABSa2CxlJY6qRSnKvALNbOOS-pFE4xzxrqSOBUcUUDdd7xRrZs4RmXAS_YHFxNc3c5fe7riWaT9rmvKw0VdMGIYFhUik6Uy6mU7IPZ5fpzHg3B5pCmmdI0NU3znaaRVWKTVCrcr33-G_2P9QVxt3iJ</recordid><startdate>20210501</startdate><enddate>20210501</enddate><creator>Zhang, Yi</creator><creator>Gao, Xing</creator><creator>Luo, Yanfeng</creator><general>Springer US</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><orcidid>https://orcid.org/0000-0003-0513-334X</orcidid><orcidid>https://orcid.org/0000-0002-2367-2497</orcidid></search><sort><creationdate>20210501</creationdate><title>Weighted infinitesimal unitary bialgebras of rooted forests, symmetric cocycles and pre-Lie algebras</title><author>Zhang, Yi ; Gao, Xing ; Luo, Yanfeng</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c314t-798f9075a2c788448f3a09ccce7275c83e3b2c1f428482f2cec4b7d36e347f063</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2021</creationdate><topic>Algebra</topic><topic>Combinatorics</topic><topic>Computer Science</topic><topic>Convex and Discrete Geometry</topic><topic>Decoration</topic><topic>Forests</topic><topic>Group Theory and Generalizations</topic><topic>Homology</topic><topic>Lattices</topic><topic>Lie groups</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Order</topic><topic>Ordered Algebraic Structures</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Zhang, Yi</creatorcontrib><creatorcontrib>Gao, Xing</creatorcontrib><creatorcontrib>Luo, Yanfeng</creatorcontrib><collection>CrossRef</collection><jtitle>Journal of algebraic combinatorics</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Zhang, Yi</au><au>Gao, Xing</au><au>Luo, Yanfeng</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Weighted infinitesimal unitary bialgebras of rooted forests, symmetric cocycles and pre-Lie algebras</atitle><jtitle>Journal of algebraic combinatorics</jtitle><stitle>J Algebr Comb</stitle><date>2021-05-01</date><risdate>2021</risdate><volume>53</volume><issue>3</issue><spage>771</spage><epage>803</epage><pages>771-803</pages><issn>0925-9899</issn><eissn>1572-9192</eissn><abstract>The concept of weighted infinitesimal unitary bialgebra is an algebraic meaning of the nonhomogenous associative Yang–Baxter equation. In this paper, we equip the space of decorated planar rooted forests with a coproduct which makes it a weighted infinitesimal unitary bialgebra. Further, we construct an infinitesimal unitary Hopf algebra on decorated planar rooted forests in the sense of Loday and Ronco. We then introduce the concept of symmetric 1-cocycle condition, which is derived from the dual of the Hochschild cohomology. We study the universal properties of the space of decorated planar rooted forests with the symmetric 1-cocycle, leading to the notation of a weighted
Ω
-cocycle infinitesimal unitary bialgebra. As an application, we obtain the initial object in the category of free cocycle infinitesimal unitary bialgebras on the undecorated planar rooted forests, which is the object studied in the well-known noncommutative Connes–Kreimer Hopf algebra. Finally, we construct a pre-Lie algebra on decorated planar rooted forests.</abstract><cop>New York</cop><pub>Springer US</pub><doi>10.1007/s10801-020-00942-7</doi><tpages>33</tpages><orcidid>https://orcid.org/0000-0003-0513-334X</orcidid><orcidid>https://orcid.org/0000-0002-2367-2497</orcidid><oa>free_for_read</oa></addata></record> |
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subjects | Algebra Combinatorics Computer Science Convex and Discrete Geometry Decoration Forests Group Theory and Generalizations Homology Lattices Lie groups Mathematics Mathematics and Statistics Order Ordered Algebraic Structures |
title | Weighted infinitesimal unitary bialgebras of rooted forests, symmetric cocycles and pre-Lie algebras |
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