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Splitting of Operads and Rota-Baxter Operators on Operads

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Abstract

This paper establishes a procedure that splits the operations in any algebraic operad, generalizing previous notions of splitting algebraic structures, from the dendriform algebra of Loday splitting the associative operation to the successors splitting binary operads. The separately treated bisuccessor and trisuccessor for binary operads are unified for general operads through the notion of configuration. Applications are provided for various n-algebras, the \(A_{\infty }\) and \(L_{\infty }\) algebras. Further, the concept of a Rota-Baxter operator, first showing its importance in the associative and Lie algebra contexts and then generalized to binary operads, is defined for all operads. The well-known connection from Rota-Baxter operators to dendriform algebras and its numerous extensions are expanded as the link from (relative) Rota-Baxter operators on operads to splittings of the operads

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Authors and Affiliations

  1. Chern Institute of Mathematics & LPMC, Nankai University, Tianjin, 300071, China

    Chengming Bai

  2. Department of Mathematics, Jiangxi Normal University, Nanchang, Jiangxi, 330022, China

    Li Guo

  3. Department of Mathematics and Computer Science, Rutgers University, Newark, NJ, 07102, USA

    Li Guo

  4. School of Mathematics and Statistics, Southwest University, Chongqing, 400715, China

    Jun Pei

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  1. Jun Pei
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  2. Chengming Bai
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  3. Li Guo
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Correspondence to Li Guo.

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Pei, J., Bai, C. & Guo, L. Splitting of Operads and Rota-Baxter Operators on Operads. Appl Categor Struct 25, 505–538 (2017). https://doi.org/10.1007/s10485-016-9431-5

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