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Compatible Structures of Nonsymmetric Operads, Manin Products and Koszul Duality

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Abstract

Various compatibility conditions among replicated copies of operations in a given algebraic structure have appeared in broad contexts in recent years. Taking a uniform approach, this paper presents an operadic study of compatibility conditions for nonsymmetric operads with unary and binary operations, and homogeneous quadratic and cubic relations. This generalizes the previous studies for binary quadratic operads. We consider three compatibility conditions, namely the linear compatibility, matching compatibility and total compatibility, with increasingly stronger restraints among the replicated copies. The linear compatibility is in Koszul duality to the total compatibility, while the matching compatibility is self dual. Further, each compatibility condition can be expressed in terms of either one or both of the two Manin square products. Finally it is shown that the operads defined by these compatibility conditions from the associative algebra and differential algebra are Koszul utilizing rewriting systems.

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Acknowledgements

This work is supported by the National Natural Science Foundation of China (No. 12071191), the Innovative Fundamental Research Group Project of Gansu Province (23JRRA684) and Natural Science Project of Gansu Province (No. 22JR11RA138). The authors thank the referee for helpful suggestions.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu, 730000, China

    Huhu Zhang & Xing Gao

  2. Gansu Provincial Research Center for Basic Disciplines of Mathematics and Statistics, Lanzhou, 730070, China

    Xing Gao

  3. School of Mathematics and Statistics, Qinghai Nationalities University, Xining, 810007, China

    Xing Gao

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

    Li Guo

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  1. Huhu Zhang
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Correspondence to Li Guo.

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Zhang, H., Gao, X. & Guo, L. Compatible Structures of Nonsymmetric Operads, Manin Products and Koszul Duality. Appl Categor Struct 32, 2 (2024). https://doi.org/10.1007/s10485-023-09760-x

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