Input/output coloring and Gröbner basis for dioperads
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866915882892001280 |
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| author | Khoroshkin, Anton |
| author_facet | Khoroshkin, Anton |
| contents | We introduce a functor $Ψ$ that associates to a dioperad $P$ acting on a vector space $V$ a two-colored operad $Ψ(P)$ acting on the pair $(V, V^*)$. The construction is based on a simple pictorial idea: by selecting one input or output and dualizing, if necessary, the remaining ones, any dioperadic tree can be ``rerooted'' as a colored operadic tree. This transformation allows one to apply the standard operadic machinery -- such as Gröbner bases and Hilbert series -- to the study of dioperads.
We illustrate the method with several examples and applications. (1) We compute the dimensions of the spaces of operations for the dioperad of Lie bialgebras. (2) We describe a Gröbner basis and construct a minimal resolution for the dioperad of triangular Lie bialgebras. (3) We perform explicit computations for the dioperad of ``algebraic string operations''. (4) We give a pictorial construction proving the existence of quadratic Gröbner bases and establishing the Koszul property for a broad class of dioperads arising from cyclic operads. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_20069 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Input/output coloring and Gröbner basis for dioperads Khoroshkin, Anton Quantum Algebra Algebraic Topology Combinatorics 18M70, 18M85, 16S15 We introduce a functor $Ψ$ that associates to a dioperad $P$ acting on a vector space $V$ a two-colored operad $Ψ(P)$ acting on the pair $(V, V^*)$. The construction is based on a simple pictorial idea: by selecting one input or output and dualizing, if necessary, the remaining ones, any dioperadic tree can be ``rerooted'' as a colored operadic tree. This transformation allows one to apply the standard operadic machinery -- such as Gröbner bases and Hilbert series -- to the study of dioperads. We illustrate the method with several examples and applications. (1) We compute the dimensions of the spaces of operations for the dioperad of Lie bialgebras. (2) We describe a Gröbner basis and construct a minimal resolution for the dioperad of triangular Lie bialgebras. (3) We perform explicit computations for the dioperad of ``algebraic string operations''. (4) We give a pictorial construction proving the existence of quadratic Gröbner bases and establishing the Koszul property for a broad class of dioperads arising from cyclic operads. |
| title | Input/output coloring and Gröbner basis for dioperads |
| topic | Quantum Algebra Algebraic Topology Combinatorics 18M70, 18M85, 16S15 |
| url | https://arxiv.org/abs/2602.20069 |