A co-preLie structure from chronological loop erasure in graph walks
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866916302646411264 |
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| author | Foissy, Loïc Giscard, Pierre-Louis Mammez, Cécile |
| author_facet | Foissy, Loïc Giscard, Pierre-Louis Mammez, Cécile |
| contents | We show that the chronological removal of cycles from a walk on a graph, known as Lawler's loop-erasing procedure, generates a preLie co-algebra on the vector space spanned by the walks. In addition, we prove that the tensor and symmetric algebras of graph walks are graded Hopf algebras, provide their antipodes explicitly and recover the preLie co-algebra from a brace coalgebra on the tensor algebra of graph walks. Finally we exhibit sub-Hopf algebras associated to particular types of walks. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_17605 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A co-preLie structure from chronological loop erasure in graph walks Foissy, Loïc Giscard, Pierre-Louis Mammez, Cécile Combinatorics Rings and Algebras 05C38, 16T30, 16T15 We show that the chronological removal of cycles from a walk on a graph, known as Lawler's loop-erasing procedure, generates a preLie co-algebra on the vector space spanned by the walks. In addition, we prove that the tensor and symmetric algebras of graph walks are graded Hopf algebras, provide their antipodes explicitly and recover the preLie co-algebra from a brace coalgebra on the tensor algebra of graph walks. Finally we exhibit sub-Hopf algebras associated to particular types of walks. |
| title | A co-preLie structure from chronological loop erasure in graph walks |
| topic | Combinatorics Rings and Algebras 05C38, 16T30, 16T15 |
| url | https://arxiv.org/abs/2306.17605 |