Algebra of Path Integrals on Digraphs
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866914363085946880 |
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| author | Yau, Shing-Tung Zhang, Mengmeng Zi, Yunpeng |
| author_facet | Yau, Shing-Tung Zhang, Mengmeng Zi, Yunpeng |
| contents | In this paper, we extend the iterated integrals from smooth manifolds to digraphs and develop the associated algebraic and geometric structures. Iterated integrals on a digraph naturally give rise to the iterated path algebra and the iterated loop algebra, both defined as quotient algebras of a shuffle algebra, with the latter carrying a canonical Hopf algebra structure.
We construct a non-degenerate pairing between elementarily equivalent classes of loops on a digraph and the iterated loop algebra. By restricting to iterated integrals that are invariant under $C_\partial$-homotopy, a distinguished subalgebra is obtained which, under this pairing, corresponds to the group algebra of the fundamental group. We further show that this subalgebra is a homotopy invariant and forms a Hopf algebra with involutive antipode. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_01531 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Algebra of Path Integrals on Digraphs Yau, Shing-Tung Zhang, Mengmeng Zi, Yunpeng Algebraic Topology Combinatorics 05C20, 16T05, 53A70, 55P35 In this paper, we extend the iterated integrals from smooth manifolds to digraphs and develop the associated algebraic and geometric structures. Iterated integrals on a digraph naturally give rise to the iterated path algebra and the iterated loop algebra, both defined as quotient algebras of a shuffle algebra, with the latter carrying a canonical Hopf algebra structure. We construct a non-degenerate pairing between elementarily equivalent classes of loops on a digraph and the iterated loop algebra. By restricting to iterated integrals that are invariant under $C_\partial$-homotopy, a distinguished subalgebra is obtained which, under this pairing, corresponds to the group algebra of the fundamental group. We further show that this subalgebra is a homotopy invariant and forms a Hopf algebra with involutive antipode. |
| title | Algebra of Path Integrals on Digraphs |
| topic | Algebraic Topology Combinatorics 05C20, 16T05, 53A70, 55P35 |
| url | https://arxiv.org/abs/2603.01531 |