{"title":"熵、共循环及其图解法","authors":"Mee Seong Im, Mikhail Khovanov","doi":"arxiv-2409.08462","DOIUrl":null,"url":null,"abstract":"The first part of the paper explains how to encode a one-cocycle and a\ntwo-cocycle on a group $G$ with values in its representation by networks of\nplanar trivalent graphs with edges labelled by elements of $G$, elements of the\nrepresentation floating in the regions, and suitable rules for manipulation of\nthese diagrams. When the group is a semidirect product, there is a similar\npresentation via overlapping networks for the two subgroups involved. M. Kontsevich and J.-L. Cathelineau have shown how to interpret the entropy\nof a finite random variable and infinitesimal dilogarithms, including their\nfour-term functional relations, via 2-cocycles on the group of affine\nsymmetries of a line. We convert their construction into a diagrammatical calculus evaluating\nplanar networks that describe morphisms in suitable monoidal categories. In\nparticular, the four-term relations become equalities of networks analogous to\nassociativity equations. The resulting monoidal categories complement existing\ncategorical and operadic approaches to entropy.","PeriodicalId":501312,"journal":{"name":"arXiv - MATH - Mathematical Physics","volume":"17 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Entropy, cocycles, and their diagrammatics\",\"authors\":\"Mee Seong Im, Mikhail Khovanov\",\"doi\":\"arxiv-2409.08462\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"The first part of the paper explains how to encode a one-cocycle and a\\ntwo-cocycle on a group $G$ with values in its representation by networks of\\nplanar trivalent graphs with edges labelled by elements of $G$, elements of the\\nrepresentation floating in the regions, and suitable rules for manipulation of\\nthese diagrams. When the group is a semidirect product, there is a similar\\npresentation via overlapping networks for the two subgroups involved. M. Kontsevich and J.-L. Cathelineau have shown how to interpret the entropy\\nof a finite random variable and infinitesimal dilogarithms, including their\\nfour-term functional relations, via 2-cocycles on the group of affine\\nsymmetries of a line. We convert their construction into a diagrammatical calculus evaluating\\nplanar networks that describe morphisms in suitable monoidal categories. In\\nparticular, the four-term relations become equalities of networks analogous to\\nassociativity equations. The resulting monoidal categories complement existing\\ncategorical and operadic approaches to entropy.\",\"PeriodicalId\":501312,\"journal\":{\"name\":\"arXiv - MATH - Mathematical Physics\",\"volume\":\"17 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-13\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Mathematical Physics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2409.08462\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Mathematical Physics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.08462","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
The first part of the paper explains how to encode a one-cocycle and a
two-cocycle on a group $G$ with values in its representation by networks of
planar trivalent graphs with edges labelled by elements of $G$, elements of the
representation floating in the regions, and suitable rules for manipulation of
these diagrams. When the group is a semidirect product, there is a similar
presentation via overlapping networks for the two subgroups involved. M. Kontsevich and J.-L. Cathelineau have shown how to interpret the entropy
of a finite random variable and infinitesimal dilogarithms, including their
four-term functional relations, via 2-cocycles on the group of affine
symmetries of a line. We convert their construction into a diagrammatical calculus evaluating
planar networks that describe morphisms in suitable monoidal categories. In
particular, the four-term relations become equalities of networks analogous to
associativity equations. The resulting monoidal categories complement existing
categorical and operadic approaches to entropy.