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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Online Access: | https://arxiv.org/abs/2409.08462 |
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| _version_ | 1866913537069154304 |
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| author | Im, Mee Seong Khovanov, Mikhail |
| author_facet | Im, Mee Seong Khovanov, Mikhail |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_08462 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Entropy, cocycles, and their diagrammatics Im, Mee Seong Khovanov, Mikhail K-Theory and Homology Information Theory Mathematical Physics Category Theory Primary: 94A17, 20J06, 18M10, 18M30, Secondary: 18B40, 37A20, 18G45 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. |
| title | Entropy, cocycles, and their diagrammatics |
| topic | K-Theory and Homology Information Theory Mathematical Physics Category Theory Primary: 94A17, 20J06, 18M10, 18M30, Secondary: 18B40, 37A20, 18G45 |
| url | https://arxiv.org/abs/2409.08462 |