Entropy, cocycles, and their diagrammatics

Abstract

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.

Figures (100)

• Figure 2.1.1: Left: a $\sigma$-labelled edge with a co-orientation. Middle and right: two possible diagrams of co-orientations at a vertex, with edges carrying $G$-labels $\sigma,\tau\in G$, and either $\sigma\tau$ or $\tau\sigma$. Vertices with such co-orientations are called type I vertices.
• Figure 2.1.2: The $G$-winding number $w(p,\mathcal{N})=\sigma_1\sigma_2^{-1}\sigma_3$ of a point $p$ relative to network $\mathcal{N}$. For this particular $G$-network labels of the remaining edges can be uniquely reconstructed from the labels $\sigma_1,\dots, \sigma_4$ that are shown.
• Figure 2.1.3: Associativity transformation for a merge (or a split) of $\sigma$, $\tau$ and $\gamma$.
• Figure 2.1.4: These transformations are essentially versions of the Figure \ref{['fig3_011']} for other co-orientations. The second transformation comes from reversing the coorientation and label of the central edge.
• Figure 2.1.5: Composition of a merge and a split of $\sigma$ and $\tau$ lines with compatible co-orientations.

Theorems & Definitions (71)

• Remark 2.1
• Proposition 2.2
• Remark 2.3
• Remark 2.4
• Proposition 2.5
• Remark 2.6
• Remark 2.7
• Remark 2.8
• Proposition 2.9
• proof