 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2007/iii/paper-2/3/a/solution

Codex  0 Created 2026-10-03 Updated 2026-10-14

H

r : H \otimes H \to k

satisfying the following identities, in Sweedler notation:

r (ab, c) r (a, b c) \sum r (a_{(1)}, b_{(1)}) a_{(2)} b_{(2)} r (1, a) = \sum r (a, c_{(1)}) r (b, c_{(2)}), = \sum r (a_{(1)}, c) r (a_{(2)}, b), = \sum b_{(1)} a_{(1)} r (a_{(2)}, b_{(2)}), = r (a, 1) = ε (a) .

Convolution invertibility means there is

\overset{r}{ˉ}

with

\sum r (a_{(1)}, b_{(1)}) \overset{r}{ˉ} (a_{(2)}, b_{(2)}) = ε (a) ε (b)

, and likewise with

r, \overset{r}{ˉ}

interchanged. This is a coquasitriangular structure. On right comodules it induces the braiding

c (v \otimes w) = \sum w_{(0)} \otimes v_{(0)} r (v_{(1)}, w_{(1)}) .

The third identity makes this a comodule map, the first two give its two hexagon identities, and the convolution inverse gives the inverse braiding.

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