The trivial universal R-matrix for the group algebra of a two-element group

The multiplication is associative because on basis elements $(g^i{g}^j)g^m=g^{i+j+m}=g^i(g^j{g}^m)$ with exponents modulo two; $1=g^0$ is a two-sided unit. The maps $\mathrm{\Delta }$ and $\epsilon$ preserve products and units on the basis $g^0,g^1$, and both iterated coproducts send $g^i$ to $g^i\otimes g^i\otimes g^i$, while both counit contractions return $g^i$. Thus $H$ is a bialgebra.

On each basis element $g^i$, both $m(S\otimes \mathrm{id})\mathrm{\Delta }(g^i)$ and $m(\mathrm{id}\otimes S)\mathrm{\Delta }(g^i)$ equal $g^{2i}=1=\epsilon (g^i)1$. Linearity gives both antipode contractions on $H$, so the stated structure is a Hopf algebra. Also $\mathrm{\Delta }(g^i)=g^i\otimes g^i$ is fixed by the flip, proving cocommutativity.

The tensor $R=1_H\otimes 1_H$ is invertible with inverse itself. For every $h\in H$, $R\mathrm{\Delta }(h)=\mathrm{\Delta }(h)={\mathrm{\Delta }}^{\mathrm{o}\mathrm{p}}(h)R$ by cocommutativity; moreover $(\mathrm{\Delta }\otimes \mathrm{id})R=1_H^{\otimes 3}=R_{13}{R}_{23}$ and ${\alpha }^{-1}(\mathrm{id}\otimes \mathrm{\Delta })R=1_H^{\otimes 3}=R_{13}{R}_{12}$. Hence $R$ is quasitriangular. Since $R_{21}R=1_H\otimes 1_H$, it is triangular.

On any pair of left modules the formula $c_{V,W}={\tau }_{V,W}({\rho }_V\otimes {\rho }_W)(R)$ reduces to $v\otimes w\mapsto w\otimes v$. For $h\in H$, the two module actions agree because $\mathrm{\Delta }(h)={\mathrm{\Delta }}^{\mathrm{o}\mathrm{p}}(h)$; the flip is natural for module maps; and each of the two hexagon composites sends a pure triple tensor to the same reordered pure tensor. The pure tensors span, so this is the symmetric module braiding associated with $R$.