almost cocommutative bialgebra

A bialgebra $A$ is called almost cocommutative if there is an unit $ℛ\in A\otimes A$ such that

$$ℛ\mathrm{\Delta }(a)={\mathrm{\Delta }}^{op}(a)ℛ$$

where ${\mathrm{\Delta }}^{op}$ is the opposite comultiplication (the usual comultiplication, composed with the flip map of the tensor product $A\otimes A$). The element $ℛ$ is often called the $ℛ$-matrix of $A$.

The significance of the almost cocommutative condition is that ${\sigma }_{V,W}=\sigma \circ ℛ:V\otimes W\to W\otimes V$ gives a natural isomorphism of bialgebra representations, where $V$ and $W$ are $A$-modules, making the category of $A$-modules into a quasi-tensor or braided monoidal category. Note that ${\sigma }_{W,V}\circ {\sigma }_{V,W}$ is not necessarily the identity (this is the braiding of the category).