almost cocommutative bialgebra

A bialgebra $A$ is called almost cocommutative if there is an unit $\mc{R}\in A\otimes A$ such that $$\mc{R}\Delta(a)=\Delta^{op}(a)\mc{R}$$ where $\Delta^{op}$ is the opposite comultiplication (the usual comultiplication, composed with the flip map of the tensor product $A\otimes A$ ). The element $\mc{R}$ is often called the $\mc{R}$ -matrix of $A$ .

The significance of the almost cocommutative condition is that $\sigma_{V,W}=\sigma\circ\mc{R}:V\otimes W\to W\otimes V$ 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).