# almost cocommutative bialgebra

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

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

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

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

Title almost cocommutative bialgebra AlmostCocommutativeBialgebra 2013-03-22 13:31:50 2013-03-22 13:31:50 bwebste (988) bwebste (988) 5 bwebste (988) Definition msc 16W30