# almost cocommutative bialgebra

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

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

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 $\mathcal{R}$ 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 |
---|---|

Canonical name | AlmostCocommutativeBialgebra |

Date of creation | 2013-03-22 13:31:50 |

Last modified on | 2013-03-22 13:31:50 |

Owner | bwebste (988) |

Last modified by | bwebste (988) |

Numerical id | 5 |

Author | bwebste (988) |

Entry type | Definition |

Classification | msc 16W30 |