On non-abelian quadrirational Yang-Baxter maps

We introduce four non-equivalent lists of families of non-abelian quadrirational Yang-Baxter maps, the so-called F, H, K and A lists. We provide the canonical form of the generic map in each list, which under various degenerations lead to the remaining members of each list. In the abelian setting all four lists constitute the well known F and H lists of quadrirational Yang-Baxter maps.