Quantum lines over non-cocommutative cosemisimple Hopf algebras

Abstract

The recognition of the importance of braided tensor categories came by the remarkable recent developments of quantum group theory. An especially important example of such a category is the braided tensor category HYD of Yetter–Drinfeld modules over a Hopf algebra H with bijective antipode. It is interesting and delightful to see that old results on ordinary Hopf algebras, such as those by Nichols [18] (1978) and by Radford [19] (1985), are reproduced in terms of the new notion of braided Hopf algebra in HYD. Given a braided Hopf algebra R in HYD, one can construct an ordinary Hopf algebra R # H of biproduct; this naturally forms a triple (R # H, ι,π) called a Hopf algebra triple over H , such that ι :H → R # H and π :R # H → H are Hopf algebra maps with π ◦ ι = id, if we define ι(h) = 1 # h, π(x # h) = e(x)h. It was essentially proved by Radford [19] and then reproduced by Majid [13] that R → R # H gives a category equivalence, called ‘bosonisation,’ from the category of braided Hopf algebras in HHYD to the category of Hopf algebra triples over H ; see Proposition 1.1.