Abstract. We complete the classification of Hopf algebras whose infinitesimal braiding is a principal Yetter-Drinfeld realization of a braided vector space of Cartan type G2 over a cosemisimple Hopf algebra.We develop a general formula for a class of liftings in which the quantum Serre relations hold. We give a detailed explanation of the procedure for finding the relations, based on the recent work of Andruskiewitsch, Angiono and Rossi Bertone.
IntroductionThis article belongs to the series initiated in [AAG], and followed by [AG2], with the purpose of computing all liftings of braided vector spaces of diagonal type (V, c) whose Nichols algebra B(V ) is finite-dimensional.In this part we focus on:That is, we assume there are N > 3, q a primitive N -th root of 1 and a basis {x 1 , x 2 } of V such that the braiding is determined by a matrix q = q q 12 q 21 q 3 ∈ k 2×2 with q 12 q 21 = q −3 as:• H a cosemisimple Hopf algebra with a principal realizationWe recall that a Hopf algebra L is called a lifting of V ∈ H H YD when gr L ≃ B(V )#H. We refer the reader to the article [AAG], the first of the series, based on [A+], for a description of the program for computing all liftings of braided vector spaces of diagonal type with a realization V ∈ H H YD. Set B = B(V )#H. In a sentence, the program consists in constructing a subset Cleft ′ (B) ⊆ Cleft(B) of right cleft objects for B in such a way that for every X ∈ Cleft ′ (B) the left Schauenburg Hopf algebra A = L(X, B), see [S], satisfies gr A ≃ B and checking that indeed, every lifting can be obtained in this way.2000 Mathematics Subject Classification. 16W30.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.