Liftings of Nichols algebras of diagonal type III. Cartan type G2

Abstract

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.

Pointed Hopf algebras: a guided tour to the liftings

“…with deg(t 2,j ) + deg(t j 2 ) < deg(t p 1 ), j ∈ I p2 , and deg t j 2 < deg t j+1 2 , j ∈ I p2−1 . As r − t p1 1 does not satisfy (20), we have that p2 j=1 t 2,j ⊗ t j 2 = 0. With the same argument as before, we may assume that t 2,p2 = 1.…”

“…Assume you want to find γ(r) satisfying (20) and you guess γ(r) = r. Then you need to compute ρ(r) and check the equality.…”

“…[20, Theorem 3.1] For each λ ∈ Λ, the Hopf algebra u q (λ) defined as the quotient of T (V )#H by the ideal generated by the relations…”