Finite-dimensional pointed Hopf algebras over $\mathbb{S}_4$

Abstract

Let k be an algebraically closed field of characteristic 0. We conclude the classification of finite-dimensional pointed Hopf algebras whose group of group-likes is S 4 . We also describe all pointed Hopf algebras over S 5 whose infinitesimal braiding is associated to the rack of transpositions.

Examples of finite-dimensional Hopf algebras with the dual Chevalley property

“…• these algebras are liftings of B(V )#kS 4 , • any lifting is isomorphic to one of these algebras, • H(Q −1 4 [t]) ≃ H(Q −1 4 [t ′ ]) iff t = 0 and t = t ′ ∈ P 1 k or if t = t ′ = (0, 0), and the same holds for H (D[t] [GG,Lemma 6.1].…”

“…The liftings of B(V )#kS 4 , where V is as above, are classified in [GG,Proposition 5.3]. Indeed, for the ql-data [GG,Def. 3 The classification of the finite-dimensional Nichols algebras over S 5 is unknown; there are two non-zero Yetter-Drinfeld modules over kS 5 with finite-dimensional Nichols algebra [FK,G2,GG] and one open case [AFGV].…”

“…Indeed, for the ql-data [GG,Def. 3 The classification of the finite-dimensional Nichols algebras over S 5 is unknown; there are two non-zero Yetter-Drinfeld modules over kS 5 with finite-dimensional Nichols algebra [FK,G2,GG] and one open case [AFGV]. The underlying rack and cocycles are (O 5 2 , −1) or (O 5 2 , χ).…”

“…Table 1. [FK, MS] S 4 Table 3 (O 5 2 , −1), (O 5 2 , χ) 8294400 [FK,G2,GG] S 5 Table 4 (T , −1) 72 [G1] A 4 × C 2 (D 3…”

Nichols algebras associated to the transpositions of the symmetric group are twist-equivalent

“…These algebras appeared first in [11,21]. For more information about these algebras, see [12], Theorem 2.4 and Proposition 2.5, or [15], Proposition 5.11.…”