On the lifting of Nichols algebras

Virasoro central charges for Nichols algebras

“…In terms of the momenta α, β ∈ C 2 of the screenings, the common condition for all these cases takes the form 2α.β + β .β = 2m (m ∈ Z). [22] ). q 11 q 12 q 21 = 1,…”

“…The Nichols algebra B(X ) (of the two-dimensional braided vector space X with basis F 1 and F 2 with braiding matrix (1)) is in this case the quotient [22] (11)…”

“…Generally, these long screenings are to produce m-plet structures in logarithmic models, similarly to how the triplet structure of the (p, 1) logarithmic models [25,26] is generated by the corresponding long screening [18]. For the current W 3 -case, the resulting octuplet algebra is outlined in Appendix W. [22] ).…”

“…All of these were listed by Heckenberger [21] (the general classification, for any rank, was achieved in [7] and was reproduced in a different and independent way in [15,16,17]). These notes are in fact a compilation of the original Heckenberger's list with explicit results on the presentation of some Nichols algebras (obtained in [15] for the standard type and in [22] in several nonstandard cases), and with several CFT constructions added. The extended algebra of a logarithmic model-the octuplet algebra extending the W 3 algebra-is offered in only one case; the other CFT constructions are merely a starting point for finding extended algebras.…”

“…3 In what follows, I therefore reproduce Heckenberger's list of rank-2 finite-dimensional Nichols algebras [21], with the only difference that I enumerate, not itemize the subitems. For several items, I also add the presentations known from [15] and explicitly borrowed from [22], including the case number in [22]. From the Nichols-algebra data, I move toward conformal field theory by analyzing the conditions on the screening momenta.…”