Logarithmic $\widehat{s\ell }(2)$ CFT models from Nichols algebras: I

Abstract

We construct chiral algebras that centralize rank-two Nichols algebras with at least one fermionic generator. This gives "logarithmic" W -algebra extensions of a fractional-level p sℓp2q algebra. We discuss crucial aspects of the emerging general relation between Nichols algebras and logarithmic CFT models: (i) the extra input, beyond the Nichols algebra proper, needed to uniquely specify a conformal model; (ii) a relation between the CFT counterparts of Nichols algebras connected by Weyl groupoid maps; and (iii) the common double bosonization UpXq of such Nichols algebras. For an extended chiral algebra, candidates for its simple modules that are counterparts of the UpXq simple modules are proposed, as a first step toward a functorial relation between UpXq and W -algebra representation categories.

The Verlinde formula in logarithmic CFT

“…There are also logarithmic conformal field theories that have been extensively studied for their own sake, in particular the (1, p) triplet models [24][25][26][27][28][29][30][31] (see [32] for a proposed lattice-theoretic realisation), their (p ′ , p) generalisations [33][34][35][36][37][38][39][40], also known as the (W-extended) logarithmic minimal models, and the fractional level Wess-Zumino-Witten models with sl (2) [17,18,41,42] and sl (3) [43] symmetry. Recently, these models have been extended to sl (2) triplets [44].…”

Rank three Nichols algebras of diagonal type over arbitrary fields

“…Woronowicz, M. Rosso, S. Majid and G. Lusztig in many different ways, see for example [37,38,32,30,29]. Besides that, Nichols algebras have interesting applications to other research fields such as Kac-Moody Lie superalgebras [1,Example 3.2] and conformal field theory [34,35,36]. Nichols algebras appeared naturally in the classification of pointed Hopf algebras by the lifting method of N. Andruskiewitsch and H.-J.…”

Representations of U¯qsℓ(2|1) at even roots of unity

“…2 The ideas of the logarithmic Kazhdan-Lusztig correspondence need to be extended to higher-rank Nichols algebras; this would show, among other things, how much of what we know in the rank-1 case is "accidental," and which features are indeed generic. Moving to higher-rank Nichols algebras was initiated in [32] and, with precisely the two Nichols algebras defined by braiding matrices (1.1), in [33].…”

“…2.1.2. The second Hopf algebra structure [33]. The Hopf algebra UpX q admits a nontrivial "twist"-an invertible normalized 2-cocycle (see B.3) Φ " 1 b 1`pq´q´1qBk bCk´1 P UpX q b UpX q twisting by which gives rise to the second coalgebra structure r ∆pxq " Φ´1∆pxqΦ.…”