Logarithmic (sl)over-cap(2) CFT models from Nichols algebras: I

We construct chiral algebras that centralize rank-2 Nichols algebras with at least one fermionic generator. This gives 'logarithmic' W-algebra extensions of a fractional-level (sl) over cap (2) algebra. We discuss crucial aspects of the emerging general relation between Nichols algebras and logarithmic conformal field theory (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 U(X) of such Nichols algebras. For an extended chiral algebra, candidates for its simple modules that are counterparts of the U(X) simple modules are proposed, as a first step toward a functorial relation between U(X) and W-algebra representation categories.