Logarithmic ^sl(2) CFT models from Nichols algebras. 1

TL;DR

This paper constructs logarithmic W-algebra extensions of fractional-level sl(2) algebras using Nichols algebras, revealing new links between algebraic structures and logarithmic conformal field theories.

Contribution

It introduces a method to build logarithmic W-algebras from Nichols algebras with fermionic generators, establishing connections to fractional-level sl(2) models.

Findings

Constructed chiral algebras centralizing rank-two Nichols algebras.

Identified relations between Nichols algebras and logarithmic CFT models.

Proposed candidates for simple modules of the extended chiral algebra.

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 ^sl(2) 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 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.