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

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.

V3: Several minor corrections and references updated/corrected, plus two important changes: 1. The presentation in Sec. 7.4 used to be missing some essential information, which is now restored and the subsection expanded. 2. Details pertaining to U(X) representations are now moved to the second paper in the series (to appear soon), where the U(X) representation theory will be expounded

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• The R-matrix of quantum doubles of Nichols algebras of diagonal type
• Representations of ${\bar{U}}_{q}s\ell (2|1)$ at even roots of unity
• Nichols Algebras and Quantum Principal Bundles
• Quasi-lisse extension of affine ${\mathfrak{sl}}_2$ à la Feigin--Tipunin
• Bimodule structure of the mixed tensor product over $U_{q}s\ell (2|1)$ and quantum walled Brauer algebra
• Algebras of Non-Local Screenings and Diagonal Nichols Algebras