Virasoro Central Charges for Nichols Algebras

Abstract

ABSTRACT. A Virasoro central charge can be associated with each Nichols algebra with diagonal braiding in a way that is invariant under the Weyl groupoid action. The central charge takes very suggestive values for some items in Heckenberger's list of rank-2 Nichols algebras. In particular, this might be viewed as an indication of the existence of reasonable logarithmic extensions of W 3 ≡ WA 2 , WB 2 , and WG 2 models of conformal field theory. In the W 3 case, the construction of an octuplet extended algebra-a counterpart of the triplet (1, p) algebra-is outlined.

Coset construction of logarithmic minimal models: branching rules and branching functions

“…From the lattice perspective, the corresponding W-extended CFTs WLM(p, p ′ ) are described [49][50][51] by the same LM(p, p ′ ) lattice models but with a special class of restricted boundary conditions which, in the continuum scaling limit, respect the W-extended symmetry. Construction of other cosets of logarithmic CFTs with Wsymmetry first appeared in [52,53], whereas the first hints of the existence of a W diagonal coset construction of the WLM(p, p ′ ) models, of an analogous form to (1.1), appeared in [54]. Specifically, it was found that the Grothendieck ring associated with W-projective representations leads to a Verlindelike formula involving coset graphs A…”

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.…”

W-algebras, higher rank false theta functions, and quantum dimensions

“…In the last ten years, a lot of research has been devoted to understanding the so called logarithmic conformal field theories and related irrational vertex algebras [2,4,[18][19][20]24,30,32], etc. In these theories, modularity either holds with some modifications, as seen in C 2 -cofinite vertex operator algebras (VOAs), or is completely absent as is the case of non C 2 -cofinite VOAs.…”

“…In particular, for Q = A 1 , W ( p) Q is precisely the (1, p)-singlet algebra discussed in [10,14]. Because of the integrality condition, the previous algebra can be maximally extended within the lattice vertex algebra leading to [2,12,18,32] …”