Virasoro Central Charges for Nichols Algebras
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
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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 W3 ≡ W A2 , W B2 , and W G2 models of conformal field theory. In the W3 case, the construction of an octuplet extended algebra—a counterpart of the triplet (1, p) algebra—is outlined.
1 Introduction In [1], we described a paradigm treating screening operators in two-dimensional conformal field theory as a braided Hopf algebra, a Nichols algebra [2–9]. This immediately suggests that the inverse relation may also exist. Is any finite-dimensional Nichols algebra with diagonal braiding an algebra of screenings in some conformal model? This is a fascinating problem, especially considering the recent remarkable development in the theory of Nichols algebras—originally a “technicality” in Andruskiewitsch and Schneider’s program of classification of pointed Hopf algebras, which has grown into a beautiful theory in and of itself (in addition to the papers cited above and the references therein, also see [8, 10–17]). Diagonal braiding is assumed in what follows. As many “inverse” problems, that of identifying a conformal field model behind a given Nichols algebra is not necessarily well defined. It is of course well known that screenings can be used to define models of conformal field theory; in particular, defining logarithmic models as kernels of screening [18] turned out to be especially useful. But passing from a Nichols algebras to screenings involves various ambiguities. Nevertheless, the central charges associated with Nichols algebras in what follows have the nice property of being invariant under the Weyl groupoid action— the natural “symmetry” up to which Nichols algebras are classified [10, 19, 20]. To proceed beyond the central charge identification, I restrict myself to Nichols algebras of rank two (already a fairly large number in terms of the possible conforA.M. Semikhatov Lebedev Physics Institute, Moscow 119991, Russia C. Bai et al. (eds.), Conformal Field Theories and Tensor Categories, Mathematical Lectures from Peking University, DOI 10.1007/978-3-642-39383-9_3, © Springer-Verlag Berlin Heidelberg 2014
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mal models). All of these were listed by Heckenberger [21, 22] (the general classification, for any rank, was achieved in [7] and was reproduced in a different and independent way in [15–17]). These notes are in fact a compilation of the original Heckenberger’s list with explicit results on the presentation of some Nichols algebras (obtained in [15] for the standard type and in [23] in several nonstandard cases), and with several CFT constructions added. The extended algebra of a logarithmic model—the octuplet algebra extending the W3 algebra—is offered in only one case; the other CFT constructions are
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