Kazhdan-Lusztig-dual quantum group for logarithimic extensions of Virasoro minimal models

Abstract

ABSTRACT. We derive and study a quantum group g p,q that is Kazhdan-Lusztig-dual to the W -algebra W p,q of the logarithmic (p, q) conformal field theory model. The algebra W p,q is generated by two currents W + (z) and W − (z) of dimension (2p−1)(2q−1), and the energy-momentum tensor T (z). The two currents generate a vertex-operator ideal R with the property that the quotient W p,q /R is the vertex-operator algebra of the (p, q) Virasoro minimal model. The number (2pq) of irreducible g p,q representations is the same as the number of irreducible W p,q -representations on which R acts nontrivially. We find the center of g p,q and show that the modular group representation on it is equivalent to the modular group representation on the W p,q characters and "pseudocharacters." The factorization of the g p,q ribbon element leads to a factorization of the modular group representation on the center. We also find the g p,q Grothendieck ring, which is presumably the "logarithmic" fusion of the (p, q) model. CONTENTS

Free fermion resolution of supergroup WZNW models

“…Their role for logarithmic extensions of minimal models was also emphasized in [76,78], mostly based on studies of the dual quantum group. It seems worth pointing out, though, that for quotients of supergroups, projective modules might not play such a prominent role, even though some of them are likely to be logarithmic as well.…”

Logarithmic extensions of minimal models: Characters and modular transformations

“…On the other hand, this "approximation" becomes the precise correspondence as regards the modular group representations: naturally associated with g p + ,p − is the SL(2, Z)-representation on its center [34], which turns out to be equivalent to the SL(2, Z)-representation on the W p + ,p − -algebra characters and generalized characters in (1.2).…”

“…(Their counterparts for the Kazhdan-Lusztig-dual quantum group [34] are indeed Verma modules, but investigation of the Verma properties of V ± r,s is a separate problem, which we do not consider here and only use the convenient and suggestive name for these modules.) We now describe their subquotients.…”

“…The structure of g p + ,p − and its relation to the logarithmic (p + , p − ) model are investigated in [34].…”

“…We next take a subalgebra inD(H) (which, unlikeD(H), is a factorizable ribbon quantum group, see [34]). …”