Affine sl(2|1) and D(2|1;alpha) as Vertex Operator Extensions of Dual Affine sl(2) Algebras

We discover a realisation of the affine Lie superalgebra sl(2|1) and of the exceptional affine superalgebra D(2|1;alpha) as vertex operator extensions of two affine sl(2) algebras with dual levels (and an auxiliary level 1 sl(2) algebra). The duality relation between the levels is (k+1)(k'+1)=1. We construct the representation of sl(2|1) at level k' on a sum of tensor products of sl(2) at level k, sl(2) at level k' and sl(2) at level 1 modules and decompose it into a direct sum over the sl(2|1) spectral flow orbit. This decomposition gives rise to character identities, which we also derive. The extension of the construction to the affine D(2|1;k') at level k is traced to properties of sl(2)+sl(2)+sl(2) embeddings into D(2|1;alpha) and their relation with the dual sl(2) pairs. Conversely, we show how the level k' sl(2) representations are constructed from level k sl(2|1) representations.Comment: 50 pages, Latex2e, 2 figures, acknowledgements adde

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

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 (2 p q) 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.Comment: 52pp., AMSLaTeX++. half a dozen minor inaccuracies (cross-refs etc) correcte

Quantum-sl(2) action on a divided-power quantum plane at even roots of unity

We describe a nonstandard version of the quantum plane, the one in the basis of divided powers at an even root of unity $q=e^{i\pi/p}$. It can be regarded as an extension of the "nearly commutative" algebra $C[X,Y]$ with $X Y =(-1)^p Y X$ by nilpotents. For this quantum plane, we construct a Wess--Zumino-type de Rham complex and find its decomposition into representations of the $2p^3$-dimensional quantum group $U_q sl(2)$ and its Lusztig extension; the quantum group action is also defined on the algebra of quantum differential operators on the quantum plane.Comment: 18 pages, amsart++, xy, times. V2: a reference and related comments adde

A differential U-module algebra for U=U_q sl(2) at an even root of unity

We show that the full matrix algebra Mat_p(C) is a U-module algebra for U = U_q sl(2), a 2p^3-dimensional quantum sl(2) group at the 2p-th root of unity. Mat_p(C) decomposes into a direct sum of projective U-modules P^+_n with all odd n, 1<=n<=p. In terms of generators and relations, this U-module algebra is described as the algebra of q-differential operators "in one variable" with the relations D z = q - q^{-1} + q^{-2} z D and z^p = D^p = 0. These relations define a "parafermionic" statistics that generalizes the fermionic commutation relations. By the Kazhdan--Lusztig duality, it is to be realized in a manifestly quantum-group-symmetric description of (p,1) logarithmic conformal field models. We extend the Kazhdan--Lusztig duality between U and the (p,1) logarithmic models by constructing a quantum de Rham complex of the new U-module algebra.Comment: 29 pages, amsart++, xypics. V3: The differential U-module algebra was claimed quantum commutative erroneously. This is now corrected, the other results unaffecte

Factorizable ribbon quantum groups in logarithmic conformal field theories

We review the properties of quantum groups occurring as Kazhdan--Lusztig dual to logarithmic conformal field theory models. These quantum groups at even roots of unity are not quasitriangular but are factorizable and have a ribbon structure; the modular group representation on their center coincides with the representation on generalized characters of the chiral algebra in logarithmic conformal field models.Comment: 27pp., amsart++, xy. v2: references added, some other minor addition

Gepner-like models and Landau-Ginzburg/sigma-model correspondence

The Gepner-like models of $k^{K}$-type is considered. When $k+2$ is multiple of $K$ the elliptic genus and the Euler characteristic is calculated. Using free-field representation we relate these models with $\sigma$-models on hypersurfaces in the total space of anticanonical bundle over the projective space $\mathbb{P}^{K-1}$

Fusion in the entwined category of Yetter--Drinfeld modules of a rank-1 Nichols algebra

We rederive a popular nonsemisimple fusion algebra in the braided context, from a Nichols algebra. Together with the decomposition that we find for the product of simple Yetter-Drinfeld modules, this strongly suggests that the relevant Nichols algebra furnishes an equivalence with the triplet W-algebra in the (p,1) logarithmic models of conformal field theory. For this, the category of Yetter-Drinfeld modules is to be regarded as an \textit{entwined} category (the one with monodromy, but not with braiding).Comment: 36 pages, amsart++, times, xy. V3: references added, an unnecessary assumption removed, plus some minor change

The Baxter Q Operator of Critical Dense Polymers

We consider critical dense polymers ${\cal L}_{1,2}$, corresponding to a logarithmic conformal field theory with central charge $c=-2$. An elegant decomposition of the Baxter $Q$ operator is obtained in terms of a finite number of lattice integrals of motion. All local, non local and dual non local involutive charges are introduced directly on the lattice and their continuum limit is found to agree with the expressions predicted by conformal field theory. A highly non trivial operator $\Psi(\nu)$ is introduced on the lattice taking values in the Temperley Lieb Algebra. This $\Psi$ function provides a lattice discretization of the analogous function introduced by Bazhanov, Lukyanov and Zamolodchikov. It is also observed how the eigenvalues of the $Q$ operator reproduce the well known spectral determinant for the harmonic oscillator in the continuum scaling limit.Comment: improved version, accepted for publishing on JSTA