Elliptic algebra U_{q,p}(^sl_2): Drinfeld currents and vertex operators
M. Jimbo, H. Konno, S. Odake, J. Shiraishi
备注：
49 pages, (AMS-)LaTeX ; added an explanation of integration contours; added comments. To appear in Comm. Math. Phys. Numbering of equations is corrected
We investigate the structure of the elliptic algebra U_{q,p}(^sl_2) introduced earlier by one of the authors. Our construction is based on a new set of generating series in the quantum affine algebra U_q(^sl_2), which are elliptic analogs of the Drinfeld currents. They enable us to identify U_{q,p}(^sl_2) with the tensor product of U_q(^sl_2) and a Heisenberg algebra generated by P,Q with [Q,P]=1. In terms of these currents, we construct an L operator satisfying the dynamical RLL relation in the presence of the central element c. The vertex operators of Lukyanov and Pugai arise as `intertwiners' of U_{q,p}(^sl_2) for level one representation, in the sense to be elaborated on in the text. We also present vertex operators with higher level/spin in the free field representation.
Drinfeld realisations are constructed for the quantum affine superalgebras of the series ${\rm\mathfrak{osp}}(1|2n)^{(1)}$,${\rm\mathfrak{sl}}(1|2n)^{(2)}$ and ${\rm\mathfrak{osp}}(2|2n)^{(2)}$. By using the realisations, we develop vertex operator representations and classify the finite dimensional irreducible representations for these quantum affine superalgebras.
For the current realization of the quantum affine algebras, Drinfeld gave a simple comultiplication of the quantum current operators. With this comultiplication, we study the related vertex operators for the case of $U_q(\hgtsl_n)$ and give an explicit bosonization of these new vertex operators. We use these vertex operators to construct the quantum current operators of $U_q(\hgtsl_n)$ and discuss its connection with quantum boson-fermion correspondence.
We give a new presentation of the Drinfeld double of the elliptic Hall algebra introduced in a previous work with I. Burban. This presentation is similar in spirit to Drinfeld's `new realization' of quantum affine algebras. This answers, in the case of elliptic curves, a question of Kapranov concerning functional relations satisfied by (principal, unramified) Eisenstein series for the groups GL(n) over a function field. It also provides proofs of some recent conjectures of Feigin, Feigin, Jimbo, Miwa and Mukhin.
The two-parameter quantum vertex operator representation of level-one is explicitly constructed for $U_{r,s}(C^{(1)}_n)$ based on its two-parameter Drinfeld realization we give. This construction will degenerate to the one-parameter case due to Jing-Koyama-Misra (\cite{JKM2}) when $rs=1$.
In \cite{S} O. Schiffmann gave a presentation of the Drinfel'd double of the elliptic Hall algebra which is similar in spirit to Drinfel'd's new realization of quantum affine algebras. Using this result together with a part of his proof we can provide such a description for the elliptic Hall algebra.
Given a tensor category C, one constructs its Drinfeld center Z(C) which is a braided tensor category, having as objects pairs (X, lambda), where X in Obj(C) and lambda is a half-braiding. For a premodular category C, we construct a new category Zel(C) which we call the Elliptic Drinfeld Center, which has objects (X, lambda1, lambda2), where the lambda i's are half-braidings that satisfy some compatibility conditions. We discuss an SL2(Z)-action on Zel(C) that is related to the anomaly appearing in Reshetikhin-Turaev theory. This construction is motivated from the study of the extended Crane-Y…
The representation theory of the Drinfeld doubles of dihedral groups is used to solve the Yang-Baxter equation. Use of the 2-dimensional representations recovers the six-vertex model solution. Solutions in arbitrary dimensions, which are viewed as descendants of the six-vertex model case, are then obtained using tensor product graph methods which were originally formulated for quantum algebras. Connections with the Fateev-Zamolodchikov model are discussed.
By using the elliptic analogue of the Drinfeld currents in the elliptic algebra U_{q,p}(\hat{sl}_N), we construct a L-operator, which satisfies the RLL-relations characterizing the face type elliptic quantum group B_{q,λ}(\hat{sl}_N). For this purpose, we introduce a set of new currents K_j(v) (1\leq j\leq N) in U_{q,p}(\hat{sl}_N). As in the N=2 case, we find a structure of U_{q,p}(\hat{sl}_N) as a certain tensor product of B_{q,λ}(\hat{sl}_N) and a Heisenberg algebra. In the level-one representation, we give a free field realization of the currents in U_{q,p}(\hat{sl}_N). Using the coalgebra…
Critical CoHAs, vertex coalgebras and Deformed Drinfeld coproducts
Kavli Institute for the Physics and Mathematics of the Universe (WPI), The University of Tokyo（东京大学汤川理论物理研究所（WPI））；The University of Tokyo Institutes for Advanced Study, The University of Tokyo（东京大学高等研究院）；Beijing Institute of Mathematical Sciences and Applications (BIMSA)（北京国际数学研究中心）
We construct a realization of the L-operator satisfying the RLL-relation of the face type elliptic quantum group B_{q,lambda}(A_2^(2)). The construction is based on the elliptic analogue of the Drinfeld currents of U_q(A_2^(2)), which forms the elliptic algebra U_{q,p}(A_2^(2)). We give a realization of the elliptic currents E(z), F(z) and K(z) as a tensor product of the Drinfeld currents of U_q(A_2^(2)) and a Heisenberg algebra. In the level-one representation, we also give a free field realization of the elliptic currents. Applying these results, we derive a free field realization of the U_{…
Drinfeld gave a current realization of the quantum affine algebras as a Hopf algebra with a simple comultiplication for the quantum current operators. In this paper, we will present a generalization of such a realization of quantum Hopf algebras. As a special case, we will choose the structure functions for this algebra to be elliptic functions to derive certain elliptic quantum groups as a Hopf algebra, which degenerates into quantum affine algebras if we take certain degeneration of the structure functions.
Colour-kinematics duality suggests that Yang-Mills (YM) theory possesses some hidden Lie algebraic structure. So far this structure has resisted understanding, apart from some progress in the self-dual sector. We show that there is indeed a Lie algebra behind the YM Feynman rules. The Lie algebra we uncover is the Drinfeld double of the Lie algebra of vector fields. More specifically, we show that the kinematic numerators following from the YM Feynman rules satisfy a version of the Jacobi identity, in that the Jacobiator of the bracket defined by the YM cubic vertex is cancelled by the contrib…
We propose a new realization of the elliptic quantum group equipped with the H-Hopf algebroid structure on the basis of the elliptic algebra U_{q,p}(\hat{sl}_2). The algebra U_{q,p}(\hat{sl}_2) has a constructive definition in terms of the Drinfeld generators of the quantum affine algebra U_q(\hat{sl}_2) and a Heisenberg algebra. This yields a systematic construction of both finite and infinite-dimensional dynamical representations and their parallel structures to U_q(\hat{sl}_2). In particular we give a classification theorem of the finite-dimensional irreducible pseudo-highest weight represe…
We put forward the exact version of two-parameter generating functions with $τ$-invariance, which allows us to give a unified and inherent definition for the Drinfeld realization of two-parameter quantum affine algebras for all the untwisted types. As verification, we first construct their level-one vertex representations of $U_{r,s}(\widehat{\mathfrak{g}})$ for simply-laced types, which in turn well-detect the effectiveness of our definitions both for $(r,s)$-generating functions and $(r,s)$-Drinfeld realization in the framework of establishing the two-parameter vertex representation theory.
Two new realizations, denoted $U_{q,x}(\widehat{gl_2})$ and $U(R_{q,x}(\widehat{gl_2}))$ of the trigonometric dynamical quantum affine algebra $U_{q,λ}(\widehat{gl_2})$ are proposed, based on Drinfeld-currents and $RLL$ relations respectively, along with a Heisenberg algebra $\left\{P,Q\right\}$, with $x=q^{2P}$. Here $P$ plays the role of the dynamical variable $λ$ and $Q=\frac{\partial}{\partial P}$. An explicit isomorphism from $U_{q,x}(\widehat{gl_2})$ to $U(R_{q,x}(\widehat{gl_2}))$ is established, which is a dynamical extension of the Ding-Frenkel isomorphism of $U_{q}(\widehat{gl_2})$ w…
In this paper we generalize Drinfeld's twisted quantum affine algebras to construct twisted quantum algebras for all simply-laced generalized Cartan matrices and present their vertex representation realizations.
In a program to formulate and develop two-dimensional conformal field theory in the framework of algebraic geometry, Beilinson and Drinfeld have recently given a notion of ``chiral algebra'' in terms of D-modules on algebraic curves. This definition consists of a ``skew-symmetry'' relation and a ``Jacobi identity'' relation in a categorical setting. In this paper, we show directly that these chiral algebras are essentially the same as vertex algebras without vacuum vector (and without grading), by establishing an equivalence between the skew-symmetry and Jacobi identity relations of Beilinson-…
As for an elliptic $R$-operator which satisfies the Yang--Baxter equation, the incoming and outgoing intertwining vectors are constructed, and the vertex--IRF correspondence for the elliptic $R$-operator is obtained. The vertex--IRF correspondence implies that the Boltzmann weights of the IRF model satisfy the star--triangle relation. By means of these intertwining vectors, the factorized L-operators for the elliptic $R$-operator are also constructed. The vertex--IRF correspondence and the factorized L-operators for Belavin's $R$-matrix are reproduced from those of the elliptic $R$-operator.
The paper is the sequel to q-alg/9704011. We extend the Drinfeld-Sokolov reduction procedure to q-difference operators associated with arbitrary semisimple Lie algebras. This leads to a new elliptic deformation of the Lie bialgebra structure on the associated loop algebra. The related classical r-matrix is explicitly described in terms of the Coxeter transformation. We also present a cross-section theorem for q-gauge transformations which generalizes a theorem due to R.Steinberg.
We introduce a new elliptic quantum toroidal algebra $U_{q,t,p}(gl_{1,tor})$. Various representations in the quantum toroidal algebra $U_{q,t}(gl_{1,tor})$ are extended to the elliptic case including the level (0,0) representation realized by using the elliptic Ruijsenaars difference operator. Intertwining operators of $U_{q,t,p}(gl_{1,tor})$-modules w.r.t. the Drinfeld comultiplication are also constructed. We show that $U_{q,t,p}(gl_{1,tor})$ gives a realization of the affine quiver $W$-algebra $W_{q,t}(Γ(\widehat{A}_0))$ proposed by Kimura-Pestun. This realization turns out to be useful to …
We associate elliptic affine Lie algebras with what are called vertex $\C((z))$-algebras and their modules in a certain category. In the course, we construct two families of Lie algebras closely related to elliptic affine Lie algebras.
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Let $\mathfrak{g}$ be the derived subalgebra of a Kac-Moody Lie algebra of finite type or affine type, $μ$ a diagram automorphism of $\mathfrak{g}$ and $L(\mathfrak{g},μ)$ the loop algebra of $\mathfrak{g}$ associated to $μ$. In this paper, by using the vertex algebra technique, we provide a general construction of current type presentations for the universal central extension $\widehat{\mathfrak{g}}[μ]$ of $L(\mathfrak{g},μ)$. The construction contains the classical limit of Drinfeld's new realization for (twisted and untwisted) quantum affine algebras ([Dr]) and the Moody-Rao-Yokonuma presen…
We study solutions to a generalized version of the classical Yang-Baxter equation (CYBE) with values in a central simple Lie algebra over a field of characteristic 0 from an algebro-geometric perspective. In particular, we describe such solutions using cohomology free sheaves of Lie algebras on projective curves. This framework leads to a new proof of the Belavin-Drinfeld trichotomy, which asserts that over the ground field $\mathbb{C}$ any non-degenerate solution of the CYBE is either elliptic, trigonometric or rational. Furthermore, we obtain new structural results for solutions of the gener…
We propose a q-difference version of the Drinfeld-Sokolov reduction scheme, which gives us q-deformations of the classical W-algebras by reduction from Poisson-Lie loop groups. We consider in detail the case of SL(2). The nontrivial consistency conditions fix the choice of the classical r-matrix defining the Poisson-Lie structure on the loop group LSL(2), and this leads to a new elliptic classical r-matrix. The reduced Poisson algebra coincides with the deformation of the classical Virasoro algebra previously defined in q-alg/9505025. We also consider a discrete analogue of this Poisson algebr…
We define quantum W-algebras generalizing the results of Reshetikhin and the second author, and Shiraishi-Kubo-Awata-Odake. The quantum W-algebra associated to sl_N is an associative algebra depending on two parameters. For special values of parameters it becomes the ordinary W-algebra of sl_N, or the q-deformed classical W-algebra of sl_N. We construct free field realizations of the quantum W-algebras and the screening currents. We also point out some interesting elliptic structures arising in these algebras. In particular, we show that the screening currents satisfy elliptic analogues of the…
An elliptic deformation of $\widehat{sl}_2$ is proposed. Our presentation of the algebra is based on the relation $RLL=LLR^*$, where $R$ and $R^*$ are eight-vertex $R$-matrices with the elliptic moduli chosen differently. In the trigonometric limit, this algebra reduces to a quotient of that proposed by Reshetikhin and Semenov-Tian-Shansky. Conjectures concerning highest weight modules and vertex operators are formulated, and the physical interpretation of $R^*$ is discussed.
Let Uq(g) be the quantum affine superalgebra associated with an affine Kac-Moody superalgebra g which belongs to the three series osp(1|2n)^(1),sl(1|2n)^(2) and osp(2|2n)^(2). We develop vertex operator constructions for the level 1 irreducible integrable highest weight representations and classify the finite dimensional irreducible representations of Uq(g). This makes essential use of the Drinfeld realisation for Uq(g), and quantum correspondences between affine Kac-Moody superalgebras, developed in earlier papers.
We introduce a twisted quantum affine algebra associated to each simply laced finite dimensional simple Lie algebra. This new algebra is a Hopf algebra with a Drinfeld-type comultiplication. We obtain this algebra by considering its vertex representation. The vertex representation quantizes the twisted vertex operators of Lepowsky-Wilson and Frenkel-Lepowsky-Meurman. We also introduce a twisted quantum loop algebra for the Kac-Moody case and give its level one representation.
Elliptic current algebras E_{q,p}(\hat{g}) for arbitrary simply laced finite dimensional Lie algebra g are defined and their co-algebraic structures are studied. It is shown that under the Drinfeld like comultiplications, the algebra E_{q,p}(\hat{g}) is not co-closed for any g. However putting the algebras E_{q,p}(\hat{g}) with different deformation parameters together, we can establish a structure of infinite Hopf family of algebras. The level 1 bosonic realization for the algebra E_{q,p}(\hat{g}) is also established.
After reviewing the recent results on the Drinfeld realization of the face type elliptic quantum group B_{q,lambda}(sl_N^) by the elliptic algebra U_{q,p}(sl_N^), we investigate a fusion of the vertex operators of U_{q,p}(sl_N^). The basic generating functions Λ_j(z) (j=1,2,.. N-1) of the deformed W_N algebra are derived explicitly.
Assuming the existence of the L-operators, we study the Hopf algebroid structure of U_{q,p}(B_N^{(1)}). As an application, we derive the type I and II vertex operators, which intertwine the U_{q,p}(B_N^{(1)})-modules of generic level, by assuming some analytic properties of the L-operators. For the level-1 case, we construct their free field realizations and show that the results satisfy the desired commutation relations with coefficients given by the elliptic dynamical R-matrices of the B_N^{(1)} type.
The Yangian double $\text{DY}_{\hbar}(\mathfrak{g}_N)$ is introduced for the classical types of $\mathfrak{g}_N=\mathfrak{o}_{2n+1}$, $\mathfrak{sp}_{2n}$, $\mathfrak{o}_{2n}$. Via the Gauss decomposition of the generator matrix, the Yangian double is given the Drinfeld presentation. In addition, bosonization of level $1$ realizations for the Yangian double $\text{DY}_{\hbar}(\mathfrak{g}_N)$ of non-simply-laced types are explicitly constructed.
We construct a realization of the elliptic quantum algebra $U_{q,p}(\hat{sl_N})$ for any given level $k$ in terms of free boson fields and their twisted partners. It can be considered as the elliptic deformation of the Wakimoto realization of the quantum affine algebra $U_{q}(\hat{sl_N})$. We also construct a family of screening currents, which commute with the currents of $U_{q,p}(\hat{sl_N})$ up to total q-differences. And we give explicit twisted expressions for the type $I$ and the type $II$ vertex operators of $U_{q,p}(\hat{sl_N})$ by twisting the known results of the type $I$ vertex oper…
We obtain explicit expressions for all genus one chiral n-point functions for free bosonic and lattice vertex operator algebras. We also consider the elliptic properties of these functions.
We construct bundles of modules of vertex operator algebras, and prove the rigidity and vanishing theorem for the Dirac operator on loop space twisted by such bundles. This result generalizes many previous results.
Let g be a complex semisimple Lie algebra, tau a point in the upper half-plane, and h a complex deformation parameter such that the image of h in the elliptic curve E_tau is of infinite order. In this paper, we give an intrinsic definition of the category of finite-dimensional representations of the elliptic quantum group E_{h,tau}(g) associated to g. The definition is given in terms of Drinfeld half-currents and extends that given by Enriquez-Felder for g=sl_2. When g=sl_n, it reproduces Felder's RLL definition via the Gauss decomposition obtained by Enriquez-Felder for n=2 and by the first a…
Introducing an H-Hopf algebroid structure into U_{q,p}(\widedhat{sl}_2), we investigate the vertex operators of the elliptic quantum group U_{q,p}(\widedhat{sl}_2) defined as intertwining operators of infinite dimensional U_{q,p}(\widedhat{sl}_2)-modules. We show that the vertex operators coincide with the previous results obtained indirectly by using the quasi-Hopf algebra B_{q,λ}(\hat{sl}_2). This shows a consistency of our H-Hopf algebroid structure even in the case with non-zero central element.
We reformurate a central extension of Felder's elliptic quantum group in the FRST formulation as a topological algebra E_{q,p}(gl_N) over the ring of formal power series in p. We then discuss the isomorphism between E_{q,p}(gl_N) and the elliptic algebra U_{q,p}(gl_N) of the Drinfeld realization. An evaluation H-algebra homomorphism from U_{q,p}(gl_N) to a dynamical extension of the quantum affine algebra U_q(gl_N) resolves the problem into the one discussed by Ding and Frenkel in the trigonometric case. We also provide some useful formulas for the elliptic quantum determinants.
A realization of the elliptic quantum algebra $U_{q,p}(\widehat{sl_2})$ for any given level $k$ is constructed in terms of three free boson fields and their accompanying twisted partners. It can be viewed as the elliptic deformation of Wakimoto realization. Two screening currents are constructed; they commute or anti-commute with $U_{q,p}(\widehat{sl_2})$ modulo total q-differences. The free fields realization for two types vertex operators nominated as the type $I$ and the type $II$ vertex operators are presented. The twisted version of the two types vertex operators are also obtained. They a…
In this note, we prove that the universal affine vertex algebra associated with a simple Lie algebra $\mathfrak{g}$ is simple if and only if the associated variety of its unique simple quotient is equal to $\mathfrak{g}^*$. We also derive an analogous result for the quantized Drinfeld-Sokolov reduction applied to the universal affine vertex algebra.
We consider the motivic Hall algebra of coherent sheaves over an irreducible reduced projective curve of arithmetic genus $1$. We introduce the composition subalgebra in the singular curve case, and show that it is isomorphic to the composition subalgebra for a smooth elliptic curve. As in the case of smooth elliptic case studied by Burban and Schiffmann, the reduced Drinfeld double of the composition subalgebra is isomorphic to the quantum toroidal algebra for $\mathfrak{gl}_1$ (also called Ding-Iohara-Miki algebra), and it inherits automorphisms induced from equivalences of the associated de…
We show that trace functions on modules of topological N=2 super vertex algebras give rise to conformal blocks on elliptic supercurves. We show that they satisfy a system of linear partial differential equations with respect to the modular parameters of the supercurves. Under some finiteness condition on the vertex algebra these differential equations can be interpreted as a connection on the vector bundle of conformal blocks. We show that this connection is equivariant with respect to a natural action of the Jacobi modular group on the modular parameters and the trace functions. In the append…
A new formula connecting the elliptic $6j$-symbols and the fusion of the vertex-face intertwining vectors is given. This is based on the identification of the $k$ fusion intertwining vectors with the change of base matrix elements from Sklyanin's standard base to Rosengren's natural base in the space of even theta functions of order $2k$. The new formula allows us to derive various properties of the elliptic $6j$-symbols, such as the addition formula, the biorthogonality property, the fusion formula and the Yang-Baxter relation. We also discuss a connection with the Sklyanin algebra based on t…
Three examples of free field constructions for the vertex operators of the elliptic quantum group ${\cal A}_{q,p}(\hat{sl}_2)$ are obtained. Two of these (for $p^{1/2}=\pm q^{3/2},p^{1/2}=-q^2$) are based on representation theories of the deformed Virasoro algebra, which correspond to the level 4 and level 2 $Z$-algebra of Lepowsky and Wilson. The third one ($p^{1/2}=q^{3}$) is constructed over a tensor product of a bosonic and a fermionic Fock spaces. The algebraic structure at $p^{1/2}=q^{3}$, however, is not related to the deformed Virasoro algebra. Using these free field constructions, an …
The matrix elements of the $2\times 2$ fusion of Baxter's elliptic $R$-matrix, $R^{(2,2)}(u)$, are given explicitly. Based on a note by Jimbo, we give a formula which show that $R^{(2,2)}(u)$ is gauge equivalent to Fateev's $R$-matrix for the 21-vertex model. Then the crossing symmetry formula for $R^{(2,2)}(u)$ is derived. We also consider the fusion of the vertex-face correspondence relation and derive a crossing symmetry relation between the fusion of the intertwining vectors and their dual vectors.
Quantum vertex algebra theory, developed by H.-S. Li, allows us to apply zeroth products of Frenkel-Jing operators, corresponding to Drinfeld realization of $U_q (\widehat{\mathfrak{sl}}_{n+1})$, on the extension of Koyama vertex operators. As a result, we obtain an infinite-dimensional space and describe its structure as a module for the associative algebra $U_q (\mathfrak{sl}_{n+1})_z$, a certain quantum analogue of $U(\mathfrak{sl}_{n+1})$ which we introduce in this paper.
In this work, we introduce Urod algebras associated to simply-laced Lie algebras as well as the concept of translation of W-algebras. Both results are achieved by showing that the quantum Hamiltonian reduction commutes with tensoring with integrable representations, that is, for $V$ and $L$ an affine vertex algebra and an integrable affine vertex algebra associated with $\mathfrak{g}$, we have the vertex algebra isomorphism $H_{DS,f}^0(V\otimes L)\cong H_{DS,f}^0(V)\otimes L$, where in the left-hand-side the Drinfeld-Sokolov reduction is taken with respect to the diagonal action of $\widehat{\…
In this paper, we propose an elliptic algebra $A_{q,p;\hatπ}(\hat{gl_2})$ which is based on the relations $RLL=LLR^{*}$, where $R$ and $R^{*}$ are the dynamical R-maxtrices of $A^{(1)}_{1}$ type face model with the elliptic moduli shifted by the center of the algebra. From the Ding-Frenkel correspondence, we find that its corresponding (Drinfeld) current algebra at level one is the algebra of screening currents for q-deformed Virasoro algebra.We realize the elliptic algebra at level one by Miki's construction from the bosonization for the type I and type II vertex operators.We also show that t…
Trace map on deformation quantized algebra leads to the algebraic index theorem. In this paper, we investigate a two-dimensional chiral analogue of the algebraic index theorem via the theory of chiral algebras developed by Beilinson and Drinfeld. We construct a trace map on the elliptic chiral homology of the free beta gamma-bc system using the BV quantization framework. As an example, we compute the trace evaluated on the unit constant chiral chain and obtain the formal Witten genus in the Lie algebra cohomology. We also construct a family of elliptic trace maps on coset models.
A generalization of the eight vertex model by means of higher spin representations of the Sklyanin algebra is investigated by the quantum inverse scattering method and the algebraic Bethe Ansatz. Under the well-known string hypothesis low-lying excited states are considered and scattering phase shifts of two physical particles are calculated. The $S$ matrix of two particle states is shown to be proportional to the Baxter's elliptic $R$ matrix with a different elliptic modulus from the original one.
In this article we describe the Hall algebra H_X of an elliptic curve X defined over a finite field and show that the group SL(2,Z) of exact auto-equivalences of the derived category D^b(Coh(X)) acts on the Drinfeld double DH_X of H_X by algebra automorphisms. Next, we study a certain natural subalgebra U_X of DH_X for which we give a presentation by generators and relations. This algebra turns out to be a flat two-parameter deformation of the ring of diagonal invariants C[x_1^{\pm 1}, ..., y_1^{\pm 1},...]^{S_{\infty}}, i.e. the ring of symmetric Laurent polynomials in two sets of countably m…
Two related constructions are associated with screening operators in models of two-dimensional conformal field theory. One is a local system constructed in terms of the braided vector space X spanned by the screening species in a given CFT model and the space of vertex operators Y and the other is the Nichols algebra B(X) and the category of its Yetter--Drinfeld modules, which we propose as an algebraic counterpart, in a "braided" version of the Kazhdan--Lusztig duality, of the representation category of vertex-operator algebras realized in logarithmic CFT models.
We introduce the notion of meromorphic tensor category and illustrate it in several examples. They include representations of quantum affine algebras, chiral algebras of Beilinson and Drinfeld, G-vertex algebras of Borcherds, and representations of GL over a local field. Hopefully the formalism will accomodate various tensor structures arising in relation to the quantized Knizhnik-Zamolodchikov equations and deformed CFT
