Bosonized q-vertex operators related to the 4-dimensional evaluation modules of the quantum affine superalgebra $U_q[\hat{sl(2|1)}]$ are constructed for arbitrary level $k=α$, where $α\neq 0, -1$ is a complex parameter appearing in the 4-dimensional evaluation representations. They are intertwiners among the level-$α$ highest weight Fock-Wakimoto modules. Screen currents which commute with the action of $U_q[\hat{sl(2|1)}]$ up to total differences are presented. Integral formulae for N-point functions of type I and type II q-vertex operators are proposed.
We calculate the exchange relations of vertex operators of $U_q(\hat{sl_2})$ at level-two from its bosonic realization. The corresponding invertibility relation of type I vertex operators is also studied.
We determine the exchange relations of the level-one q-vertex operators of the quantum affine superalgebra $U_q[\hat{gl(N|N)}]$. We study in details the level-one irreducible highest weight representations of $U_q[\hat{gl(2|2)}]$, and compute the characters and supercharacters associated with these irreducible modules.
We study the level-one irreducible highest weight representations of $U_q[\hat{gl(1|1)}]$ and associated q-vertex operators. We obtain the exchange relations satisfied by the vertex operators. The characters and supercharacters associated with these irreducible representations are calculated.
We construct the screening currents of the quantum superalgebra $U_q(\hat{sl}(N|1))$ for an arbitrary level $k \neq -N+1$. We show that these screening currents commute with the superalgebra modulo total difference. We propose bosonizations of the vertex operators by using the screening currents. We check that these vertex operators are the intertwiners among the Fock-Wakimoto representation and the typical representation for rank $N \leq 4$.
We classify the irreducible restricted modules for the affine Nappi-Witten Lie algebra $\hat{H}_{4}$ with some natural conditions. It turns out the representation theory of $\hat{H}_{4}$ is quite different from the theory of representations of Heisenberg algebras. We also study the extension of the vertex operator algebra $V_{\hat{H}_{4}}(\ell,0)$ by the even lattice $L$. We give the structure of the extension $V_{\hat{H}_{4}}(\ell,0)\otimes \C[L]$ and its irreducible modules via irreducible representations of $V_{\hat{H}_{4}}(\ell,0)$ viewed as a vertex algebra.
Level-one representations of the quantum affine superalgebra $U_q[\hat{gl(N|N)}]$ associated to the appropriate non-standard system of simple roots and $q$-vertex operators (intertwining operators) associated with the level-one modules are constructed explicitly in terms of free bosonic fields.
We study the level-one irreducible highest weight representations of the quantum affine superalgebra $U_q[\hat{sl(N|1)}]$, and calculate their characters and supercharacters. We obtain bosonized q-vertex operators acting on the irreducible $U_q[\hat{sl(N|1)}]$-modules and derive the exchange relations satisfied by the vertex operators. We give the bosonization of the multi-component super $t-J$ model by using the bosonized vertex operators.
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…
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…
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.
For the Klein group $K$, $k\in\mathbb{Z}_{\geqslant 1}$ and $m\in\mathbb{Z}_{\geqslant 4}$, we study the representations of the orbifold vertex operator algebra $L_{\hat{\mathfrak{sl}_2}}(k,0)^{K}$ and the commutant vertex operator algebra of $L_{\hat{\mathfrak{so}_m}}(3,0)$ in $L_{\hat{\mathfrak{so}_m}}(1,0)^{\otimes 3}$ which can be realized as the orbifold vertex operator subalgebra $L_{\hat{\mathfrak{sl}_2}}(2m,0)^{K}$ or its extension. All the irreducible modules for $L_{\hat{\mathfrak{sl}_2}}(k,0)^{K}$ and $C_{{L_{\hat{\mathfrak{so}_m}}(1,0)}^{\otimes 3}}({L_{\hat{\mathfrak{so}_m}}(3,0)}…
Free field relization of vertex operators for lvel two modules of $U_q(\hat{sl(2)})$ is shown through the free field relization of the modules given by Idzumi in Ref.[4,5]. We constructed types I and II vertex operators when the spin of the addociated evaluation modules is 1/2 and typ II's for the spin 1.
We study graded nonlocal $\underline{\mathsf{q}}$-vertex algebras and we prove that they can be generated by certain sets of vertex operators. As an application, we consider the family of graded nonlocal $\underline{\mathsf{q}}$-vertex algebras $V_{c,1}$, $c\geq 1$, associated with the principal subspaces $W(cΛ_0)$ of the integrable highest weight $U_q (\hat{\mathfrak{sl}}_2)$-modules $L(cΛ_0)$. Using quantum integrability, we derive combinatorial bases for $V_{c,1}$ and compute the corresponding character formulae.
In this paper, we construct a Q-operator as a trace of a representation of the universal R-matrix of $U_q(\hat{sl}_2)$ over an infinite-dimensional auxiliary space. This auxiliary space is a four-parameter generalization of the q-oscillator representations used previously. We derive generalized T-Q relations in which 3 of these parameters shift. After a suitable restriction of parameters, we give an explicit expression for the Q-operator of the 6-vertex model and show the connection with Baxter's expression for the central block of his corresponding operator.
In this paper, we continue the study on toroidal vertex algebras initiated in \cite{LTW}, to study concrete toroidal vertex algebras associated to toroidal Lie algebra $L_{r}(\hat{\frak{g}})=\hat{\frak{g}}\otimes L_r$, where $\hat{\frak{g}}$ is an untwisted affine Lie algebra and $L_r=$\mathbb{C}[t_{1}^{\pm 1},\ldots,t_{r}^{\pm 1}]$. We first construct an $(r+1)$-toroidal vertex algebra $V(T,0)$ and show that the category of restricted $L_{r}(\hat{\frak{g}})$-modules is canonically isomorphic to that of $V(T,0)$-modules.Let $c$ denote the standard central element of $\hat{\frak{g}}$ and set $S…
The diagonalisation of the transfer matrices of solvable vertex models with alternating spins is given. The crystal structure of (semi-)infinite tensor products of finite-dimensional $U_q(\hat{sl}_2)$ crystals with alternating dimensions is determined. Upon this basis the vertex models are formulated and then solved by means of $U_q(\hat{sl}_2)$ intertwiners.
The admissible modules for $\hat{sl}_2$ are studied from the point of view of vertex operator algebra. If $l$ is rational such that $l+2={p\over q}$ for some coprime positive integers $p\ge 2$ and $q$, Kac and Wakimoto found finitely many distinguished irreducible representations for $\hat{sl}_2$, called admissible representations. In this paper we prove that the vertex operator algebra $L(l,0)$ associated to irreducible highest weight representation of $l$ is not rational if $l$ is not a positive integer. However if we change the Virasoro algebra in certain way, $L(l,0)$ becomes a rational ve…
Based on the screened vertex operators associated with the affine screening operators, we introduce the formal power series f^{hat{gl}_N}(x,p|s,kappa|q,t) which we call the non-stationary Ruijsenaars function. We identify it with the generating function for the Euler characteristics of the affine Laumon spaces. When the parameters s and kappa are suitably chosen, the limit t rightarrow q of f^{hat{gl}_N}(x,p|s,kappa|q,q/t) gives us the dominant integrable characters of hat{sl}_N multiplied by 1/(p^N;p^N)_infty (i.e. the hat{gl}_1 character). Several conjectures are presented for f^{hat{gl}_N}(…
We construct a bosonization of the quantum superalgebra $U_q(\hat{sl}(N|1))$ for an arbitrary level $k$. We construct the screening that commutes with the quantum superalgebra for an arbitrary level $k \neq -N+1$. We propose a bosonization of the vertex operator that gives the intertwiner among the Wakimoto realization and the typical representation.
We consider sets of screening operators with fermionic screening currents. We study sums of vertex operators which formally commute with the screening operators assuming that each vertex operator has rational contractions with all screening currents with only simple poles. We develop and use the method of $qq$-characters which are combinatorial objects described in terms of deformed Cartan matrix. We show that each qq-character gives rise to a sum of vertex operators commuting with screening operators and describe ways to understand the sum in the case it is infinite. We discuss combinatorics …
We give a criterion for the complete reducibility of modules satisfying a composability condition for a meromorphic open-string vertex algebra $V$ using the first cohomology of the algebra. For a $V$-bimodule $M$, let $\hat{H}^{1}_{\infty}(V, M)$ be the first cohomology of $V$ with the coefficients in $M$. Let $\hat{Z}^{1}_{\infty}(V, M)$ be the subspace of $\hat{H}^{1}_{\infty}(V, M)$ canonically isomorphic to the space of derivations obtained from the zero mode of the right vertex operators of weight $1$ elements such that the difference between the skew-symmetric opposite action of the left…
Let $L$ be an even positive definite lattice with no roots, i.e., $L(2)=\{x\in L\mid (x|x)=2\}=\emptyset$. Let $g\in O(L)$ be an isometry of order $4$ such that $g^2=-1$ on $L$. In this article, we determine the full automorphism group of the orbifold vertex operator algebra $V_L^{\hat{g}}$. As our main result, we show that $Aut(V_L^{\hat{g}})$ is isomorphic to $N_{Aut(V_L)}(\langle \hat{g}\rangle)/ \langle\hat{g}\rangle $ unless $L\cong \sqrt{2}E_8$ or $BW_{16}$.
We construct a vertex representation for the quantum toroidal algebra through the quantum general linear algebra. Using a new realization of the quantum general linear algebra we construct vertex operators for root vectors on the basic representation of the affine Lie algebra $gl_n$ and show that the simple generators give rise a realization of the quantum toroidal algebra with two parameters.
For the simple Lie algebra $ \frak{so}_m$, we study the commutant vertex operator algebra of $ L_{\hat{\frak{so}}_{m}}(n,0)$ in the $n$-fold tensor product $ L_{\hat{\frak{so}}_{m}}(1,0)^{\otimes n}$. It turns out that this commutant vertex operator algebra can be realized as a fixed point subalgebra of $L_{\hat{\frak{so}}_{n}}(m,0)$ (or its simple current extension) associated with a certain abelian group. This result may be viewed as a version of level-rank duality.
The following integrability theorem for vertex operator algebras V satisfying some finiteness conditions(C_2-cofinite and CFT-type) is proved: the vertex operator subalgebra generated by a simple Lie subalgebra {\frak g} of the weight one subspace V_1 is isomorphic to the irreducible highest weight \hat{\frak g}-module L(k, 0) for a positive integer k, and V is an integrable \hat{\frak g}-module. The case in which {\frak g} is replaced by an abelian Lie subalgebra is also considered, and several consequences of integrability are discussed.
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 …
We study and classify systems of certain screening operators arising in a generalized vertex operator algebra, or more generally an abelian intertwining algebra with an associated vertex operator (super)algebra. Screening pairs arising from weight one primary vectors acting commutatively on a lattice vertex operator algebra (the vacuum module) are classified into four general types, one type of which has been shown to play an important role in the construction and study of certain important families of $\mathcal{W}$-vertex algebras. These types of screening pairs we go on to study in detail th…
Let $\mathfrak{g}$ be a simple finite-dimensional Lie superalgebra with a non-degenerate supersymmetric even invariant bilinear form, $f$ a nilpotent element in the even part of $\mathfrak{g}$, $Γ$ a good grading of $\mathfrak{g}$ for $f$ and $\mathcal{W}^{k}(\mathfrak{g},f;Γ)$ the $\mathcal{W}$-algebra associated with $\mathfrak{g},f,k,Γ$ defined by the generalized Drinfeld-Sokolov reduction. In this paper, we present each $\mathcal{W}$-algebra as the intersection of kernels of the screening operators, acting on the tensor vertex superalgebra of an affine vertex superalgebra and a neutral fre…
The $U_q(\hat{gl}(N|N))$-analog of the half-infinite $t-J$ model with a boundary is considered by using the vertex operator approach. We find explicit bosonic formula of the boundary state in the integrable highest-weight module over the quantum superalgebra $U_q(\hat{gl}(N|N))$.
The Lie superalgebra SD of regular differential operators on the super circle has a universal central extension \hat{SD}. For each c\in C, the vacuum module M_c(\hat{SD}) of central charge c admits a vertex superalgebra structure, and M_c(\hat{SD}) \cong M_{-c}(\hat{SD}). The irreducible quotient V_c(\hat{SD}) of the vacuum module is known as the super W_{1+\infty} algebra. We show that for each integer n>0, V_n(\hat{SD}) has a minimal strong generating set consisting of 4n fields, and we identify it with a W-algebra associated to the purely odd simple root system of gl(n|n). Finally, we reali…
We bosonize certain components of level $\ell$ $U_q(\hat{sl}_2)$-intertwiners of $(\ell + 1)$-dimensions. For $\ell = 2$, these intertwiners, after certain modification by bosonic vertex operators, are added to the algebra $U_q(\hat{sl}_2)$ at level 2 to construct all irreducible highest weight representations of level 1 for the quantum affine algebra $U_q(\hat{sp}_4)$.
We present a vertex operator algebra which is an extension of the level $k$ vertex operator algebra for the $\hat{sl}_2$ conformal field theory. We construct monomial basis of its irreducible representations.
We re-examine the level-one irreducible highest weight representations of the quantum affine superalgebra $U_q(\hat{sl}(2|1))$ and derive the characters and supercharacters associated with these representations. We calculate the exchange relations of the vertex operators and find that these vertex operators satisfy the graded Faddeev-Zamolodchikov algebra. We give an integral expression of the correlation functions of the q-deformed supersymmetric t-J model and derive the difference equations which they satisfy.
We study a vertex operator algebra (VOA) V related to the M(3,p) Virasoro minimal series. This VOA reduces in the simplest case p=4 to the level two integrable vacuum module of $\hat{sl}_2$. On V there is an action of a commutative current a(z), which is an analog of the current e(z) of $\hat{sl}_2$. Our main concern is the subspace W generated by this action from the highest weight vector of V. Using the Fourier components of a(z), we present a monomial basis of W and a semi-infinite monomial basis of V. We also give a Gordon type formula for their characters.
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.
For an affine Lie algebra $\hat{\mathfrak g}$ the coefficients of certain vertex operators which annihilate level $k$ standard $\hat{\mathfrak g}$-modules are the defining relations for level $k$ standard modules. In this paper we study a combinatorial structure of the leading terms of these relations for level $k=2$ standard $\hat{\mathfrak g}$-modules for affine Lie algebras of type $C_{n}\sp{(1)}$ and the main result is a construction of combinatorially parameterized relations among the coefficients of annihilating fields. It is believed that the constructed relations among relations will p…
We present a general vertex operator construction based on the Fock space for an affine Lie algebras of type $A$. This construction allows us to give a unified treatment for both the homogeneous and principle realizations of the affine Lie algebras $\hat{gl}_N$ as well as for some extended affine Lie algebras coordinatized by certain quantum tori.
We find sufficient conditions for the construction of vertex algebraic intertwining operators, among generalized Verma modules for an affine Lie algebra $\hat{\mathfrak{g}}$, from $\mathfrak{g}$-module homomorphisms. When $\mathfrak{g}=\mathfrak{sl}_2$, these results extend previous joint work with J. Yang, but the method used here is different. Here, we construct intertwining operators by solving Knizhnik-Zamolodchikov equations for three-point correlation functions associated to $\hat{\mathfrak{g}}$, and we identify obstructions to the construction arising from the possible non-existence of …
A level-one representation of the quantum affine superalgebra $\U_q(\hat{\frak{sl}}(M+1|N+1))$ and vertex operators associated with the fundamental representations are constructed in terms of free bosonic fields. Character formulas of level-one irreducible highest weight modules of $\U_q(\hat{\frak{sl}}(2|1))$ are conjectured.
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…
Let $L$ be an even (positive definite) lattice and $g\in O(L)$. In this article, we prove that the orbifold vertex operator algebra $V_{L}^{\hat{g}}$ has group-like fusion if and only if $g$ acts trivially on the discriminant group $\mathcal{D}(L)=L^*/L$ (or equivalently $(1-g)L^*<L$). We also determine their fusion rings and the corresponding quadratic space structures when $g$ is fixed point free on $L$. By applying our method to some coinvariant sublattices of the Leech lattice $Λ$, we prove a conjecture proposed by G. Höhn. In addition, we also discuss a construction of certain holomorphic…
In this paper, it is shown that the diagonal coset vertex operator algebra $C(L_{\mathfrak{g}}(k+2,0),L_{\mathfrak{g}}(k,0)\otimes L_{\mathfrak{g}}(2,0))$ is rational and $C_2$-cofinite in case $\mathfrak{g}=so(2n), n\geq 3$ and $k$ is an admissible number for $\hat{\mathfrak{g}}$. It is also shown that the diagonal coset vertex operator algebra $C(L_{sl_2}(k+4,0),L_{sl_2}(k,0)\otimes L_{sl_2}(4,0))$ is rational and $C_2$-cofinite in case $k$ is an admissible number for $\hat{sl_2}$. Furthermore, irreducible modules of $C(L_{sl_2}(k+4,0),L_{sl_2}(k,0)\otimes L_{sl_2}(4,0))$ are classified in c…
We study the representation theory of the N=1 super Heisenberg-Virasoro vertex algebra at level zero, which extends the previous work on the Heisenberg-Virasoro vertex algebra arXiv:math/0201314, arXiv:1405.1707 and arXiv:1703.00531 to the super case. We calculated all characters of irreducible highest weight representations by investigating certain Fock space representations. Quite surprisingly, we found that the maximal submodules of certain Verma modules are generated by subsingular vectors. The formulas for singular and subsingular vectors are obtained using screening operators appearing i…
For a finite-dimensional simple Lie algebra $\mathfrak{g}$, we use the vertex tensor category theory of Huang and Lepowsky to identify the category of standard modules for the affine Lie algebra $\hat{\mathfrak{g}}$ at a fixed level $\ell\in\mathbb{N}$ with a certain tensor category of finite-dimensional $\mathfrak{g}$-modules. More precisely, the category of level $\ell$ standard $\hat{\mathfrak{g}}$-modules is the module category for the simple vertex operator algebra $L_{\hat{\mathfrak{g}}}(\ell,0)$, and as is well known, this category is equivalent as an abelian category to $\mathbf{D}(\ma…
Using vertex operators, we construct explicitly Lusztig's $\mathbb Z[q, q^{-1}]$-lattice for the level one irreducible representations of quantum affine algebras of ADE type. We then realize the level one irreducible modules at roots of unity and show that the q-dimension is still given by the Weyl-Kac character formula. As a consequence we also answer the corresponding question of realizing the affine Kac-Moody Lie algebras of simply laced type at level one in finite characteristic.
Based on the vertex-face correspondence, we give an algebraic analysis formulation of correlation functions of the $k\times k$ fusion eight-vertex model in terms of the corresponding fusion SOS model. Here $k\in Z_{>0}$. A general formula for correlation functions is derived as a trace over the space of states of lattice operators such as the corner transfer matrices, the half transfer matrices (vertex operators) and the tail operator. We give a realization of these lattice operators as well as the space of states as objects in the level $k$ representation theory of the elliptic algebra $U_{q,…
We introduce the quasi-Hopf superalgebras which are $Z_2$ graded versions of Drinfeld's quasi-Hopf algebras. We describe the realization of elliptic quantum supergroups as quasi-triangular quasi-Hopf superalgebras obtained from twisting the normal quantum supergroups by twistors which satisfy the graded shifted cocycle condition, thus generalizing the quasi-Hopf twisting procedure to the supersymmetric case. Two types of elliptic quantum supergroups are defined, that is the face type $B_{q,λ}(G)$ and the vertex type $A_{q,p}[\hat{sl(n|n)}]$ (and $A_{q,p}[\hat{gl(n|n)}]$), where $G$ is any Kac-…
We study the 19-vertex model associated with the quantum group $U_q(\hat{sl_2})$ at critical regime $|q|=1$. We give the realizations of the type-I vertex operators in terms of free bosons and free fermions. Using these free field realizations, we give the integral representations for the correlation functions.
We study the explicit formula of Lusztig's integral forms of the level one quantum affine algebra $U_q(\hat{sl}_2)$ in the endomorphism ring of symmetric functions in infinitely many variables tensored with the group algebra of $\mathbb Z$. Schur functions are realized as certain orthonormal basis vectors in the vertex representation associated to the standard Heisenberg algebra. In this picture the Littlewood-Richardson rule is expressed by integral formulas, and is used to define the action of Lusztig's $\mathbb Z[q, q]$-form of $U_q(\hat{sl}_2)$ on Schur polynomials.
We write the integral formula of Tarasov-Varchenko type for the solutions to the quantum Knizhnik-Zamolodchikov associated with a tensor product the of vector representations of sl_n. We consider the case where the deformation parameter q satisfies |q|=1. We use the bosonization of the type II vertex operators in order to find the hypergeometric pairing in this setting.
In this note we prove the Davies-Foda-Jimbo-Miwa-Nakayashiki conjecture on the asymptotics of the composition of n quantum vertex operators for the quantum affine algebra U_q(\hat sl_2), as n goes to infinity. For this purpose we define and study the leading eigenvalue and eigenvector of the product of two components of the quantum vertex operator. This eigenvector and the corresponding eigenvalue were recently computed by M.Jimbo. The results of his computation are given in Section 4.
An analog of the minimal unitary series representations for the deformed Virasoro algebra is constructed using vertex operators of the quantum affine algebra $U_q(\hat{sl}_2)$. A similar construction is proposed for the elliptic algebra $A_{q,p}(\hat{sl}_2)$.
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…
We prove that certain screening operators in conformal field theory obey the algebra relations of a corresponding Nichols algebra with diagonal braiding. Our result proves in particular a long-standing expectation that the Borel parts of small quantum groups appear as the algebra of screening operators. The proof is based on a novel, intimate relation between Hopf algebras, vertex algebras and a class of multivalued analytic special functions, which are generalizations of Selberg integrals. We prove that the zeroes of these special functions correspond to the algebra relations of the respectiv…
We use the theory of vertex operator algebras and intertwining operators to obtain systems of q-difference equations satisfied by the graded dimensions of the principal subspaces of certain level k standard modules for \hat{\goth{sl}(3)}. As a consequence we establish new formulas for the graded dimensions of the principal subspaces corresponding to the highest-weights iΛ_1+(k-i)Λ_2, where 1 \leq i \leq k and Λ_1 and Λ_2 are fundamental weights of \hat{\goth{sl}(3)}.
