Given a finite group $Γ$ and a virtual character $\wt$ on it, we construct a Fock space and associated vertex operators in terms of representation ring of wreath products $Γ\sim S_n$. We recover the character tables of wreath products $Γ\sim S_n$ by vertex operator calculus. When $Γ$ is a finite subgroup of $SU_2$, our construction yields a group theoretic realization of the basic representations of the affine and toroidal Lie algebras of $ADE$ type, which can be regarded as a new form of McKay correspondence.
We introduce two-parameter quantum toroidal algebras of simply laced types and provide their group theoretic realization using finite subgroups of $SL_2(\mathbb C)$ via McKay correspondence. In particular our construction contains a realization of the vertex representation of the two-parameter quantum affine algebras of $ADE$ types.
We establish a $q$-analog of our recent work on vertex representations and the McKay correspondence. For each finite group $Γ$ we construct a Fock space and associated vertex operators in terms of wreath products of $Γ\times \mathbb C^{\times}$ and the symmetric groups. An important special case is obtained when $Γ$ is a finite subgroup of $SU_2$, where our construction yields a group theoretic realization of the representations of the quantum affine and quantum toroidal algebras of $ADE$ type.
Let $N$ be a normal subgroup of a finite group $G$ and $V$ be a fixed finite-dimensional $G$-module. The Poincaré series for the multiplicities of induced modules and restriction modules in the tensor algebra $T(V)=\oplus_{k \geq 0}V^{\otimes k}$ are studied in connection with the McKay-Slodowy correspondence. In particular, it is shown that the closed formulas for the Poincaré series associated with the distinguished pairs of subgroups of $\mathrm{SU}_2$ give rise to the exponents of all untwisted and twisted affine Lie algebras except ${\rm A}_{2n}^{(1)}$.
We continue our study of the McKay Correspondence for grading preserving actions of semisimple Hopf algebras H on (noncommutative) Artin-Schelter regular algebras A. Here, we establish correspondences between module categories over A^H, over A\#H, and over \End_{A^H} A. We also study homological properties of (endomorphism rings of) maximal Cohen-Macaulay modules over A^H.
We consider a class of quasi-Hopf algebras which we call \emph{generalized twisted quantum doubles}. They are abelian extensions $H = \mb{C}[\bar{G}] \bowtie \mb{C}[G]$ ($G$ is a finite group and $\bar{G}$ a homomorphic image), possibly twisted by a 3-cocycle, and are a natural generalization of the twisted quantum double construction of Dijkgraaf, Pasquier and Roche. We show that if $G$ is a subgroup of $SU_2(\mb{C})$ then $H$ exhibits an orbifold McKay Correspondence: certain fusion rules of $H$ define a graph with connected components indexed by conjugacy classes of $\bar{G}$, each connecte…
In establishing a more general version of the McKay correspondence, we prove Auslander's theorem for actions of semisimple Hopf algebras H on noncommutative Artin-Schelter regular algebras A of global dimension two, where A is a graded H-module algebra, and the Hopf action on A is inner faithful with trivial homological determinant. We also show that each fixed ring A^H under such an action arises an analogue of a coordinate ring of a Kleinian singularity.
Let $G$ be a finite subgroup in SU(2), and $Q$ the corresponding affine Dynkin diagram. In this paper, we review the relation between the categories of $G$-equivariant sheaves on $P^1$ and $Rep Q_h$, where $h$ is an orientation of $Q$, constructing an explicit equivalence of corresponding derived categories.
Various algebraic structures have recently appeared in a parallel way in the framework of Hilbert schemes of points on a surface and respectively in the framework of equivariant K-theory [N1,Gr,S2,W], but direct connections are yet to be clarified to explain such a coincidence. We provide several non-trivial steps toward establishing our main conjecture on the isomorphism between the Hilbert quotient of the affine space $\C^{2n}$ by the wreath product $\G ~ S_n$ and Hilbert schemes of points on the minimal resolution of a simple singularity $\C^2 /\G$. We discuss further various implications o…
E7 part: In this paper, we study McKay's E7 observation on the Baby Monster. By investigating so called derived c=7/10 Virasoro vectors, we show that there is a natural correspondence between dihedral subgroups of the Baby Monster and certain subalgebras of the Baby Monster vertex operator algebra which are constructed by the nodes of the affine E7 diagram. This allows us to reinterpret McKay's E7 observation via the theory of vertex operator algebras. For a class of vertex operator algebras including the Moonshine module, we will show that the product of two Miyamoto involutions associated to…
The goal of this paper is to classify ``finite subgroups in U_q sl(2)'' where $q=e^{\piı/l}$ is a root of unity. We propose a definition of such a subgroup in terms of the category of representations of U_q sl(2); we show that this definition is a natural generalization of the notion of a subgroup in a reductive group, and that it is also related with extensions of the chiral (vertex operator) algebra corresponding to sl^(2) at level k=l-2. We show that ``finite subgroups in U_q sl(2)'' are classified by Dynkin diagrams of types A_n, D_{2n}, E_6, E_8 with Coxeter number equal to $l$, give a de…
For each finite subgroup G of SL(n, C), we introduce the generalized Cartan matrix C_{G} in view of McKay correspondence from the fusion rule of its natural representation. Using group theory, we show that the generalized Cartan matrices have similar favorable properties such as positive semi-definiteness as in the classical case of affine Cartan matrices (the case of SL(2,C)). The complete McKay quivers for SL(3,C) are explicitly described and classified based on representation theory.
This paper is a continuation of our paper math.QA/0403010 at which several coset subalgebras of the lattice VOA $V_{\sqrt{2}E_8}$ were constructed and the relationship between such algebras with the famous McKay observation on the extended E_8 diagram and the Monster simple group were discussed. In this article, we shall provide the technical details. We completely determine the structure of the coset subalgebras constructed and show that they are all generated by two conformal vectors of central charge 1/2. We also study the representation theory of these coset subalgebras and show that the p…
We study McKay's observation on the Monster simple group, which relates the 2A-involutions of the Monster simple group to the extended E_8 diagram, using the theory of vertex operator algebras (VOAs). We first consider the sublattices L of the E_8 lattice obtained by removing one node from the extended E_8 diagram at each time. We then construct a certain coset (or commutant) subalgebra U associated with L in the lattice VOA V_{\sqrt{2}E_8}. There are two natural conformal vectors of central charge 1/2 in U such that their inner product is exactly the value predicted by Conway. The Griess alge…
Global dimensions for fusion categories defined by a pair (G,k), where G is a Lie group and k a positive integer, are expressed in terms of Lie quantum superfactorial functions. The global dimension is defined as the square sum of quantum dimensions of simple objects, for the category of integrable modules over an affine Lie algebra at some level. The same quantities can also be defined from the theory of quantum groups at roots of unity or from conformal field theory WZW models. Similar results are also presented for those associated module-categories that can be obtained via conformal embedd…
We establish a new group-theoretic realization of the basic representations of the twisted affine and twisted toroidal algebras of ADE types in the same spirit of our new approach to the McKay correspondence. Our vertex operator construction provides a unified description to the character tables for the spin cover of the wreath product of the twisted hyperoctahedral groups and an arbitrary finite group.
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 boson-fermion correspondence of type A is an isomorphism between two super vertex algebras (and so has singularities in the operator product expansions only at $z = w$). The boson-fermion correspondence of type B plays similarly important role in many areas, including representation theory, integrable systems, random matrix theory and random processes. But the vertex operators describing it have singularities in their operator product expansions at both $z = w$ and $z = -w$, and thus need a more general notion than that of a super vertex algebra. In this paper we present such a notion: the…
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,…
Inspired by the quantum McKay correspondence, we consider the classical $ADE$ Lie theory as a quantum theory over $\mathfrak{sl}_2$. We introduce anti-symmetric characters for representations of quantum groups and investigate the Fourier duality to study the spectral theory. In the $ADE$ Lie theory, there is a correspondence between the eigenvalues of the Coxeter element and the eigenvalues of the adjacency matrix. We formalize related notions and prove such a correspondence for representations of Verlinde algebras of quantum groups: this includes the quiver of any module category acted on by …
The boson-fermion correspondences are an important phenomena on the intersection of several areas in mathematical physics: representation theory, vertex algebras and conformal field theory, integrable systems, number theory, cohomology. Two such correspondences are well known: the types A and B (and their super extensions). As a main result of this paper we present a new boson-fermion correspondence, of type D-A. Further, we define a new concept of twisted vertex algebra of order $N$, which generalizes super vertex algebra. We develop the bicharacter construction which we use for constructing …
Let $V$ be a rational, selfdual, $C_2$-cofinite vertex operator algebra of CFT type, and $G$ a finite automorphism group of $V.$ It is proved that the kernel of the representation of the modular group on twisted conformal blocks associated to $V$ and $G$ is a congruence subgroup. In particular, the $q$-character of each irreducible twisted module is a modular function on the same congruence subgroup. In the case $V$ is the Frenkel-Lepowsky-Meurman's moonshine vertex operator algebra and $G$ is the monster simple group, the generalized McKay-Thompson series associated to any commuting pair in t…
Formulae expressing the trace of the composition of several (up to five) adjoint actions of elements of the Griess algebra of a vertex operator algebra are derived under certain assumptions on the action of the automorphism group. They coincide, when applied to the moonshine module $V^\natural$, with the trace formulae obtained in a different way by S.Norton, and the spectrum of idempotents related to 2A, 2B, 3A and 4A element of the Monster is determined by the representation theory of Virasoro algebra at $c=1/2$, $W_3$ algebra at $c=4/5$ or $W_4$ algebra at $c=1$. The generalization to the t…
In this paper, we present a general construction of 3-transposition groups as automorphism groups of vertex operator algebras. Applying to the moonshine vertex operator algebra, we establish the Conway-Miyamoto correspondences between Fischer 3-transposition groups $\mathrm{Fi}_{23}$ and $\mathrm{Fi}_{22}$ and $c=25/28$ and $c=11/12$ Virasoro vectors of subalgebras of the moonshine vertex operator algebra.
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.
