Ching Hung Lam, Hiroshi Yamauchi
In this article, we show that a framed vertex operator algebra V satisfying the conditions: (1) V is holomorphic (i.e., V is the only irreducible V-module); (2) V is of rank 24; and (3) V_1=0; is isomorphic to the moonshine vertex operator algebra constructed by Frenkel-Lepowsky-Meurman.
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For a simple vertex operator algebra whose Virasoro element is a sum of commutative Virasoro elements of central charge 1/2, two codes are introduced and studied. It is proved that such vertex operator algebras are rational. For lattice vertex operator algebras and related ones, decompositions into direct sums of irreducible modules for the product of the Virasoro algebras of central charge 1/2 are explicitly described. As an application, the decomposition of the moonshine vertex operator algebra is obtained for a distinguished system of 48 Virasoro algebras.
In this paper, we study the residue codes of extremal Type II Z_4-codes of length 24 and their relations to the famous moonshine vertex operator algebra. The main result is a complete classification of all residue codes of extremal Type II Z_4-codes of length 24. Some corresponding results associated to the moonshine vertex operator algebra are also discussed.
For $p = 3,5,7,13$, we consider a ${\mathbb Z}_p$-orbifold construction of the Moonshine vertex operator algebra $V^\natural$. We show that the vertex operator algebra obtained by the ${\mathbb Z}_p$-orbifold construction on the Leech lattice vertex operator algebra $V_Λ$ and a lift of a fixed-point-free isometry of order $p$ is isomorphic to the Moonshine vertex operator algebra $V^\natural$. We also describe the relationship between those ${\mathbb Z}_p$-orbifold constructions and the ${\mathbb Z}_2$-orbifold construction in a uniform manner. In Appendix, we give a characterization of the Mo…
Borcherds's Lie algebra and $C_2$-cofiniteness of vertex operator algebras of moonshine type
本文研究 moonshine 型顶点算子代数的 $C_2$-余有限性，精确刻画中心荷为 24 时 Zhu Poisson 代数的分次结构及 $C_2(V)$ 的显式形式。
A vertex operator algebra of lattice type ADE has a standard integral form which extends a Chevalley basis for its degree 1 Lie algebra. This integral form may be used to define a vertex algebra over a commutative ring $R$ and to get a Chevalley group over $R$ of the same type, acting as automorphisms of this vertex algebra. We define vertex algebras of types BCFG over a commutative ring and certain reduced VAs, then get analogous results about automorphism groups. In characteristics 2 and 3, there are exceptionally large automorphism groups. A covering algebra idea of Frohardt and Griess for …
It is proved that a vertex operator algebra is isomorphic to the moonshine VOA of Frenkel-Lepowsky-Meurman if it satisfies certain conditions. Our two main theorems establish a weak version of the FLM uniqueness conjecture for the moonshine vertex operator algebra. We believe that these are the first such results.
In this article, we describe some maximal $3$-local subgroups of the Monster simple group using vertex operator algebras (VOA). We first study the holomorphic vertex operator algebra obtained by applying the orbifold construction to the Leech lattice vertex operator algebra and a lift of a fixed-point free isometry of order $3$ of the Leech lattice. We also consider some of its special subVOAs and study their stabilizer subgroups using the symmetries of the subVOAs. It turns out that these stabilizer subgroups are $3$-local subgroups of its full automorphism group. As one of our main results, …
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.
We describe a natural structure of an abelian intertwining algebra (in the sense of Dong and Lepowsky) on the direct sum of the untwisted vertex operator algebra constructed {}from the Leech lattice and its (unique) irreducible twisted module. When restricting ourselves to the moonshine module, we obtain a new and conceptual proof that the moonshine module has a natural structure of a vertex operator algebra. This abelian intertwining algebra also contains an irreducible twisted module for the moonshine module with respect to the obvious involution. In addition, it contains a vertex operator s…
In this article, we study the moonshine vertex operator algebra starting with the tensor product of three copies of the vertex operator algebra $V_{\sqrt2E_8}^+$, and describe it by the quadratic space over $\F_2$ associated to $V_{\sqrt2E_8}^+$. Using quadratic spaces and orthogonal groups, we show the transitivity of the automorphism group of the moonshine vertex operator algebra on the set of all full vertex operator subalgebras isomorphic to the tensor product of three copies of $V_{\sqrt2E_8}^+$, and determine the stabilizer of such a vertex operator subalgebra. Our approach is a vertex o…
Vertex operator expressions for Lie algebras of physical states
本研究针对中心荷24的特定顶点算子代数对应的物理态李代数，借助弦论无鬼定理用顶点代数运算表示李括号，在月光模特例下解答了Borcherds关于魔群月光猜想的问题。
We give a new construction of the moonshine VOA V^{\natural} over the real number field. We proved that V^{\natural} has a positive definite invariant bilinear form and its full automorphism group is the Monster simple group. We also construct an infinite series of meromorphic VOAs whose full automorphism groups are finite. We calculate the trace form on V^{\natural} for some element of the Monster.
Given a holomorphic $C_2$-cofinite vertex operator algebra $V$ with graded dimension $j-744$, Borcherds's proof of the monstrous moonshine conjecture implies any finite order automorphism of $V$ has graded trace given by a "completely replicable function", and by work of Cummins and Gannon, these functions are principal moduli of genus zero modular groups. The action of the monster simple group on the monster vertex operator algebra produces 171 such functions, known as the monstrous moonshine functions. We show that 154 of the 157 non-monstrous completely replicable functions cannot possibly …
Moonshine relates three fundamental mathematical objects: the Monster sporadic simple group, the modular function j, and the moonshine module vertex operator algebra. Examining the relationship between modular functions and the representation theory of vertex operator algebras reveals rich structure. In particular, C2-cofiniteness (also called Zhu's finiteness condition) implies the existence of finite generating sets and Poincare-Birkhoff-Witt-like spanning sets for vertex operator algebras and their modules. These spanning sets feature desirable ordering restrictions, e.g., a difference-one …
In this article we prove that the full automorphism group of the baby-monster vertex operator superalgebra constructed by Hoehn is isomorphic to 2xB, where B is the baby-monster sporadic finite simple group and determine irreducible modules for the baby-monster vertex operator algebra. Our result has many corollaries. In particular, we can prove that the Z_2-orbifold construction with respect to a 2A-involution of the Monster applied to the moonshine vertex operator algebra yields the moonshine vertex operator algebra itself again.
In this paper, we study Virasoro vertex algebras and affine vertex algebras over a general field of characteristic $p>2$. More specifically, we study certain quotients of the universal Virasoro and affine vertex algebras by ideals related to the $p$-centers of the Virasoro algebra and affine Lie algebras. Among the main results, we classify their irreducible $\mathbb{N}$-graded modules by explicitly determining their Zhu algebras and show that these vertex algebras have only finitely many irreducible $\mathbb{N}$-graded modules and they are $C_2$-cofinite.
In this paper, we study a new kind of vertex operator algebra related to the twisted Heisenberg-Virasoro algebra, which we call the twisted Heisenberg-Virasoro vertex operator algebra, and its modules. Specifically, we present some results concerning the relationship between the restricted module categories of twisted Heisenberg-Virasoro algebras of rank one and rank two and several different kinds of module categories of their corresponding vertex algebras. We also study fully the structures of the twisted Heisenberg-Virasoro vertex operator algebra, give a characterization of it as a tensor …
In this paper, extensions of nonunitary rational Virasoro vertex operator algebras corresponding to some exceptional modular invariants are constructed. The uniqueness of these extensions is also established.
We give an abstract construction, based on the Belavin-Polyakov-Zamolodchikov equations, of a family of vertex operator algebras of rank $26$ associated to the modified regular representations of the Virasoro algebra. The vertex operators are obtained from the tensor products of intertwining operators for a pair of Virasoro algebras. We explicitly determine the structure coefficients that yield the axioms of VOAs. In the process of our construction, we obtain new hypergeometric identities.
We consider all genus two correlation functions for the Virasoro vacuum descendants of a vertex operator algebra. These are described in terms of explicit generating functions that can be combinatorially expressed in terms of a sequence of globally defined differential operators on which the genus two Siegel modular group $Sp(4,\mathbb{Z})$ has a natural action.
In this paper, we study the structure of a general framed vertex operator algebra. We show that the structure codes (C,D) of a framed VOA V satisfy certain duality conditions. As a consequence, we prove that every framed VOA is a simple current extension of the associated binary code VOA V_C. This result would give a prospect on the classification of framed vertex operator algebras. In addition, the pointwise frame stabilizer of V is studied. We completely determine all automorphisms in this pointwise stabilizer, which are of order 1, 2 or 4. The 4A-twisted sector and the 4A-twisted orbifold t…
We consider all genus zero and genus one correlation functions for the Virasoro vacuum descendants of a vertex operator algebra. These are described in terms of explicit generating functions that can be combinatorially expressed in terms of graph theory related to derangements in the genus zero case and to partial permutations in the genus one case.
In this article, we give a sufficient and necessary condition for the $C_2$-cofiniteness of the 2-cycle permutation orbifold model $(V\otimes V)^σ$ for a $C_2$-cofinite vertex operator algebra and the 2-cycle permutation $σ$ of $V\otimes V$. As an application, we show that the 2-cycle permutation orbifold model of the simple Virasoro vertex operator algebra $L(c,0)$ of minimal central charge $c$ is $C_2$-cofinite.
In this paper, we study a certain deformation $D$ of the Virasoro algebra that was introduced and called $q$-Virasoro algebra by Nigro,in the context of vertex algebras. Among the main results, we prove that for any complex number $\ell$, the category of restricted $D$-modules of level $\ell$ is canonically isomorphic to the category of quasi modules for a certain vertex algebra of affine type. We also prove that the category of restricted $D$-modules of level $\ell$ is canonically isomorphic to the category of $\mathbb{Z}$-equivariant $ϕ$-coordinated quasi modules for the same vertex algebra.…
Sub-structure in module category of Virasoro vertex operator algebras $L(c_{p,q}, 0)$
本文研究了Virasoro顶点算子代数 $L(c_{p,q},0)$ 模 category 的子结构，确定了预模和模张量子category。
In this paper we complete the proof of Ryba's modular moonshine conjectures. We do this by applying Hodge theory to the cohomology of the monster Lie algebra over the ring of p-adic integers in order to calculate the Tate cohomology groups of elements of the monster acting on the monster vertex algebra.
We show using Borcherds products that for any fixed-point free automorphism of the Leech lattice satisfying a "no massless states" condition, the corresponding cyclic orbifold of the Leech lattice vertex operator algebra is isomorphic to the Monster vertex operator algebra. This induces an "orbifold duality" bijection between algebraic conjugacy classes of fixed-point free automorphisms of the Leech lattice satisfying this condition and algebraic conjugacy classes of non-Fricke elements in the Monster. We use the duality to show that non-Fricke Monstrous Lie algebras are Borcherds-Kac-Moody Li…
We first determine the automorphism group of the twisted Heisenberg-Virasoro vertex operator algebra $V_{\mathcal{L}}(\ell_{123},0)$.Then, for any integer $t>1$, we introduce a new Lie algebra $\mathcal{L}_{t}$, and show that $σ_{t}$-twisted $V_{\mathcal{L}}(\ell_{123},0)$($\ell_{2}=0$)-modules are in one-to-one correspondence with restricted $\mathcal{L}_{t}$-modules of level $\ell_{13}$, where $σ_{t}$ is an order $t$ automorphism of $V_{\mathcal{L}}(\ell_{123},0)$. At the end, we give a complete list of irreducible $σ_{t}$-twisted $V_{\mathcal{L}}(\ell_{123},0)$($\ell_{2}=0$)-modules.
A class of vertex operator algebras generated by Virasoro vectors
本文研究一类由简单 Virasoro 向量生成的简单 OZ 型顶点算子代数，证明其由 Griess 代数唯一确定，给出了自同构群及为酉代数的必要条件。
In this note we associate to each Frobenius algebra a vertex algebra, the simplest example being the Virasoro vertex algebra. This construction is analogous to the procedure which associates to a Lie algebra with an invariant bilinear form the vacuum module over the corresponding Kac-Moody Lie algebra.
This paper consists of two parts: (1) Using a Z[1/2]-form of Virasoro vertex operator algebra L(1/2,0) with central charge 1/2, we obtain a modular vertex operator algebra over any field F of finite characteristic different from 2. We determine the generators and classify the irreducible modules for this vertex operator algebra. (2) We investigate modular framed vertex operator algebras. In particular, the rationality of modular framed vertex operator algebras is established. For a modular code vertex operator algebra, the irreducible modules are constructed and classified. Moreover, a Z[1/2]-…
In this paper, we study certain Virasoro frames for lattice vertex operator algebras and their $\Z_2$-orbifolds using linear codes over $\Z_4$. We also compute the corresponding frame stabilizer from the view point of binary codes and $\Z_4$-codes. As an application, we determine the frame stabilizers of several Virasoro frames of the vertex operator algebra $V_{E_8}$ and the moonshine vertex operator algebra $V^\natural$.
Modular Zhu algebra theory and Virasoro vertex algebras
该研究利用高阶朱代数和模转换代数发展顶点代数表示理论，给出正特征域上莫比乌斯顶点代数合理性条件，证明特定离散系列维拉索罗顶点算子代数的合理性，还研究了特征$p>2$时简单商的全纯性情况。
We provide a novel and simple description of Schellekens' seventy-one affine Kac-Moody structures of self-dual vertex operator algebras of central charge 24 by utilizing cyclic subgroups of the glue codes of the Niemeier lattices with roots. We also discuss a possible uniform construction procedure of the self-dual vertex operator algebras of central charge 24 starting from the Leech lattice. This also allows us to consider the uniqueness question for all non-trivial affine Kac-Moody structures. We finally discuss our description from a Lorentzian viewpoint.
The vertex operator algebras and modules associated to the highest weight modules for the Virasoro algebra over an arbitrary field F whose characteristic is not equal to 2 are studied. The irreducible modules of vertex operator algebra L(1/2,0)_F are classified. The rationality of L(1/2,0)_F is established if the characteristic of F is different from 7.
In this article, we study isomorphisms between simple current extensions of a simple VOA. For example, we classify the isomorphism classes of simple current extensions of some VOAs. Moreover, we consider the same simple current extension and describe the normalizer of the abelian automorphism group associated with this extension. In particular, we regard the moonshine module as simple current extensions of five subVOAs V_L^+ for 2-elementary totally even lattices L, and describe corresponding five normalizers of elementary abelian 2-group in the automorphism group of the moonshine module in te…
Selbstduale Vertexoperatorsuperalgebren und das Babymonster (Self-dual Vertex Operator Super Algebras and the Baby Monster)
本文研究自对偶顶点算子超代数，构造中心荷23.5的较短月光模，宝贝怪兽群作用其上，并引入极值SVOAs概念，给出存在性结果。
Zhu's algebra and the $C_2$-algebra of a classically free vertex operator algebra
证明经典自由顶点代数的$C_2$-代数与Zhu代数的分次代数自然同构，并应用该结果证明非平凡全纯CFT型VOA不是经典自由的，包括$E_8$ level 1仿射VOA和月光VOA。
We construct integral forms containing the conformal vector $ω$ in certain tensor powers of the Virasoro vertex operator algebra $L(\frac{1}{2},0)$, and we construct integral forms in certain modules for these algebras. When a triple of modules for a tensor power of $L(\frac{1}{2},0)$ have integral forms, we classify which intertwining operators among these modules respect the integral forms. As an application, we explore how these results might be used to obtain integral forms in framed vertex operator algebras.
Herein we study conformal vectors of a Z-graded vertex algebra of (strong) CFT type. We prove that the full vertex algebra automorphism group transitively acts on the set of the conformal vectors of strong CFT type if the vertex algebra is simple. The statement is equivalent to the uniqueness of self-dual vertex operator algebra structures of a simple vertex algebra. As an application, we show that the full vertex algebra automorphism group of a simple vertex operator algebra of strong CFT type uniquely decomposes into the product of certain two subgroups and the vertex operator algebra automo…
We consider exceptional vertex operator algebras for which particular Casimir vectors constructed from the primary vectors of lowest conformal weight are Virasoro descendants of the vacuum. We discuss constraints on these theories that follow from an analysis of appropriate genus zero and genus one two point correlation functions. We find explicit differential equations for the partition function in the cases where the lowest weight primary vectors form a Lie algebra or a Griess algebra. Examples include the Wess-Zumino-Witten model for Deligne's exceptional Lie algebras and the Moonshine Modu…
The work is devoted to a probably new connection between deformed Virasoro algebra and quantum $\widehat{\mathfrak{sl}}_2$. We give an explicit realization of Virasoro current via vertex operators of level 1 integrable representation of $\widehat{\mathfrak{sl}}_2$. The same is done for a twisted version of deformed Virasoro algebra.
In this paper we prove the fusion rules of $L(c_{1,q},0)$ for all $q\geq1$. Roughly speaking, we consider $L(c_{1,q},0)$ as the limitation of $L(c_{n,nq-1},0)$, where $n\rightarrow\infty$, and the fusion rules of $L(c_{1,q},0)$ follow as the limitation of the fusion rules of $L(c_{n,nq-1},0)$.
We study permutation orbifolds of the $2$-fold and $3$-fold tensor product for the Virasoro vertex algebra $\mathcal{V}_c$ of central charge $c$. In particular, we show that for all but finitely many central charges $\left(\mathcal{V}_c^{\otimes 3}\right)^{\mathbb{Z}_3}$ is a $W$-algebra of type $(2, 4, 5, 6^3 , 7, 8^3 , 9^3 , 10^2 )$. We also study orbifolds of their simple quotients and obtain new realizations of certain rational affine $W$-algebras associated to a principal nilpotent element. Further analysis of permutation orbifolds of the celebrated $(2,5)$-minimal vertex algebra $\mathca…
We investigate a general theory of the Z_2-twisted representations of vertex operator superalgebras. Certain one-to-one correspondence theorems are established. We also give an explicit realization of the Ising model SVOA and its Z_2-twisted modules. As an application, we obtain the Gerald Hoehn's Babymonster SVOA VB and its Z_2-twisted module VB_{tw} from the moonshine VOA V^\nat by cutting off the Ising models. It is also shown in this paper that Aut (VB) is finite.
In this article, we construct explicitly certain moonshine type vertex operator algebras generated by a set of Ising vectors $I$ such that (1) for any $e\neq f\in I$, the subVOA $\mathrm{VOA}(e,f)$ generated by $e$ and $f$ is isomorphic to either $U_{2B}$ or $U_{3C}$; and (2)the subgroup generated by the corresponding Miyamoto involutions $\{τ_e|\,e\in I\}$ is isomorphic to the Weyl group of a root system of type $A_n$, $D_n$, $E_6$, $E_7$ or $E_8$. The structures of the corresponding vertex operator algebras and their Griess algebras are also studied. In particular, the central charge of thes…
Rationality of vertex operator superalgebras with rational conformal weights
本文证明了容许级别下的仿射顶点代数及特定仿射顶点算子超代数、极小 $W$-代数关于一族 Virasoro 元的有理性，并得出弱模范畴特定子范畴半单的结论。
This paper focuses on the connection of holomorphic two-dimensional factorization algebras and vertex algebras which has been made precise in the forthcoming book of Costello-Gwilliam. We provide a construction of the Virasoro vertex algebra starting from a local Lie algebra on the complex plane. Moreover, we discuss an extension of this factorization algebra to a factorization algebra on the category of Riemann surfaces. The factorization homology of this factorization algebra is computed as are the correlation functions. We provide an example of how the Virasoro factorization algebra impleme…
For vertex operator algebra V_{\sqrt{2}A_l} associated to the even lattice \sqrt{2}A_l which is \sqrt{2} times root lattice of type A_l, it was shown by Dong-Li-Maosn-Norton that the Virasoro vector is a sum of l+1 mutually orthogonal conformal vectors with central charges c_i=1-6/(i+2)(i+3) for i=1,...,l and c_{l+1}=2l/(l+3) and the subalgebra T generated by these vectors is a tensor product of Virasoro vertex operator algebras L(c_i,0). In this paper we determine the decomposition of V_{\sqrt{2}A_3} into the sum of irreducible T-modules completely.
In this article, we study Griess algebras and vertex operator subalgebras generated by Ising vectors in a moonshine type VOA such that the subgroup generated by the corresponding Miyamoto involutions has the shape $3^2{:}2$ and any two Ising vectors generate a 3C subVOA $U_{3C}$. We show that such a Griess algebra is uniquely determined, up to isomorphisms. The structure of the corresponding vertex operator algebra is also discussed. In addition, we give a construction of such a VOA inside the lattice VOA $V_{E_8^3}$, which gives an explicit example for Majorana representations of the group $3…
Primary fields of the $q$-deformed Virasoro algebra are constructed. Commutation relations among the primary fields are studied. Adjoint actions of the deformed Virasoro current on the primary fields are represented by the shift operator $Θ_ξ f(x)=f(ξx)$. Four point functions of the primary fields enjoy the connection formula associated with the Boltzmann weights of the fusion Andrews-Baxter-Forrester model.
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…
In this article we study a VOA with two Miyamoto involutions generating $S_3$. In \cite{M3}, Miyamoto showed that a VOA generated by two conformal vectors whose Miyamoto involutions generate an automorphism group isomorphic to $S_3$ is isomorphic to one of the four candidates he listed. We construct one of them and prove that our VOA is actually the same as $\mathrm{VA}(e,f)$ studied by Miyamoto. We also show that there is an embedding into the moonshine VOA. Using our VOA, we can define the 3A-triality in the Monster.
This paper studies restricted modules of gap-$p$ Virasoro algebra $Ł$ and their intrinsic connection to twisted modules of certain vertex algebras. We first establish an equivalence between the category of restricted $Ł$-modules of level $\el$ and the category of twisted modules of vertex algebra $V_{\mathcal{N}_{p}}(\el,0)$, where $\mathcal{N}_{p}$ is a new Lie algebra, $\el:=(\ell_{0},0,\cdots,0)\in\C^{\halfp+1}$, $\ell_{0}\in\C$ is the action of the Virasoro center. Then we focus on the construction and classification of simple restricted $Ł$-modules of level $\el$. More explicitly, we give…
We define an automorphism of VOA of order 3 by using a sub VOA isomorphic to a direct sum of 3-state Potts models $L(\ff,0)$ and an its module $L(\ff,3)$. This automorphism is a 3A element of the monster simple group if $V$ is the moonshine VOA $V^{\natural}$.
Let $\mathcal{F}_{25}$ be the family of irreducible highest weight modules for the Virasoro algebra of central charge $25$ which are not isomorphic to Verma modules. Let $L(25,0)$ be the Virasoro vertex operator algebra of central charge 25. We prove that the fusion rules for the $L(25,0)$-modules in $\mathcal{F}_{25}$ are in correspondence with the tensor rules for the irreducible finite dimensional representations of $sl(2, \mathbb{C})$, extending the known correspondence between modules for the Virasoro algebras of dual central charges 1 and 25.
In the references [HL1]--[HL5] and [H1], a theory of tensor products of modules for a vertex operator algebra is being developed. To use this theory, one first has to verify that the vertex operator algebra satisfies certain conditions. We show in the present paper that for any vertex operator algebra containing a vertex operator subalgebra isomorphic to a tensor product algebra of minimal Virasoro vertex operator algebras (vertex operator algebras associated to minimal models), the tensor product theory can be applied. In particular, intertwining operators for such a vertex operator algebra s…
We identify the algebra of matrix elements of big projective modules in category O with the regular functions on the big Bruhat cell of G. Analogous extensions of the regular representations of the affine Lie and Virasoro algebras yield vertex operator algebras, equipped with two commuting actions with special values of the total central charge. In the generic case, we identify the structure of these VOAs, compute their semi-infinite cohomology. Its superalgebra structure encodes the fusion rules for the associated tensor categories. Full details are provided for the affine sl(2) and Virasoro …
We establish a natural connection of the $q$-Virasoro algebra $D_{q}$ introduced by Belov and Chaltikian with affine Kac-Moody Lie algebras. More specifically, for each abelian group $S$ together with a one-to-one linear character $χ$, we define an infinite-dimensional Lie algebra $D_{S}$ which reduces to $D_{q}$ when $S=\mathbb{Z}$. Guided by the theory of equivariant quasi modules for vertex algebras, we introduce another Lie algebra ${\mathfrak{g}}_{S}$ with $S$ as an automorphism group and we prove that $D_{S}$ is isomorphic to the $S$-covariant algebra of the affine Lie algebra $\widehat{…
In these notes, based on lectures given in Istanbul, we give an introduction both to Monstrous Moonshine and to the classification of rational conformal field theories, using this as an excuse to explore several related structures and go on a little tour of modern math. We will discuss Lie algebras, modular functions, the finite simple group classification, vertex operator algebras, Fermat's Last Theorem, category theory, (generalised) Kac-Moody algebras, denominator identities, the A-D-E meta-pattern, representations of affine algebras, Galois theory, etc. This work is informal and pedagogica…
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