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Let g be a symmetrisable Kac–Moody algebra and V an integrable g –module in category O . We show that the monodromy of the (normally ordered) rational Casimir connection on V can be made equivariant with respect to the Weyl group W of g , and therefore defines an action of the braid group B W on V . We then prove that this action is canonically equivalent to the quantum Weyl group action of B W on a quantum deformation of V , that is an integrable, category O module V over the quantum group U ħ g such that V / ħ V is isomorphic to V . This extends a result of the second author which is valid for g semisimple. [ABSTRACT FROM AUTHOR]

Inspired by Sugawara operators, we introduce a new basis for the universal non-degenerate Whittaker module of level κ over the classical affine Kac-Moody algebra of type B N (1) , C N (1) or D N (1). This basis is different from the one derived from the PBW basis and would help us to understand the structure of Whittaker modules. As a result, we classify simple non-degenerate Whittaker modules for the classical affine Kac-Moody algebras whether at the critical or noncritical level. [ABSTRACT FROM AUTHOR]

We construct new string vacua featuring Brane Supersymmetry Breaking, based on T4/ℤN orientifolds with O5+ planes and D 5 ¯ branes. Differently from the original construction, in these vacua the cancellation of twisted R-R charges of D-branes and orientifold planes is achieved in a non-trivial way, and results in rigid configurations with few open-string moduli, which highly restrict the possible deformations of the model. The breaking of supersymmetry and, consequently, the non-vanishing untwisted and twisted NS-NS tadpoles, generates a rich potential both for the dilaton and the blown-up moduli. We also uncover the stringy origin of the J form entering the gauge kinetic functions in the low-energy effective action, and display its relation to the NS-NS tadpoles for the scalars in the tensor multiplets. As a result, the J form can be consistently identified also when supersymmetry is broken, thus solving an embarrassing puzzle related to its very existence. We also discuss the unitarity constraints for one-dimensional defects in these vacua, where the Kač-Moody algebra for D9 and D 5 ¯ can be realised both in the left-moving and right-moving sectors. [ABSTRACT FROM AUTHOR]

We derive a new system of integrable derivative non-linear Schrödinger equations with an L operator, quadratic in the spectral parameter with coefficients belonging to the Kac–Moody algebra A2(1). The construction of the fundamental analytic solutions of L is outlined and they are used to introduce the scattering data, thus formulating the scattering problem for the Lax pair L,M.

Recent studies on the holographic descriptions of Kerr black holes indicate that the conformal or the warped conformal symmetries are responsible for the Kerr black hole physics at both background and perturbation levels. In the present paper, we extend the validity of these studies to the case of accelerating Kerr black hole. By invoking a set of non-trivial diffeomorphisms near the horizon bifurcation surface of the accelerating Kerr black hole, the Dirac brackets among charges of the diffeomorphisms form the symmetry algebra of a warped CFT which consists of one Virasoro and one Kac-Moody algebra with central extensions. This provides the evidence for warped CFTs being possible holographic dual to accelerating Kerr black holes. The thermal entropy formula of the warped CFT fixed by modular parameters and vacuum charges reproduces the entropy of the rotating black hole with acceleration. [ABSTRACT FROM AUTHOR]

We derive a new system of integrable derivative non-linear Schrödinger equations with an L operator, quadratic in the spectral parameter with coefficients belonging to the Kac–Moody algebra A 2 (1) . The construction of the fundamental analytic solutions of L is outlined and they are used to introduce the scattering data, thus formulating the scattering problem for the Lax pair L , M . [ABSTRACT FROM AUTHOR]

Let U q − be the negative half of the quantum group associated to the Kac-Moody algebra g , and U _ q − the quantum group obtained by a folding of g. Let A = Z [ q , q − 1 ]. McNamara showed that U _ q − is categorified over a certain extension ring A ˜ of A , by using the folding theory of KLR algebras. He posed a question whether A ˜ coincides with A or not. In this paper, we give an affirmative answer for this problem. [ABSTRACT FROM AUTHOR]

In this paper, we study the finite dimensional modules over indefinite Kac–Moody Lie algebras. We prove that any Kac–Moody Lie algebra with indecomposable indefinite Cartan matrix has no non-trivial finite dimensional simple module. This result would be indispensable for researching finite dimensional modules over GIM Lie algebras. [ABSTRACT FROM AUTHOR]

The Miura transformation plays a crucial role in the study of integrable systems. There have been various extensions of the Miura transformation, which have been used to relate different kinds of integrable equations and to classify the bi‐Hamiltonian structures. In this paper, we are mainly concerned with the geometric aspects of the Miura transformation. The generalized Miura transformations from the mKdV‐type hierarchies to the KdV‐type hierarchies are constructed under both algebraic and geometric settings. It is shown that the Miura transformations not only relate integrable curve flows in different geometries but also induce the transition between different moving frames. Moreover, the Miura transformation gives the factorization of generating operators of constraint Gelfand–Dickey hierarchy. Other geometric formulations are also investigated. [ABSTRACT FROM AUTHOR]

In this paper, we study a family of infinite-dimensional Lie algebras X ˆ S , where X stands for the type: A , B , C , D , and S is an abelian group, which generalize the A , B , C , D series of trigonometric Lie algebras. Among the main results, we identify X ˆ S with what are called the covariant algebras of the affine Lie algebra L S ˆ with respect to some automorphism groups, where L S is an explicitly defined associative algebra viewed as a Lie algebra. We then show that restricted X ˆ S -modules of level ℓ naturally correspond to equivariant quasi modules for affine vertex algebras related to L S. Furthermore, for any finite cyclic group S , we completely determine the structures of these four families of Lie algebras, showing that they are essentially affine Kac-Moody Lie algebras of certain types. [ABSTRACT FROM AUTHOR]

In this paper, we characterize simple smooth modules over some infinite-dimensional Z -graded Lie algebras. More precisely, we prove that if one specific positive root element of a Z -graded Lie algebra g locally finitely acts on a simple g -module V , then V is a smooth g -module. These infinite-dimensional Z -graded Lie algebras include the Virasoro algebra, affine-Virasoro algebras, the (twisted, mirror) Heisenberg-Virasoro algebras, the planar Galilean conformal algebra, and many others. This result for untwisted affine Kac-Moody algebras holds unless we change the condition from "locally finitely" to "locally nilpotently". We also show that these are not the case for the Heisenberg algebra. [ABSTRACT FROM AUTHOR]

In 2014, Braverman, Kazhdan, Patnaik and Bardy-Panse, Gaussent and Rousseau associated Iwahori-Hecke algebras to Kac-Moody groups over non-Archimedean local fields. In a previous paper, we defined and studied their principal series representations. In 1982, Kato provided an irreducibility criterion for these representations, in the reductive case. We had obtained partially this criterion in the Kac-Moody case. In this paper, we prove this criterion in the Kac-Moody case. [ABSTRACT FROM AUTHOR]

A net in P 2 is a configuration of lines A and points X satisfying certain incidence properties. Nets appear in a variety of settings, ranging from quasigroups to combinatorial design to classification of Kac–Moody algebras to cohomology jump loci of hyperplane arrangements. For a matroid M and rank r, we associate a monomial ideal (a monomial variant of the Orlik–Solomon ideal) to the set of flats of M of rank ≤ r . In the context of line arrangements in P 2 , applying Alexander duality to the resulting ideal yields insight into the combinatorial structure of nets. [ABSTRACT FROM AUTHOR]

The classification of Nilpotent Lie algebras is an open and difficult problem. The aim of the present work is to give some classes of Nilpotent Lie algebras using Kac-Moody Lie al gebras. [ABSTRACT FROM AUTHOR]

In finite-dimensional simple Lie algebras and affine Kac-Moody Lie algebras, Chevalley involutions are crucial ingredients of the modular theory. Towards establishing the modular theory for extended affine Lie algebras, we investigate the existence of "Chevalley involutions" for Lie tori and extended affine Lie algebras. We first discuss how to lift a Chevalley involution from the centerless core which is characterized to be a centerless Lie torus to the core and then to the entire extended affine Lie algebra. We then prove by a type-dependent argument the existence of Chevalley involutions for centerless Lie tori. [ABSTRACT FROM AUTHOR]

Recently the quantum toroidal superalgebra ℰ m | n associated with g l m | n was introduced by L. Bezerra and E. Mukhin, which is not a quantum Kac–Moody algebra. The quantum toroidal superalgebra ℰ m | n exploits infinite sequences of generators and relations of the form, which are called Drinfeld realization. In this paper, we use only finite set of generators and relations to define an associative algebra ℰ m | n ′ and show that there exists an epimorphism from ℰ m | n ′ to the quantum toroidal superalgebra ℰ m | n . In particular, the structure of ℰ m | n ′ enjoys some properties like Drinfeld–Jimbo realization. [ABSTRACT FROM AUTHOR]

We propose a refined version of the existing conjectural asymptotic formula for the moments of the family of quadratic Dirichlet L -functions over rational function fields. Our prediction is motivated by two natural conjectures that provide sufficient information to determine the analytic properties (meromorphic continuation, location of poles, and the residue at each pole) of a certain generating function of moments of quadratic L -functions. The number field analogue of our asymptotic formula can be obtained by a similar procedure, the only difference being the contributions coming from the archimedean and even places, which require a separate analysis. To avoid this additional technical issue, we present, for simplicity, the asymptotic formula only in the rational function field setting. This has also the advantage of being much easier to test. [ABSTRACT FROM AUTHOR]

Based on the realization of quantum Borcherds–Bozec algebra U∼$\operatorname{\widetilde{\bf U}}\nolimits$ and quantum‐generalized Kac–Moody algebra BU∼${}^{\texttt {B}}\operatorname{\widetilde{\bf U}}\nolimits$ via semi‐derived Ringel–Hall algebra of a quiver with loops, we deduce the braid group actions of U∼$\operatorname{\widetilde{\bf U}}\nolimits$ introduced by Fan and Tong recently and establish braid group actions for BU∼${}^{\texttt {B}}\operatorname{\widetilde{\bf U}}\nolimits$ by applying the Bernstein‐Gelfand‐Ponomarev (BGP)‐reflection functors to semi‐derived Ringel–Hall algebras. [ABSTRACT FROM AUTHOR]

We study the spaces of twisted conformal blocks attached to a $\Gamma$ -curve $\Sigma$ with marked $\Gamma$ -orbits and an action of $\Gamma$ on a simple Lie algebra $\mathfrak {g}$ , where $\Gamma$ is a finite group. We prove that if $\Gamma$ stabilizes a Borel subalgebra of $\mathfrak {g}$ , then the propagation theorem and factorization theorem hold. We endow a flat projective connection on the sheaf of twisted conformal blocks attached to a smooth family of pointed $\Gamma$ -curves; in particular, it is locally free. We also prove that the sheaf of twisted conformal blocks on the stable compactification of Hurwitz stack is locally free. Let $\mathscr {G}$ be the parahoric Bruhat–Tits group scheme on the quotient curve $\Sigma /\Gamma$ obtained via the $\Gamma$ -invariance of Weil restriction associated to $\Sigma$ and the simply connected simple algebraic group $G$ with Lie algebra $\mathfrak {g}$. We prove that the space of twisted conformal blocks can be identified with the space of generalized theta functions on the moduli stack of quasi-parabolic $\mathscr {G}$ -torsors on $\Sigma /\Gamma$ when the level $c$ is divisible by $|\Gamma |$ (establishing a conjecture due to Pappas and Rapoport). [ABSTRACT FROM AUTHOR]

In [12] , the authors classified a class of simple modules over untwised affine Kac-Moody Lie algebras, on these modules each weight vector of the positive parts of affine Kac-Moody Lie algebras acts locally finitely. In this paper, for all affine-Virasoro algebras we also classify all simple modules with the similar property. We determine that there are precisely three classes of simple modules on which each weight vector of the positive part of any affine-Virasoro algebra acts locally finitely: simple highest weight or Whittaker Virasoro algebra modules, simple highest weight or Whittaker affine Lie algebra modules, and simple highest weight or Whittaker affine-Virasoro algebra modules which are neither simple Virasoro algebra modules nor simple affine Lie algebra modules. We also obtain three equivalent conditions to characterize such simple modules over affine-Virasoro algebras. [ABSTRACT FROM AUTHOR]

Let U ħ (g ^) be the untwisted affinization of a symmetrizable quantum Kac-Moody algebra U ħ (g) . For ℓ ∈ C , we construct an ħ -adic quantum vertex algebra V g ^ , ħ (ℓ , 0) , and establish a one-to-one correspondence between ϕ -coordinated V g ^ , ħ (ℓ , 0) -modules and restricted U ħ (g ^) -modules of level ℓ . Suppose that ℓ is a positive integer. We construct a quotient ħ -adic quantum vertex algebra L g ^ , ħ (ℓ , 0) of V g ^ , ħ (ℓ , 0) , and establish a one-to-one correspondence between certain ϕ -coordinated L g ^ , ħ (ℓ , 0) -modules and restricted integrable U ħ (g ^) -modules of level ℓ . Suppose further that g is of finite type. We prove that L g ^ , ħ (ℓ , 0) / ħ L g ^ , ħ (ℓ , 0) is isomorphic to the simple affine vertex algebra L g ^ (ℓ , 0) . [ABSTRACT FROM AUTHOR]

Any symmetric closed subset of a finite crystallographic root system must be a closed subroot system. This is not, in general, true for real affine root systems. In this paper, we determine when this is true and also give a very explicit description of symmetric closed subsets of real affine root systems. At the end, using our results, we study the correspondence between symmetric closed subsets of real affine root systems and the regular subalgebras generated by them. [ABSTRACT FROM AUTHOR]

We provide an explicit combinatorial description of highest weights of simple highest weight modules over the simple affine vertex algebra L κ (sl n + 1) with n ∈ N of admissible level κ. For admissible simple highest weight modules corresponding to the principal, subregular and maximal parabolic nilpotent orbits we give a realization using the Gelfand–Tsetlin theory, which also allows us to obtain a realization of certain classes of simple admissible sl 2 -induced modules in these orbits. In particular, simple admissible sl 2 -induced modules are fully classified for the principal nilpotent orbit. [ABSTRACT FROM AUTHOR]

The article discusses the concept of faithful representations of finite conformal Lie algebras. It explores the question of whether every finite conformal Lie algebra, freely generated as an H-module, has a faithful representation on a finitely generated free H-module. The article presents theorems and proofs related to this question, as well as the relationship between the Ado theorem and the structure of universal enveloping associative algebras. It also discusses the radical splitting problem in finite-dimensional Lie algebras and its adaptation to the conformal case. The article concludes with a classification theorem for semisimple associative conformal algebras with finite faithful representation. [Extracted from the article]

We use semi-derived Ringel-Hall algebras of quivers with loops to realize the whole quantum Borcherds-Bozec algebras and quantum generalized Kac-Moody algebras. [ABSTRACT FROM AUTHOR]

We discuss the emergence of a new symmetry generator in a Hamiltonian realisation of four-dimensional gauge theories in the flat space foliated by retarded (advanced) time. It generates an asymptotic symmetry that acts on the asymptotic fields in a way different from the usual large gauge transformations. The improved canonical generators, corresponding to gauge and asymptotic symmetries, form a classical Kac-Moody charge algebra with a non-trivial central extension. In particular, we describe the case of electromagnetism, where the charge algebra is the U(1) current algebra with a level proportional to the coupling constant of the theory, κ = 4π2/e2. We construct bilinear generators yielding Virasoro algebras on the null boundary. We also provide a non-Abelian generalization of the previous symmetries by analysing the evolution of Yang-Mills theory in Bondi coordinates. [ABSTRACT FROM AUTHOR]

Let g be a hyperbolic Kac-Moody algebra of rank 2. We give a polyhedral realization of the crystal basis for the extremal weight module of extremal weight λ , where λ is an integral weight whose Weyl group orbit has neither a dominant integral weight nor an antidominant integral weight. [ABSTRACT FROM AUTHOR]

In this paper, we give an explicit realization of all irreducible modules in Chari's category \widetilde {\mathcal O} for the affine Kac-Moody algebra A_{1}^{(1)} by using the idea of free fields. We work on a much more general setting which also gives us explicit realizations of all simple weight modules for certain current algebra of \mathfrak {sl}_2(\mathbb {C}) with finite weight multiplicities, including the polynomial current algebra \mathfrak {sl}_2(\mathbb {C})\otimes \mathbb {C}[t], the loop algebra \mathfrak {sl}_2(\mathbb {C})\otimes \mathbb {C}[t,t^{-1}] and the three-point Lie algebra \mathfrak {sl}_2(\mathbb {C})\otimes \mathbb {C}[t,t^{-1},(t-1)^{-1}] arisen in the work by Kazhdan-Lusztig. [ABSTRACT FROM AUTHOR]

We study the connection of Moyal deformations of self-dual gravity and self-dual Yang-Mills theory to chiral higher-spin theories, and also to deformations of operator algebras in celestial holography. The relation to Moyal deformations illuminates various aspects of the structure of chiral higher-spin theories. For instance, the appearance of the self-dual kinematic algebra in all the theories considered here leads via the double copy to vanishing tree-level scattering amplitudes. Regarding celestial holography, the Moyal deformation of self-dual gravity was recently shown to lead to the loop algebra of W∧, and we obtain here the analogous deformation of a Kac-Moody algebra corresponding to Moyal-deformed self-dual Yang-Mills theory. We also introduce the celestial algebras for various chiral higher-spin theories. [ABSTRACT FROM AUTHOR]

We investigate two dimensional (2d) quantum field theories which exhibit Non-Lorentzian Kač-Moody (NLKM) algebras as their underlying symmetry. Our investigations encompass both 2d Galilean (speed of light c → ∞) and Carrollian (c → 0) CFTs with additional number of infinite non-Abelian currents, stemming from an isomorphism between the two algebras. We alternate between an intrinsic and a limiting analysis. Our NLKM algebra is constructed first through a contraction and then derived from an intrinsically Carrollian perspective. We then go on to use the symmetries to derive a Non-Lorentzian (NL) Sugawara construction and ultimately write down the NL equivalent of the Knizhnik Zamolodchikov equations. All of these are also derived from contractions, thus providing a robust cross-check of our analyses. [ABSTRACT FROM AUTHOR]

Dedicated to the memory of Lars Brink (1943–2022) for his help and friendship over many years In this paper, we argue that the existence of solitons in theories in which local symmetries are spontaneously broken requires space–time to be enlarged by additional coordinates that are associated with large local transformations. In the context of gravity theories, the usual coordinates of space–time can be thought of arising in this way. E theory automatically contains such an enlarged space–time. We propose that space–time appears in an underlying theory when the local symmetries are spontaneously broken. [ABSTRACT FROM AUTHOR]

A map φ on a Lie algebra L is called commuting if [ φ (x) , x ] = 0 for all x ∈ L. Let g be a Kac-Moody algebra over an algebraically closed field of characteristic 0. In this paper, we determine the skew-symmetric biderivations and the linear commuting maps of g . As applications, the commuting derivations and the commuting automorphisms of g are gained. [ABSTRACT FROM AUTHOR]

String functions are important building blocks of characters of integrable highest modules over affine Kac–Moody algebras. Kac and Peterson computed string functions for affine Lie algebras of type A 1 (1) in terms of Dedekind eta functions. We obtain new symmetries for string functions by exploiting their natural setting of Hecke-type double-sums, where special double-sums are expressed in terms of Appell–Lerch functions and theta functions, where we point out that Appell–Lerch functions are the building blocks of Ramanujan's classical mock theta functions. We then demonstrate the utility of the new symmetries by giving new proofs of classical string function identities. [ABSTRACT FROM AUTHOR]

We use induction from parabolic subalgebras with infinite-dimensional Levi factor to construct new families of irreducible representations for arbitrary affine Kac-Moody algebras. Our first construction defines a functor from the category of Whittaker modules over the Levi factor of a parabolic subalgebra to the category of modules over the affine Lie algebra. The second functor sends tensor products of a module over the affine part of the Levi factor (in particular any weight module) and of a Whittaker module over the complement Heisenberg subalgebra to the affine Lie algebra modules. Both functors preserve irreducibility when the central charge is nonzero. [ABSTRACT FROM AUTHOR]

In this paper, for an arbitrary Kac-Moody Lie algebra {\mathfrak g} and a diagram automorphism \mu of {\mathfrak g} satisfying certain natural linking conditions, we introduce and study a \mu-twisted quantum affinization algebra {\mathcal U}_\hbar \left (\hat {\mathfrak g}_\mu \right) of {\mathfrak g}. When {\mathfrak g} is of finite type, {\mathcal U}_\hbar \left (\hat {\mathfrak g}_\mu \right) is Drinfeld's current algebra realization of the twisted quantum affine algebra. When \mu =\mathrm {id} and {\mathfrak g} in affine type, {\mathcal U}_\hbar \left (\hat {\mathfrak g}_\mu \right) is the quantum toroidal algebra introduced by Ginzburg, Kapranov and Vasserot. As the main results of this paper, we first prove a triangular decomposition for {\mathcal U}_\hbar \left (\hat {\mathfrak g}_\mu \right). Second, we give a simple characterization of the affine quantum Serre relations on restricted {\mathcal U}_\hbar \left (\hat {\mathfrak g}_\mu \right)-modules in terms of "normal order products". Third, we prove that the category of restricted {\mathcal U}_\hbar \left (\hat {\mathfrak g}_\mu \right)-modules is a monoidal category and hence obtain a topological Hopf algebra structure on the "restricted completion" of {\mathcal U}_\hbar \left (\hat {\mathfrak g}_\mu \right). Last, we study the classical limit of {\mathcal U}_\hbar \left (\hat {\mathfrak g}_\mu \right) and abridge it to the quantization theory of extended affine Lie algebras. In particular, based on a classification result of Allison-Berman-Pianzola, we obtain the \hbar-deformation of all nullity 2 extended affine Lie algebras. [ABSTRACT FROM AUTHOR]

A systematic construction of a class of integrable hierarchy is discussed in terms of the twisted affine A 2 r (2) Lie algebra. The zero curvature representation of the time evolution equations is shown to be classified according to its algebraic structure and according to its vacuum solutions. It is shown that a class of models admit both zero and constant (non-zero) vacuum solutions. Another consists essentially of integral non-local equations and can be classified into two sub-classes, one admitting only zero vacuum and another of constant strictly non-zero vacuum solutions. The two-dimensional gauge potentials in the vacuum play a crucial ingredient and are shown to be expanded in powers of the vacuum parameter v 0 . Soliton solutions are constructed from vertex operators, which for the non-zero vacuum solutions correspond to deformations characterized by v 0 . [ABSTRACT FROM AUTHOR]

Let X be a Cartan datum of symmetric type, with an admissible automorphism σ on X , and X _ the Cartan datum induced from (X , σ). Let U q − (resp. U _ q −) be the negative part of the quantized enveloping algebra associated to X (resp. X _). Lusztig constructed the canonical basis B of U q − and the canonical signed basis B _ ˜ of U _ q − by making use of the geometric theory of quivers. By normalizing the sign of B _ ˜ , he obtained the canonical basis B _ of U _ q − , and a natural bijection ▪. In this paper, assuming the existence of B for U q − , we construct the canonical basis B _ of U _ q − , and a bijection ▪, by an elementary method, subject to the condition that the order of σ is odd. In the case where the order is even, we obtain the corresponding result for the canonical signed basis. [ABSTRACT FROM AUTHOR]

By using the notion of extension of Kac–Moody algebras for higher-dimensional compact manifolds recently introduced, we show that for the two-torus 1 × 1 and the two-sphere 2 , these extensions, as well as extensions of the Virasoro algebra can be obtained naturally from the usual Kac–Moody and Virasoro algebras. Explicit fermionic realizations are proposed. In order to have well-defined generators, beyond the usual normal ordering prescription, we introduce a regulator and regularize infinite sums by means of Riemann ζ -function. [ABSTRACT FROM AUTHOR]

We pursue our study of generalized Kac–Moody and Virasoro algebras defined on compact homogeneous manifolds. Extending the well-known vertex operator in the case of the two-torus or the two-sphere, we obtain explicit bosonic realizations of the semi-direct product of the extension of Kac–Moody and Virasoro algebras on 1 × 1 and 2 , respectively. As for the fermionic realization previously constructed, in order to have well defined algebras, we introduce, beyond the usual normal ordering prescription, a regulator and regularize infinite sums by means of the Riemann ζ -function. [ABSTRACT FROM AUTHOR]

Let F q be a finite field with q not powers of 2 or 3. In this paper, we mainly clarify whether the Lie graph defined by a finite-dimensional generalized Kac–Moody algebra is Cayley graph or not. More precisely, we prove that the graph L (M 1 , 3 , q) is a Cayley graph on a group of dihedral type, neither L (M 2 , 4 , q) nor L (M 3 , 6 , q) is vertex transitive. Moreover, L (M 1 , 3 , q) is a Hamiltonian graph. As an application, we construct an infinite family of semisymmetric (i.e., regular, edge-transitive and non-vertex-transitive graphs) graphs. [ABSTRACT FROM AUTHOR]

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We explicitly construct, in terms of Gelfand–Tsetlin tableaux, a new family of simple positive energy representations for the simple affine vertex algebra in the minimal nilpotent orbit of. These representations are quotients of induced modules over the affine Kac–Moody algebra and include in particular all admissible simple highest weight modules and all simple modules induced from. Any such simple module in the minimal nilpotent orbit has bounded weight multiplicities. [ABSTRACT FROM AUTHOR]

We discuss the properties of recently constructed "single-valued" celestial four-gluon amplitudes. We show that the amplitude factorizes into the "current" part and the "scalar" part. The current factor is given by the group-dependent part of the Wess-Zumino-Witten correlator of four holomorphic currents with a non-vanishing level of Kač-Moody algebra. The scalar factor can be expressed in terms of a complex integral of the Koba-Nielsen form, similar to the integrals describing four-point correlators in Coulomb gas models and, more generally, in the infinite central charge limit of Liouville theory. The scalar part can be also obtained by a dimensional reduction of a single D = 4 conformal block and the shadow block from Minkowski space to the celestial sphere. [ABSTRACT FROM AUTHOR]

A linear map φ on a Lie algebra g is called a triple derivation if φ ([ [ x , y ] , z ]) = [ [ φ (x) , y ] , z ] + [ [ x , φ (y) ] , z ] + [ [ x , y ] , φ (z) ] for all x , y , z ∈ g. Let g be a Kac-Moody algebra over an algebraically closed field of characteristic 0, and p an arbitrary parabolic subalgebra of g. In this paper, we prove that any triple derivation of p is a derivation, which generalizes the corresponding existing result for finite-dimensional simple Lie algebra. [ABSTRACT FROM AUTHOR]

We construct a tau cover of the generalized Drinfeld–Sokolov hierarchy associated with an arbitrary affine Kac–Moody algebra with gradations s⩽1$\mathrm{s}\leqslant \mathbb {1}$ and derive its Virasoro symmetries. By imposing the Virasoro constraints we obtain solutions of the Drinfeld–Sokolov hierarchy of Witten–Kontsevich and of Brezin–Gross–Witten types, and of those characterized by certain ordinary differential equations of Painlevé type. We also show the existence of affine Weyl group actions on the space of solutions of such ordinary differential equations, which generalizes the theory of Noumi and Yamada on affine Weyl group symmetries of the Painlevé‐type equations. [ABSTRACT FROM AUTHOR]

GIM Lie algebras are the generalizations of Kac–Moody Lie algebras. However, the structures of GIM Lie algebras are more complex than the latter, so they have encountered many new difficulties to study their representation theory. In this paper, we classify all finite dimensional simple modules over the GIM Lie algebra Q n + 1 (2 , 1) as well as those over Θ 2 n + 1 . [ABSTRACT FROM AUTHOR]

The paper is devoted to real Hamiltonian forms of -dimensional Toda field theories related to exceptional simple Lie algebras, and to the spectral theory of the associated Lax operators. Real Hamiltonian forms are a special type of "reductions " of Hamiltonian systems, similar to real forms of semisimple Lie algebras. Examples of real Hamiltonian forms of affine Toda field theories related to exceptional complex untwisted affine Kac–Moody algebras are studied. Along with the associated Lax representations, we also formulate the relevant Riemann–Hilbert problems and derive the minimal sets of scattering data that uniquely determine the scattering matrices and the potentials of the Lax operators. [ABSTRACT FROM AUTHOR]

We give the first positive formulas for the weights of every simple highest weight module L (λ) over an arbitrary Kac–Moody algebra. Under a mild condition on the highest weight, we also express the weights of L (λ) as an alternating sum similar to the Weyl–Kac character formula. To obtain these results, we show the following data attached to a highest weight module are equivalent: (i) its integrability, (ii) the convex hull of its weights, (iii) the Weyl group symmetry of its character, and (iv) when a localization theorem is available, its behavior on certain codimension one Schubert cells. We further determine precisely when the above datum determines the weights themselves. Moreover, we use condition (iv) to relate localizations of the convex hull of the weights with the introduction of poles of the corresponding D -module on certain divisors, which answers a question of Brion. Many of these results are new even in finite type. We prove similar assertions for highest weight modules over a symmetrizable quantum group. [ABSTRACT FROM AUTHOR]

Let \mathfrak {g} be an affine Kac-Moody Lie algebra and let \lambda, \mu be two dominant integral weights for \mathfrak {g}. We prove that under some mild restriction, for any positive root \beta, V(\lambda)\otimes V(\mu) contains V(\lambda +\mu -\beta) as a component, where V(\lambda) denotes the integrable highest weight (irreducible) \mathfrak {g}-module with highest weight \lambda. This extends the corresponding result by Kumar from the case of finite dimensional semisimple Lie algebras to the affine Kac-Moody Lie algebras. One crucial ingredient in the proof is the action of Virasoro algebra via the Goddard-Kent-Olive construction on the tensor product V(\lambda)\otimes V(\mu). Then, we prove the corresponding geometric results including the higher cohomology vanishing on the \mathcal {G}-Schubert varieties in the product partial flag variety \mathcal {G}/\mathcal {P}\times \mathcal {G}/\mathcal {P} with coefficients in certain sheaves coming from the ideal sheaves of \mathcal {G}-sub-Schubert varieties. This allows us to prove the surjectivity of the Gaussian map. [ABSTRACT FROM AUTHOR]

We introduce the notion of a cylindrical bialgebra, which is a quasitriangular bialgebra H endowed with a universal K-matrix, i.e. , a universal solution of a generalized reflection equation, yielding an action of cylindrical braid groups on tensor products of its representations. We prove that new examples of such universal K-matrices arise from quantum symmetric pairs of Kac-Moody type and depend upon the choice of a pair of generalized Satake diagrams. In finite type, this yields a refinement of a result obtained by Balagović and Kolb, producing a family of non-equivalent solutions interpolating between the quasi-K-matrix originally due to Bao and Wang and the full universal K-matrix. Finally, we prove that this construction yields formal solutions of the generalized reflection equation with a spectral parameter in the case of finite-dimensional representations over the quantum affine algebra U_qL\mathfrak {sl}_{2}. [ABSTRACT FROM AUTHOR]