Generalized Kac-Moody Algebras Research Articles

Vertex operator for generalized Kac–Moody algebras associated to the two-sphere and the two-torus

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 [Formula: see text] and [Formula: see text], 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 [Formula: see text]-function.

Semisymmetric Graphs Defined by Finite-Dimensional Generalized Kac–Moody Algebras

Semisymmetric Graphs Defined by Finite-Dimensional Generalized Kac–Moody Algebras

Rigged configurations and the ⁎-involution for generalized Kac–Moody algebras

We construct a uniform model for highest weight crystals and B(∞) for generalized Kac–Moody algebras using rigged configurations. We also show an explicit description of the ⁎-involution on rigged configurations for B(∞): that the ⁎-involution interchanges the rigging and the corigging. We do this by giving a recognition theorem for B(∞) using the ⁎-involution. As a consequence, we also characterize B(λ) as a subcrystal of B(∞) using the ⁎-involution.

$K3$ surfaces with involution and analytic torsion

This is the abstruct of the revised paper. We study the equivariant analytic torsion for K3 surfaces with an anti-symplectic involution with the invariant lattice M (such a surface is called a 2-elementary K3 surface of type M in this paper), and show that it (together with the analytic torsion of the fixed curves) can be identified with the automorphic form on the moduli space characterizing the discriminant locus. Three lattices A_1, II_{1,1}(2), II_{1,9}(2) are of particular interest, because they consist of the building blocks of 2-elementary lattices. An explicit formula is given for them. In particular, if M is twice the Enriques lattice, the automorphic form coincides with Borcherds's Phi-function which confirms an observation by Jorgenson-Todorov and Harvey-Moore. Some other examples are shown to be related to Borcherds's product and generalized Kac-Moody algebras.

BPS algebras, genus zero and the heterotic Monster

In this note, we expand on some technical issues raised in (Paquette et al 2016 Commun. Number Theory Phys. 10 433–526) by the authors, as well as providing a friendly introduction to and summary of our previous work. We construct a set of heterotic string compactifications to 0 + 1 dimensions intimately related to the Monstrous moonshine module of Frenkel, Lepowsky, and Meurman (and orbifolds thereof). Using this model, we review our physical interpretation of the genus zero property of Monstrous moonshine. Furthermore, we show that the space of (second-quantized) BPS-states forms a module over the Monstrous Lie algebras —some of the first and most prominent examples of Generalized Kac–Moody algebras—constructed by Borcherds and Carnahan. In particular, we clarify the structure of the module present in the second-quantized string theory. We also sketch a proof of our methods in the language of vertex operator algebras, for the interested mathematician.

Quasi Affine Generalized Kac Moody Algebras QAGGD<sub>3</sub><sup>(2)</sup>: Dynkin diagrams and root multiplicities for a class of QAGGD<sub>3</sub><sup>(2)</sup>

Objectives: To define the class of quasi affine generalized Kac-Moody algebras QAGGD3 (2), completely classify the non isomorphic, connected Dynkin diagrams associated with QAGGD3 (2) and compute some root multiplicities for this family. Methods: The representation theory of Kac-Moody algebras is applied to compute the multiplicities of roots for a quasi affine family in QAGGD3 (2). Findings: The quasi affine generalized Kac-Moody algebras associated with symmetrizable Generalized Generalized Cartan Matrices (GGCM) of quasi affine type, obtained from the affine family D3 (2), are defined; The connected, non-isomorphic Dynkin diagrams associated with this particular family are completely classified’; Multiplicities of roots of a class GKM algebras QAGGD3 (2), with one simple imaginary root are then determined using the representation theory of Kac Moody algebras; Application: Generalized Kac-Moody algebras find interesting applications in bosonic string theory, classifications in vertex operator theory , monstrous moonshine theory etc.Keywords: Dynkin Diagram, Generalized Kac Moody Algebras (GKM), Imaginary Roots, Quasi Affine, Root Multiplicity

Some constructions of modular forms for the Weil representation of SL2(ℤ)

AbstractModular forms for the Weil representation of SL2(ℤ) play an important role in the theory of automorphic forms on orthogonal groups. In this paper we give some explicit constructions of these functions. As an application, we construct new examples of generalized Kac-Moody algebras whose denominator identities are holomorphic automorphic products of singular weight. They correspond naturally to the Niemeier lattices with root systemsand to the Leech lattice.

Some constructions of modular forms for the Weil representation of SL2(ℤ)

AbstractModular forms for the Weil representation of SL2 (ℤ) play an important role in the theory of automorphic forms on orthogonal groups. In this paper we give some explicit constructions of these functions. As an application, we construct new examples of generalized Kac-Moody algebras whose denominator identities are holomorphic automorphic products of singular weight. They correspond naturally to the Niemeier lattices with root systems and to the Leech lattice.

On a class of generalized Kac–Moody algebras arising from 2-fold affinizations

We study a class of generalized Kac–Moody algebras which are fixed point subalgebras of the covering Kac–Moody algebras of some generalized intersection matrix (GIM) algebras of Slodowy.

Demazure Character Formulas for Generalized Kac–Moody Algebras

We introduce a family of -submodules V w (λ) of the irreducible highest weight U q (𝔤)-module V(λ) of dominant integral highest weight λ for a generalized Kac–Moody algebra 𝔤. We prove that the module V w (λ) is spanned by its global basis, and then give a character formula for V w (λ), which generalizes the Demazure character formula for ordinary Kac–Moody algebras.

A fundamental representation of quantum generalized Kac-Moody algebras with one imaginary simple root

We consider the Borcherds-Cartan matrix obtained from a symmetrizable generalized Cartan matrix by adding one imaginary simple root. We extend the result of Gebert and Teschner (Lett. Math. Phys., 1994, 31: 327-334) to the quantum case. Moreover, we give a connection between the irreducible dominant representations of quantum Kac-Moody algebras and those of quantum generalized Kac-Moody algebras. As the result, a large class of irreducible dominant representations of quantum generalized Kac-Moody algebras were obtained from representations of quantum Kac-Moody algebras through tensor algebras.

Dual Perfect Bases and Dual Perfect Graphs

We introduce the notion of dual perfect bases and dual perfect graphs. We show that every integrable highest weight module $V_q(\lambda)$ over a quantum generalized Kac-Moody algebra $U_{q}(\mathcal{g})$ has a dual perfect basis and its dual perfect graph is isomorphic to the crystal $B(\lambda)$. We also show that the negative half $U_{q}^{-}(\mathcal{g})$ has a dual perfect basis whose dual perfect graph is isomorphic to the crystal $B(\infty)$. More generally, we prove that all the dual perfect graphs of a given dual perfect space are isomorphic as abstract crystals. Finally, we show that the isomorphism classes of finitely generated graded projective indecomposable modules over a Khovanov-Lauda-Rouquier algebra and its cyclotomic quotients form dual perfect bases for their Grothendieck groups.

(-1, 1) and Generalized Kac-Moody Algebras

This study determines all third power-associative (-1, 1) algebras, whose commutator algebras are generalized Kac-Moody algebras.

A generalized Kac–Moody algebra of rank 14

We construct a vertex algebra of central charge 26 from a lattice orbifold vertex operator algebra of central charge 12. The BRST-cohomology group of this vertex algebra is a new generalized Kac–Moody algebra of rank 14. We determine its root space multiplicities and a set of simple roots.

Root multiplicities of hyperbolic Kac–Moody algebras and Fourier coefficients of modular forms

In this paper we consider the hyperbolic Kac–Moody algebra \(\mathcal {F}\) associated with the generalized Cartan matrix Open image in new window. Its connection to Siegel modular forms of genus 2 was first studied by A. Feingold and I. Frenkel. The denominator function of \(\mathcal{F}\) is not an automorphic form. However, Gritsenko and Nikulin extended \(\mathcal{F}\) to a generalized Kac–Moody algebra whose denominator function is a Siegel modular form. Using the Borcherds denominator identity, the denominator function can be written as an infinite product. The exponents that appear in the product are given by Fourier coefficients of a weak Jacobi form. P. Niemann also constructed a generalized Kac–Moody algebra which contains \(\mathcal {F}\) and whose denominator function is related to a product of Dedekind η-functions. In particular, root multiplicities of the generalized Kac–Moody algebra are determined by Fourier coefficients of a modular form. As the main results of this paper, we compute asymptotic formulas for these Fourier coefficients using the method of Hardy–Ramanujan–Rademacher, and obtain an asymptotic bound for root multiplicities of the algebra \(\mathcal{F}\). Our method can be applied to other hyperbolic Kac–Moody algebras and to other modular forms as demonstrated in the later part of the paper.

Path model for representations of generalized Kac–Moody algebras

In [JL], Joseph and Lamprou (2009) generalized Littelmannʼs path model for Kac–Moody algebras to the case of generalized Kac–Moody algebras. We show that Joseph–Lamprouʼs path model can be embedded into Littelmannʼs path model for a certain Kac–Moody algebra constructed from the Borcherds–Cartan datum of a given generalized Kac–Moody algebra; note that this is not an embedding of crystals. Using this embedding, we give a new proof of the isomorphism theorem for path crystals, obtained in Joseph and Lamprou (2009) [JL, §7.4]. Moreover, for Joseph–Lamprouʼs path crystals, we give a decomposition rule for tensor product and a branching rule for restriction to Levi subalgebras. Also, we obtain a characterization of standard paths in terms of a certain monoid which can be thought of as a generalization of a Weyl group.

CATEGORIFICATION OF QUANTUM GENERALIZED KAC–MOODY ALGEBRAS AND CRYSTAL BASES

We construct and investigate the structure of the Khovanov-Lauda–Rouquier algebras R and their cyclotomic quotients Rλ which give a categorification of quantum generalized Kac–Moody algebras. Let U𝔸(𝔤) be the integral form of the quantum generalized Kac–Moody algebra associated with a Borcherds–Cartan matrix A = (aij)i, j ∈ I and let K0(R) be the Grothendieck group of finitely generated projective graded R-modules. We prove that there exists an injective algebra homomorphism [Formula: see text] and that Φ is an isomorphism if aii ≠ 0 for all i ∈ I. Let B(∞) and B(λ) be the crystals of [Formula: see text] and V(λ), respectively, where V(λ) is the irreducible highest weight Uq(𝔤)-module. We denote by 𝔅(∞) and 𝔅(λ) the isomorphism classes of irreducible graded modules over R and Rλ, respectively. If aii ≠ 0 for all i ∈ I, we define the Uq(𝔤)-crystal structures on 𝔅(∞) and 𝔅(λ), and show that there exist crystal isomorphisms 𝔅(∞) ≃ B(∞) and 𝔅(λ) ≃ B(λ). One of the key ingredients of our approach is the perfect basis theory for generalized Kac–Moody algebras.

A FAMILY OF GENERALIZED KAC–MOODY ALGEBRAS AND DEFORMATION OF MODULAR FORMS

We develop an analogue of Gindikin–Karpelevich formula for a family of generalized Kac–Moody algebras, attached to Borcherds–Cartan matrices consisting of only one positive entry in the diagonal. As an application, we obtain a deformation of Fourier coefficients of modular forms such as the modular j-function and Ramanujan τ-function.

Automorphic products, generalized Kac-Moody algebras and string amplitudes

We review automorphic products and generalized Kac-Moody algebras from a physics point of view. We discuss the appearance of automorphic products in BPS-saturated one-loop quantities in heterotic string theory. At particular points in moduli space, with enhanced gauge symmetry \U0001d525, these products can be used to define a generalized Kac-Moody algebra \U0001d50a(\U0001d524++) as an automorphic correction of the Lorentzian Kac-Moody algebra \U0001d524++, which is obtained through double extension of the complement \U0001d524 = (\U0001d5228 ⊕ \U0001d5228)/\U0001d525. The root multiplicities of \U0001d50a(\U0001d524++) are then encoded in the Fourier coefficients of certain modular forms, which appear directly in the integrand of the one-loop quantities. We review particular examples of this extension for compactifications with \U0001d4a9 = 2 and \U0001d4a9 = 4 space-time supersymmetry.

Nakajima Monomials, Strip Bundles, and the Crystals B(∞) Over U q (B ∞), and U q (C ∞), and U q (D ∞)

In this article, we give two new realizations of the crystal base B(∞) of the negative part of the quantum generalized Kac–Moody algebra U q (𝔤) (𝔤 =B ∞, C ∞ or D ∞) using Nakajima monomials and strip bundles, respectively. Furthermore, we discuss the relation between monomial realization and strip bundle realization.