Hyperbolic Kac-Moody Algebras Research Articles

A hyperbolic Kac-Moody Calogero model

A new kind of quantum Calogero model is proposed, based on a hyperbolic Kac-Moody algebra. We formulate nonrelativistic quantum mechanics on the Minkowskian root space of the simplest rank-3 hyperbolic Lie algebra AE3 with an inverse-square potential given by its real roots and reduce it to the unit future hyperboloid. By stereographic projection this defines a quantum mechanics on the Poincaré disk with a unique potential. Since the Weyl group of AE3 is a ℤ2 extension of the modular group PSL(2,ℤ), the model is naturally formulated on the complex upper half plane, and its potential is a real modular function. We present and illustrate the relevant features of AE3, give some approximations to the potential and rewrite it as an (almost everywhere convergent) Poincaré series. The standard Dunkl operators are constructed and investigated on Minkowski space and on the hyperboloid. In the former case we find that their commutativity is obstructed by rank-2 subgroups of hyperbolic type (the simplest one given by the Fibonacci sequence), casting doubt on the integrability of the model. An appendix with Don Zagier investigates the computability of the potential. We foresee applications to cosmological billards and to quantum chaos.

Polyhedral realization of the crystal bases for extremal weight modules over quantized hyperbolic Kac-Moody algebras of rank 2

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.

Crystals of Lakshmibai-Seshadri paths and extremal weight modules over quantum hyperbolic Kac-Moody algebras of rank 2

Let be a hyperbolic Kac-Moody algebra of rank 2, and let λ be an arbitrary integral weight. We denote by the crystal of all Lakshmibai-Seshadri paths of shape λ. Let be the extremal weight module of extremal weight λ generated by the (cyclic) extremal weight vector of weight λ, and let be the crystal basis of with the element corresponding to We prove that the connected component of containing is isomorphic, as a crystal, to the connected component of containing the straight line Furthermore, we prove that if λ satisfies a special condition, then the crystal basis is isomorphic, as a crystal, to the crystal As an application of these results, we obtain an algorithm for computing the number of elements of weight μ in where are the fundamental weights, in the case that is symmetric.

English

In this paper, we study a special class of indefinite Kac-Moody algebras. Based on the study of hyperbolic Kac-Moody algebras, we give the definition of type Kac-Moody algebras and study some properties of this special type Kac-Moody algebras.

Connectedness of Lakshmibai-Seshadri path crystals for hyperbolic Kac-Moody algebras of rank 2

Let be a hyperbolic Kac-Moody algebra of rank 2. We give a necessary and sufficient condition for the crystal graph of the Lakshmibai-Seshadri path crystal to be connected for an arbitrary integral weight λ.

Path model for an extremal weight module over the quantized hyperbolic Kac-Moody algebra of rank 2

Let be a hyperbolic Kac-Moody algebra of rank 2, and set where are the fundamental weights. Denote by the extremal weight module of extremal weight λ with the extremal weight vector, and by the crystal basis of with the element corresponding to We prove that (i) is connected, (ii) the subset of elements of weight μ in is a finite set for every integral weight μ, and (iii) every extremal element in is contained in the Weyl group orbit of (iv) is irreducible. Finally, we prove that the crystal basis is isomorphic, as a crystal, to the crystal of Lakshmibai-Seshadri paths of shape λ.

Lakshmibai–Seshadri paths for hyperbolic Kac–Moody algebras of rank 2

ABSTRACTLet 𝔤 be a hyperbolic Kac–Moody algebra of rank 2, and set , where are the fundamental weights for 𝔤; note that λ is neither dominant nor antidominant. Let 𝔹(λ) be the crystal of all Lakshmibai–Seshadri paths of shape λ. We prove that (the crystal graph of) 𝔹(λ) is connected. Furthermore, we give an explicit description of Lakshmibai–Seshadri paths of shape λ.

On Brane Solutions with Intersection Rules Related to Lie Algebras

The review is devoted to exact solutions with hidden symmetries arising in a multidimensional gravitational model containing scalar fields and antisymmetric forms. These solutions are defined on a manifold of the form M = M0 x M1 x . . . x Mn , where all Mi with i >= 1 are fixed Einstein (e.g., Ricci-flat) spaces. We consider a warped product metric on M. Here, M0 is a base manifold, and all scale factors (of the warped product), scalar fields and potentials for monomial forms are functions on M0 . The monomial forms (of the electric or magnetic type) appear in the so-called composite brane ansatz for fields of forms. Under certain restrictions on branes, the sigma-model approach for the solutions to field equations was derived in earlier publications with V.N.Melnikov. The sigma model is defined on the manifold M0 of dimension d0 ≠ 2 . By using the sigma-model approach, several classes of exact solutions, e.g., solutions with harmonic functions, S-brane, black brane and fluxbrane solutions, are obtained. For d0 = 1 , the solutions are governed by moduli functions that obey Toda-like equations. For certain brane intersections related to Lie algebras of finite rank—non-singular Kac–Moody (KM) algebras—the moduli functions are governed by Toda equations corresponding to these algebras. For finite-dimensional semi-simple Lie algebras, the Toda equations are integrable, and for black brane and fluxbrane configurations, they give rise to polynomial moduli functions. Some examples of solutions, e.g., corresponding to finite dimensional semi-simple Lie algebras, hyperbolic KM algebras: H2(q, q) , AE3, HA(1)2, E10 and Lorentzian KM algebra P10 , are presented.

Quantum supersymmetric cosmological billiards and their hidden Kac-Moody structure

We study the quantum fermionic billiard defined by the dynamics of a quantized supersymmetric squashed three-sphere (Bianchi IX cosmological model within $D=4$ simple supergravity). The quantization of the homogeneous gravitino field leads to a 64-dimensional fermionic Hilbert space. We focus on the 15- and 20-dimensional subspaces (with fermion numbers ${N}_{F}=2$ and ${N}_{F}=3$) where there exist propagating solutions of the supersymmetry constraints that carry (in the small-wavelength limit) a chaotic spinorial dynamics generalizing the Belinskii-Khalatnikov-Lifshitz classical ``oscillatory'' dynamics. By exactly solving the supersymmetry constraints near each one of the three dominant potential walls underlying the latter chaotic billiard dynamics, we compute the three operators that describe the corresponding three potential-wall reflections of the spinorial state describing, in supergravity, the quantum evolution of the universe. It is remarkably found that the latter, purely dynamically-defined, reflection operators satisfy generalized Coxeter relations which define a type of spinorial extension of the Weyl group of the rank-3 hyperbolic Kac-Moody algebra $A{E}_{3}$.

Weyl groups of some hyperbolic Kac–Moody algebras

We use the theory of Clifford algebras and Vahlen groups to study Weyl groups of hyperbolic Kac–Moody algebras Tn++, obtained by a process of double extension from a Cartan matrix of finite type Tn, whose corresponding generalized Cartan matrices are symmetric.

A combinatorial approach to root multiplicities of rank 2 hyperbolic Kac–Moody algebras

ABSTRACTIn this paper we study root multiplicities of rank 2 hyperbolic Kac–Moody algebras using the combinatorics of Dyck paths.

Mixmaster model associated to a Borcherds algebra

The problem of integrability of the mixmaster model as a dynamical system with finite degrees of freedom is studied. The model belongs to the class of pseudo-Euclidean generalized Toda chains. It is presented as a quasi-homogeneous system after transformations of phase variables. Application of the method of getting Kovalevskaya exponents to the model leads to the generalized Adler–van Moerbeke formula for root vectors. A generalized Cartan matrix is constructed using simple root vectors inMinkowski space. The mixmaster model is associated to a Borcherds algebra. The known hyperbolic Kac–Moody algebra of the Chitre´ billiard model is obtained by using three spacelike root vectors.

Dynkin Diagrams and Root Systems of Indefinite Quasi-Hyperbolic Kac-Moody Algebra QHA<sub>4</sub><sup>(2)</sup>

Objectives: To obtain the complete classification of a particular class of indefinite type of quasi hyperbolic Kac-Moody algebra QHA4 (2) and to study the properties of imaginary roots. Methods: Pure theoretical approach for the classification of Dynkin diagrams and an analytical approach for the root system are applied. Findings: The complete classification of Dynkin diagrams associated to the Generalized Cartan Matrix of quasi hyperbolic indefinite type of Kac-Moody algebra QHA4 (2) is obtained. Here, the number of connected, non isomorphic Dynkin diagrams associated with QHA4 (2) is 858. The properties of strictly imaginary and purely imaginary roots are also studied for QHA4 (2). Applications: Kac-Moody algebra has applications in various fields of mathematics and mathematical physics such as combinatorics, number theory, partial differential equations, quantum physics etc.Keywords: Dynkin Diagram, Indefinite Quasi Hyperbolic, Kac-Moody Algebras, Purely Imaginary Roots, Strictly Imaginary Roots

Quantum billiards with branes on product of Einstein spaces

We consider a gravitational model in dimension D with several forms, l scalar fields and a Lambda-term. We study cosmological-type block-diagonal metrics defined on a product of an 1-dimensional interval and n oriented Einstein spaces. As an electromagnetic composite brane ansatz is adopted and certain restrictions on the branes are imposed the conformally covariant Wheeler-DeWitt (WDW) equation for the model is studied. Under certain restrictions, asymptotic solutions to the WDW equation are found in the limit of the formation of the billiard walls. These solutions reduce the problem to the so-called quantum billiard in (n + l - 1)-dimensional hyperbolic space. Several examples of quantum billiards in the model with electric and magnetic branes, e.g. corresponding to hyperbolic Kac-Moody algebras, are considered. In the case n=2 we find a set of basis asymptotic solutions to the WDW equation and derive asymptotic solutions for the metric in the classical case.

Quantum billiards in multidimensional models with fields of forms on a product of Einstein spaces

The gravitational D-dimensional model is considered, with l scalar fields, a cosmological constant and several forms. When a cosmological block-diagonal metric, defined on a product of an 1-dimensional interval and n oriented Einstein spaces, is chosen, an electromagnetic composite brane ansatz is adopted, and certain restrictions on the branes are imposed, the conformally covariant Wheeler–DeWitt (WDW) equation for the model is studied. Under certain restrictions, asymptotic solutions to the WDWequation are found in the limit of the formation of billiard walls which reduce the problem to the socalled quantum billiard on (n + l - 1)-dimensional hyperbolic space. Several examples of billiards in the model with {pmn} non-intersecting electric branes, e.g., corresponding to hyperbolic Kac–Moody algebras, are considered. In the classical case, any of these billiards describe a never-ending oscillating behavior of scale factors while approaching to the singularity, which is either spacelike or timelike. For n = 2 the model is completely integrable in the asymptotic regime in the clasical and quantum cases.

Canonical structure of theE10model and supersymmetry

A coset model based on the hyperbolic Kac-Moody algebra E10 has been conjectured to underly eleven-dimensional supergravity and M theory. In this note we study the canonical structure of the bosonic model for finite- and infinite-dimensional groups. In the case of finite-dimensional groups like GL(n) we exhibit a convenient set of variables with Borel-type canonical brackets. The generalisation to the Kac-Moody case requires a proper treatment of the imaginary roots that remains elusive. As a second result, we show that the supersymmetry constraint of D=11 supergravity can be rewritten in a suggestive way using E10 algebra data. Combined with the canonical structure, this rewriting explains the previously observed association of the canonical constraints with null roots of E10. We also exhibit a basic incompatibility between local supersymmetry and the K(E10) `R symmetry', that can be traced back to the presence of imaginary roots and to the unfaithfulness of the spinor representations occurring in the present formulation of the E10 worldline model, and that may require a novel type of bosonisation/fermionisation for its resolution. This appears to be a key challenge for future progress with E10.

Weakly holomorphic modular forms and rank two hyperbolic Kac-Moody algebras

In this paper, we compute basis elements of certain spaces of weight 0 0 weakly holomorphic modular forms and consider the integrality of Fourier coefficients of the modular forms. We use the results to construct automorphic correction of the rank 2 2 hyperbolic Kac-Moody algebras H ( a ) \mathcal H(a) , a = 4 , 5 , 6 a=4,5,6 , through Hilbert modular forms explicitly given by Borcherds lifts of the weakly holomorphic modular forms. We also compute asymptotics of the Fourier coefficients as they are related to root multiplicities of the rank 2 2 hyperbolic Kac-Moody algebras. This work is a continuation of an earlier work of the first and second authors, where automorphic correction was constructed for H ( a ) \mathcal H(a) , a = 3 , 11 , 66 a=3, 11, 66 .

Root systems for Lorentzian Kac–Moody algebras in rank 3

Sometimes a hyperbolic Kac–Moody algebra (KMA) admits an automorphic correction, meaning a generalized KMA with the same real simple roots and whose denominator function has good automorphic properties. These, for example, allow one to work out the root multiplicities. Gritsenko and Nikulin have formalized this in their theory of Lorentzian Lie algebras and shown that the real simple roots must satisfy certain conditions in order for the algebra to admit an automorphic correction. We classify the hyperbolic root systems of rank 3 that satisfy their conditions and have only finite many simple roots, or equivalently a timelike Weyl vector. There are 994 of them, with as many as 24 simple roots. Patterns in the data suggest that some of the non‐obvious cases may be the richest.

Quantum supersymmetric Bianchi IX cosmology

We study the quantum dynamics of a supersymmetric squashed three-sphere by dimensionally reducing (to one timelike dimension) the action of $D=4$ simple supergravity for a $SU(2)$-homogeneous (Bianchi IX) cosmological model. The quantization of the homogeneous gravitino field leads to a 64-dimensional fermionic Hilbert space. After imposition of the diffeomorphism constraints, the wave function of the Universe becomes a 64-component spinor of spin(8,4) depending on the three squashing parameters, which satisfies Dirac-like, and Klein-Gordon-like, wave equations describing the propagation of a ``quantum spinning particle'' reflecting off spin-dependent potential walls. The algebra of the supersymmetry constraints and of the Hamiltonian one is found to close. One finds that the quantum Hamiltonian is built from operators that generate a 64-dimensional representation of the (infinite-dimensional) maximally compact subalgebra of the rank-3 hyperbolic Kac-Moody algebra $A{E}_{3}$. The (quartic-in-fermions) squared-mass term ${\stackrel{^}{\ensuremath{\mu}}}^{2}$ entering the Klein-Gordon-like equation has several remarkable properties: (i) it commutes with all the other (Kac-Moody-related) building blocks of the Hamiltonian; (ii) it is a quadratic function of the fermion number ${N}_{F}$; and (iii) it is negative in most of the Hilbert space. The latter property leads to a possible quantum avoidance of the singularity (``cosmological bounce''), and suggests imposing the boundary condition that the wave function of the Universe vanish when the volume of space tends to zero (a type of boundary condition which looks like a final-state condition when considering the big crunch inside a black hole). The space of solutions is a mixture of ``discrete-spectrum states'' (parametrized by a few constant parameters, and known in explicit form) and of continuous-spectrum states (parametrized by arbitrary functions entering some initial-value problem). The predominantly negative values of the squared-mass term lead to a ``bottle effect'' between small-volume universes and large-volume ones, and to a possible reduction of the continuous spectrum to a discrete spectrum of quantum states looking like excited versions of the Planckian-size universes described by the discrete states at fermionic levels ${N}_{F}=0$ and 1.

Rank 2 symmetric hyperbolic Kac–Moody algebras and Hilbert modular forms

In this paper we study rank 2 symmetric hyperbolic Kac–Moody algebras H(a) with the Cartan matrices (2−a−a2), a⩾3, and their automorphic correction in terms of Hilbert modular forms. We associate a family of H(a)'s to the quadratic field Q(p) for each odd prime p and show that there exists a chain of embeddings in each family. When p=5,13,17, we show that the first H(a) in each family, i.e. H(3), H(11), H(66), is contained in a generalized Kac–Moody superalgebra whose denominator function is a Hilbert modular form given by a Borcherds product. Hence, our results provide automorphic correction for those H(a)'s. We also compute asymptotic formulas for the root multiplicities of the generalized Kac–Moody superalgebras using the fact that the exponents in the Borcherds products are Fourier coefficients of weakly holomorphic modular forms of weight 0.