A Weyl group for the Virasoro andN=1 super-Virasoro algebras

We introduce a Weyl group for the highest weight modules over the Virasoro algebra and the Neveu-Schwarz and Ramond superalgebras. Using this group we rewrite the character formulae for the irreducible highest weight modules over these algebras in the form of the classical Weyl character formula for the finite-dimensional irreducible representations of semi-simple Lie algebras (and also of the Weyl-Kac character formula for the integrable highest weight modules over affine Kac-Moody algebras). This is the same group we introduced recently in order to rewrite in a similar manner the characters of the singular highest weight modules over the affine Kac-Moody algebraA ₁ ⁽¹⁾ .     

An attempt to relate area-preserving diffeomorphisms to Kac-Moody algebras

In the same way as the Virasoro algebra can be connected with Kac-Moody algebras defined on the S ¹ circle, the area-preserving diffeomorphism algebra SDiff(), where is a two-dimensional surface, acts as a derivation algebra on super Kac-Moody algebras with one or two supersymmetries. Then a Sugawara-like construction with fermions of the nonextended SDiff() algebra is discussed.     

The A-D-E classification of minimal andA 1 (1) conformal invariant theories

We present a detailed and complete proof of our earlier conjecture on the classification of minimal conformal invariant theories. This is based on an exhaustive construction of all modular invariant sesquilinear forms, with positive integral coefficients, in the characters of the Virasoro or of theA ₁ ⁽¹⁾ Kac-Moody algebras, which describe the corresponding partition functions on a torus. A remarkable correspondence emerges with simply laced Lie algebras.     

Kac-Moody current algebras of D = 2 massless gauge theories,their representations and applications

We give a classification of the Kac-Moody current algebras of all the possible massless fermion-gauge theories in two dimensions. It is shown that only Kac-Moody algebras based on A_N, B_N, C_N, and D_N in the Cartan classification with all possible central charge occur. The representation of local fermion fields and simply laced Kac-Moody algebras with minimal central charge in terms of free boson fields on a compactified space is discussed in detail, where stress is laid on the role played by the boundary conditions on the various collective modes. Fractional solitons and the possible soliton representation of certain nonsimply laced algebras is also analysed. We briefly discuss the relationship between the massless bound state sector of these two-dimensioned gauge theories and the critically coupled two-dimensional nonlinear sigma model, which share the same current algebra. Finally we briefly discuss the relevance of Sp(n) Kac-Moody algebras to the physics of monopole-fermion systems.     

The uniqueness theorem for the universal R-matrix

Up to now, the universal R-matrix for quantized Kac-Moody algebras is believed to be uniquely determined (for some ansatz) by properties of a quasi-cocommutativity and a quasi-triangularity. We prove here that the universal R-matrix (for the same ansatz) is uniquely determined by the property of the quasi-cocommutativity only. Thus, the quasi-triangular property (and the Yang-Baxter equation!) for the universal R-matrix is a consequence of the linear equation of the quasi-cocommutativity. The proof is based on properties of singular vectors in the tensor product of the Verma modules and the structure of extremal projector for quantized algebras. Explicit expressions of the universal R-matrix for quantized algebras U _q (A _inf1 ^sup(1) ) and U _q (A _inf2 ^sup(2) ) are given.

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On the relevance of some constructions of highest weight modules over (super-) Kac-Moody algebras (in connection with a paper by Wakimoto)

We correct and improve results of Wakimoto on the (ir)reducibility of his construction ofA ₁ ⁽¹⁾ highest weight modules (HWM). For a very large class of (super-) Kac-Moody algebras we argue that such a HWM is most relevant when it is isomorphic to a proper factor-module of the corresponding reducible Verma module with the same highest weight. In the same situation we present a general procedure to check the reducibility of the HWM in consideration.     

Unitary positive energy representations of the gauge group

We investigate the positive energy representations (also called highest weight representations) of the gauge groupC (T ^v,G ₀),G ₀ being a compact simple Lie group, and discuss their unitarity, using the technique of Verma modules constructed from generalized loop algebras (a simple generalization of Kac-Moody affine Lie algebras). We show that the unitarity of the representation imposes severa restrictions in it. In particular, we show, as a part of a more general result, that the gauge group does not admit faithful unitary positive energy representations.Allocataire du MRT.     

Extremal projectors for contragredient Lie (super)symmetries (short review)

A brief review of the extremal projectors for contragredient Lie (super)symmetries (finite-dimensional simple Lie algebras, basic classical Lie superalgebras, infinite-dimensional affine Kac-Moody algebras and superalgebras, as well as their quantum q analogs) is given. Some bibliographic comments on the applications of extremal projectors are presented.     

The spectrum of the transfer matrices connected with Kac-Moody algebras

The spectrum of the transfer matrices corresponding to trigonometrical Bazhanov-Jimbo R matrices is found. The Bethe equations characterizing the eigenvalues of the transfer matrices are written down in terms of root systems. Using the generalization of the Bethe equations for Kac-Moody algebras D _inf4 ^sup(3) , G _inf2 ^sup(1) , E _inf6 ^sup(1) and E _inf6 ^sup(2) , we give conjectures for the eigenvalues of the corresponding transfer matrices.     

Generalized Drinfel'd-Sokolov hierarchies

In this paper we examine the bi-Hamiltonian structure of the generalized KdV-hierarchies. We verify that both Hamiltonian structures take the form of Kirillov brackets on the Kac-Moody algebra, and that they define a coordinated system. Classical extended conformal algebras are obtained from the second Poisson bracket. In particular, we construct theW _n ^(l) algebras, first discussed for the casen=3 andl=2 by Polyakov and Bershadsky.     

New Weyl groups forA 1 (1) and characters of singular highest weight modules

We consider singular Verma modules overA ₁ ⁽¹⁾ , i.e., Verma modules for which the central charge is equal to minus the dual Coxeter number. We calculate the characters of certain factor modules of these Verma modules. In one class of cases we are able to prove that these factor modules are actually the irreducible highest modules for those highest weights. We introduce new Weyl groups which are infinitely generated abelian groups and are proper subgroups or isomorphic between themselves. Using these Weyl groups we can rewrite the character formulae obtained in the paper in the form of the classical Weyl character formula for the finite-dimensional irreducible representations of semisimple Lie algebras (respectively Weyl-Kac character formula for the integrable highest weight modules over affine Kac-Moody algebras) so that the new Weyl groups play the role of the usual Weyl group (respectively affine Weyl group).     

On the fundamental representation of borcherds algebras with one imaginary simple root

Borcherds algebras represent a new class of Lie algebras which have almost all the properties that ordinary Kac-Moody algebras have, but the only major difference is that these generalized Kac-Moody algebras are allowed to have imaginary simple roots. The simplest nontrivial examples one can think of are those where one adds by hand one imaginary simple root to an ordinary Kac-Moody algebra. We study the fundamental representation of this class of examples and prove that an irreducible module is given by the full tensor algebra over some integrable highest weight module of the underlying Kac-Moody algebra. We also comment on possible realizations of these Lie algebras in physics as symmetry algebras in quantum field theory.Supported by Konrad-Adenauer-Stiftung e.V.Supported by Deutsche Forschungsgemeinschaft.     

Level one representations of the simple affine Kac-Moody algebras in their homogeneous gradations

Using the central charge of the Virasoro algebra as a clue, we recall the known constructions of theA, D, E algebras and discuss new Bosonic constructions of the non simply laced affine Kac-Moody algebras: the twistedA, D, E and theB, C, F, andG algebras. These involve interacting Fermions and a generalization of the Frenkel-Kac sign operators which do not form a 2-cocycle when the horizontal algebra has more than one short simple root.     

Simple currents and extensions of vertex operator algebras

We consider how a vertex operator algebra can be extended to an abelian interwining algebra by a family of weak twisted modules which aresimple currents associated with semisimple weight one primary vectors. In the case that the extension is again a vertex operator algebra, the rationality of the extended algebra is discussed. These results are applied to affine Kac-Moody algebras in order to construct all the simple currents explicitly (except forE ₈) and to get various extensions of the vertex operator algebras associated with integrable representations.Supported by NSF grant DMS-9303374 and a research grant from the Committee on Research, UC Santa Cruz.Supported by NSF grant DMS-9401272 and a research grant from the Committee on Research, UC Santa Cruz.     

Continuum analogues of contragredient Lie algebras (Lie algebras with a Cartan operator and nonlinear dynamical systems)

We present an axiomatic formulation of a new class of infinitedimensional Lie algebras-the generalizations ofZ-graded Lie algebras with, generally speaking, an infinite-dimensional Cartan subalgebra and a contiguous set of roots. We call such algebras continuum Lie algebras. The simple Lie algebras of constant growth are encapsulated in our formulation. We pay particular attention to the case when the local algebra is parametrized by a commutative algebra while the Cartan operator (the generalization of the Cartan matrix) is a linear operator. Special examples of these algebras are the Kac-Moody algebras, algebras of Poisson brackets, algebras of vector fields on a manifold, current algebras, and algebras with differential or integro-differential cartan operator. The nonlinear dynamical systems associated with the continuum contragredient Lie algebras are also considered.     

A superintegrable time-dependent system with kac-moody symmetry

We investigate the Hamiltonian H _KL with a time-dependent potential in N-dimensional space that is a special combination of a Kepler and a harmonic-oscillator potential. The corresponding classical system has an angular-momentum tensor and a time-dependent analog of the Laplace-Runge-Lenz vector, which commute with the “quasi-Hamiltonian” H _c . These quantities are conserved on the orbits of H _KL, and their Poisson brackets yield a realization of twisted or untwisted centerless Kac-Moody algebras of so(N+1). The corresponding quantum-mechanical operators and their commutators yield a representation of the positive subalgebras of the above Kac-Moody algebras.     

Algebraic structure of Toda systems

Following the Leningrad school an operator $\text{P}$ is constructed which guarantees the classical complete integrability of the Toda molecule and Toda lattice equations. This quantity depends in a uniform way upon the root system of the underlying algebra, respectively the simple Lie algebras and the affine euclidean Kac-Moody algebras.     

On Minimal Affinizations of Representations of Quantum Groups

In this paper we study minimal affinizations of representations of quantum groups (generalizations of Kirillov-Reshetikhin modules of quantum affine algebras introduced in [Cha1]). We prove that all minimal affinizations in types A, B, G are special in the sense of monomials. Although this property is not satisfied in general, we also prove an analog property for a large class of minimal affinizations in types C, D, F. As an application, the Frenkel-Mukhin algorithm [FM1] works for these modules. For minimal affinizations of type A, B we prove the thin property (the l-weight spaces are of dimension 1) and a conjecture of [NN1] (already known for type A). The proof of the special property is extended uniformly for more general quantum affinizations of quantum Kac-Moody algebras.     

Free fermion and parafermion theories in two dimensions

We show that the spectrum and the observables of a free massless O(N) parafermion theory of order Q in two dimensions is related to, but not identical with, a free fermion theory with O(N)×O(Q) symmetry. The Virasoro and the O(N) Kac-Moody algebras of these theories are given.     

Solitons and vertex operators in twisted affine Toda field theories

Affine Toda field theories in two dimensions constitute families of integrable, relativistically invariant field theories in correspondence with the affine Kac-Moody algebras. The particles which are the quantum excitations of the fields display interesting patterns in their masses and coupling which have recently been shown to extend to the classical soliton solutions arising when the couplings are imaginary. Here these results are extended from the untwisted to the twisted algebras. The new soliton solutions and their masses are found by a folding procedure which can be applied to the affine Kac-Moody algebras themselves to provide new insights into their structures. The relevant foldings are related to inner automorphisms of the associated finite dimensional Lie group which are calculated explicitly and related to what is known as the twisted Coxeter element. The fact that the twisted affine Kac-Moody algebras possess vertex operator constructions emerges naturally and is relevant to the soliton solutions.