Affine Kac Research Articles

Principal vertex operator representations for toroidal Lie algebras

We introduce the principal vertex operator representations for the toroidal Lie algebras generalizing the construction for the affine Kac–Moody algebras. We also represent the derivations of the toroidal algebras and introduce analogs of the Sugawara operators.

Γ -conformal algebras

Γ -conformal algebra is an axiomatic description of the operator product expansion of chiral fields with simple poles at finitely many points. We classify these algebras and their representations in terms of Lie algebras and their representations with an action of the group Γ. To every Γ-conformal algebra and a character of Γ we associate a Lie algebra generated by fields with the OPE with simple poles. Examples include twisted affine Kac–Moody algebras, the sin algebra (which is a “Γ-conformal” analogue of the general linear algebra) and its analogues, the algebra of pseudodifferential operators on the circle, etc.

Some Infinite-Dimensional Algebras Arising in Spin Systems and in Particle Physics and their Grand Algebra

We indicate similarities in the structure of two types of infinite-dimensional algebras, one introduced 28 years ago in connection with the mass problem of elementary particles and the other seven years ago in connection with spin systems (XY models). We show that these algebras can be considered as representations of a single ‘Grand Algebra’, the enveloping algebra of an affine Kac–Moody algebra built on the Poincare Lie algebra. As an associative and coassociative bialgebra of operators, the latter representation of the grand algebra is a preferred nontrivial deformation of the Ising case bialgebra.

Systems of PDEs obtained from factorization in loop groups

We propose a generalization of a Drinfeld–Sokolov scheme of attaching integrable systems of PDEs to affine Kac–Moody algebras. With every affine Kac–Moody algebra \(\mathfrak{g} \) and a parabolic subalgebra \($$\) , we associate two hierarchies of PDEs. One, called positive, is a generalization of the KdV hierarchy, the other, called negative, generalizes the Toda hierarchy. We prove a coordinatization theorem which establishes that the number of functions needed to express all PDEs of the the total hierarchy equals the rank of\(\mathfrak{g} \) . The choice of functions, however, is shown to depend in a noncanonical way on \($$\). We employ a version of the Birkhoff decomposition and a ‘2-loop’ formulation which allows us to incorporate geometrically meaningful solutions to those hierarchies. We illustrate our formalism for positive hierarchies with a generalization of the Boussinesq system and for the negative hierarchies with the stationary Bogoyavlenskii equation.

An affine string vertex operator construction at an arbitrary level

An affine vertex operator construction at an arbitrary level is presented which is based on a completely compactified chiral bosonic string whose momentum lattice is taken to be the (Minkowskian) affine weight lattice. This construction is manifestly physical in the sense of string theory, i.e., the vertex operators are functions of Del Giudice–Di Vecchia–Fubini (DFF) “oscillators” and the Lorentz generators, both of which commute with the Virasoro constraints. We therefore obtain explicit representations of affine highest weight modules in terms of physical (DDF) string states. This opens new perspectives on the representation theory of affine Kac–Moody algebras, especially in view of the simultaneous treatment of infinitely many affine highest weight representations of arbitrary level within a single state space as required for the study of hyperbolic Kac–Moody algebras. A novel interpretation of the affine Weyl group as the “dimensional null reduction” of the corresponding hyperbolic Weyl group is given, which follows upon re-expression of the affine Weyl translations as Lorentz boosts.

Affine realizations of sphere algebras

The group Map(M,G) of smooth mappings from a compact manifold M to a simple Lie group G is an infinite-dimensional Lie group. The simplest case is when M=S1. The central extension of the corresponding Lie algebra Map(S1,g) (with g the Lie algebra of G) is an affine Kac–Moody algebra. The representations of Map(S1,g) and of its central extension have been thoroughly investigated. Less work has been done for higher-dimensional M. We discuss the particular cases when M is the two- and three-spheres Sn, n=2,3. In this work we construct particular realizations of the centrally extended Map(Sn,g) from a given realization of an affine Kac–Moody algebra.

Constrained KP models as integrable matrix hierarchies

We formulate the constrained KP hierarchy (denoted by cKPK+1,M) as an affine sl̂(M+K+1) matrix integrable hierarchy generalizing the Drinfeld–Sokolov hierarchy. Using an algebraic approach, including the graded structure of the generalized Drinfeld–Sokolov hierarchy, we are able to find several new universal results valid for the cKP hierarchy. In particular, our method yields a closed expression for the second bracket obtained through Dirac reduction of any untwisted affine Kac–Moody current algebra. An explicit example is given for the case sl̂(M+K+1), for which a closed expression for the general recursion operator is also obtained. We show how isospectral flows are characterized and grouped according to the semisimple non-regular element E of sl(M+K+1) and the content of the center of the kernel of E.

Tau-functions and dressing transformations for zero-curvature affine integrable equations

The solutions of a large class of hierarchies of zero-curvature equations that includes Toda- and KdV-type hierarchies are investigated. All these hierarchies are constructed from affine (twisted or untwisted) Kac–Moody algebras 𝔤. Their common feature is that they have some special “vacuum solutions” corresponding to Lax operators lying in some Abelian (up to the central term) subalgebra of 𝔤; in some interesting cases such subalgebras are of the Heisenberg type. Using the dressing transformation method, the solutions in the orbit of those vacuum solutions are constructed in a uniform way. Then, the generalized tau-functions for those hierarchies are defined as an alternative set of variables corresponding to certain matrix elements evaluated in the integrable highest-weight representations of 𝔤. Such definition of tau-functions applies for any level of the representation, and it is independent of its realization (vertex operator or not). The particular important cases of generalized mKdV and KdV hierarchies as well as the Abelian and non-Abelian affine Toda theories are discussed in detail.

Quasi-Hopf Deformations of Quantum Groups

The search for elliptic quantum groups leads to a modified quantum Yang–Baxter relation and to a special class of quasi-triangular quasi-Hopf algebras. This Letter calculates deformations of standard quantum groups (with or without spectral parameter) in the category of quasi-Hopf algebras. An earlier investigation of the deformations of quantum groups, in the category of Hopf algebras, showed that quantum groups are generically rigid: Hopf algebra deformations exist only under some restrictions on the parameters. In particular, affine Kac–Moody algebras are more rigid than their loop algebra quotients and only the latter (in the case of sl(n)) can be deformed to elliptic Hopf algebras. The generalization to quasi-Hopf deformations lifts this restriction. The full elliptic quantum groups (with central extension) associated with sl(n) are thus quasi-Hopf algebras. The universal R-matrices satisfy a modified Yang–Baxter relation and are calculated more or less explicitly. The modified classical Yang–Baxter relation is obtained and the elliptic solutions are worked out explicitly.

Quantization of a loop extended SU(2) affine Kac–Moody algebra

A quantization of a Lie bialgebra structure on a loop extended SU■(2) algebra is considered. An asymptotic expansion for R-matrix is given.

The Affine symmetry of self-dual gravity

Self-dual gravity may be reformulated as the two dimensional principal chiral model with the group of area preserving diffeomorphisms as its gauge group. Using this formulation, it is shown that self-dual gravity contains an infinite dimensional hidden symmetry whose generators form the Affine (Kac–Moody) algebra associated with the Lie algebra of area preserving diffeomorphisms. This result provides an observable algebra and a solution generating technique for self-dual gravity.

Automorphism modular invariants of current algebras

We consider those two-dimensional rational conformal field theories (RCFTs) whose chiral algebras, when maximally extended, are isomorphic to the current algebra formed from some affine non-twisted Kac--Moody algebra at fixed level. In this case the partition function is specified by an automorphism of the fusion ring and corresponding symmetry of the Kac--Peterson modular matrices. We classify all such partition functions when the underlying finite-dimensional Lie algebra is simple. This gives all possible spectra for this class of RCFTs. While accomplishing this, we also find the primary fields with second smallest quantum dimension.

Universal central extensions of elliptic affine Lie algebras

Let 𝔤 be a simple complex (finite dimensional) Lie algebra, and let R be the ring of regular functions on a compact complex algebraic curve with a finite number of points removed. Lie algebras of the form 𝔤⊗CR are considered; these generalize Kac–Moody loop algebras since for a curve of genus zero with two punctures R≂C[t,t−1]. The universal central extension of 𝔤⊗R is analogous to an untwisted affine Kac–Moody algebra. By Kassel’s theorem the kernel of the universal central extension is linearly isomorphic to the Kähler differentials of R modulo exact differentials. The dimension of the kernel for any R is determined first. Restricting to hyperelliptic curves with 2, 3, or 4 special points removed, a basis for the kernel is determined. Restricting further to an elliptic curve with punctures at two points (of orders one and two in the group law) we explicitly determine the cocycles which give the commutation relations for the universal central extension. The results involve Pollaczek polynomials, which are a genus-one generalization of ultraspherical (Gegenbauer) polynomials. © 1994 American Institute of Physics.

Additional symmetries of generalized integrable hierarchies

The non-isospectral symmetries of a general class of integrable hierarchies are found, generalizing the Galilean and scaling symmetries of the Korteweg--de Vries equation and its hierarchy. The symmetries arise in a very natural way from the semi-direct product structure of the Virasoro algebra and the affine Kac--Moody algebra underlying the construction of the hierarchy. In particular, the generators of the symmetries are shown to satisfy a subalgebra of the Virasoro algebra. When a tau-function formalism is available, the infinitesimal symmetries act directly on the tau-functions as moments of Virasoro currents. Some comments are made regarding the r\^ole of the non-isospectral symmetries and the form of the string equations in matrix-model formulations of quantum gravity in two-dimensions and related systems.

Constrained Hamiltonian systems on Lie groups, moment map reductions, and central extensions

The canonical symplectic structure on the cotangent bundle T*G of a Lie group, whose algebra gc admits a central extension gc, is modified so that the moment map associated to right translations generates a representation of gc. Zero moment map constraints are applied to a class of Hamiltonian systems on T*G that are not necessarily right invariant to yield reduced systems on coadjoint orbits in gc*. The Hamiltonian formulation of the WZW model is developed as an example in which G = LG0 is a loop group and the reduced space is the group of based loops ΩG0, identified as a coadjoint orbit of the affine Kac–Moody algebra [Formula: see text]. An extension of the geometric quantization method leads to representations of [Formula: see text] on Hilbert spaces of sections of holomorphic vector bundles over ΩG0.

On the characters of affine Kac–Moody groups

Let G be an affine Kac–Moody group over ℂ, and V∞ an integrable simple quotient of a Verma module for g. Let Gmin be the subgroup of G generated by the maximal algebraic torus T, and the real root subgroups.It is shown that (the least positive imaginary root) gives a character δ∈Hom(G, ℂ*) such that the pointwise character χ∞ of V∞ may be defined on Gmin ∩ G>1.

Inönü–Wigner contraction of Kac–Moody algebras

Inönü–Wigner contractions of affine Kac–Moody algebras are discussed here. The Sugawara construction for the contracted affine algebra exists only for a fixed value of the level k, which is determined in terms of the dimension of the uncontracted part of the starting Lie algebra, and the quadratic Casimir in the adjoint representation. Further, contractions of the G/H coset spaces are discussed, and an affine translation algebra is obtained, which yields a Virasoro algebra [via a Goddard–Kent–Olive (GKO) construction] with a central charge given by dim(G/H).

Weyl orbits and their expansions in irreducible representations for affine Kac–Moody algebras

Weyl orbits of affine rank 2 and 3 algebras are obtained analytically, along with the depths of their weights, in terms of an integrity basis consisting of sets of compatible (or adjacent) weights, one from each fundamental orbit. The fundamental orbits and others of the same level and congruence class are then decomposed into irreducible representations (IR). Since the orbit →IR matrix is triangular, it will be easy to invert it to get orbit multiplicities and subalgebra branching rules.

A superversion of quasifree second quantization. I. Charged particles

A formalism comprising and extending quasifree second quantization of charged bosons and fermions is presented. The second quantization of one-particle observables leads to current superalgebras, and a super-Schwinger term shows up. Anticommuting parameters are introduced in order to construct super-Bogoliubov transformations mixing bosons and fermions. As an application, representations of Lie superalgebras are given which are semidirect products of extensions of affine Kac–Moody algebras and the Virasoro algebra, and of the super-Virasoro algebra.

Sugawara-type constructions of the Virasoro algebra based on the twisted affine Kac–Moody superalgebras

Explicit realizations of representations of the Virasoro algebra are obtained using highest-weight representations of the twisted affine Kac–Moody superalgebras A(2)(2l−1/0) (for l≥2), A(4)(2l/0) (for l≥1), and C(2)(l+1) (for l≥1) in a generalization of the Sugawara construction. It is also shown that A(2)(2l−1/0) (for l≥2) and A(4)(2l/0) (for l≥1) can be modified to produce ‘‘Neveu–Schwarz’’ type superalgebras, and the corresponding Sugawara construction is given for these. For C(2)(l+1) (for l≥1) it is demonstrated that no such ‘‘Neveu–Schwarz’’ modification is possible.