Representations Of Affine Lie Algebras Research Articles

Heisenberg Algebra and Hilbert Schemes of Points on Projective Surfaces

The purpose of this paper is to throw a bridge between two seemingly unrelated subjects. One is the Hilbert scheme of points on projective surfaces, which has been intensively studied by various people (see e.g. [I], [Br], [ES], [G61], [G62]). The other is the infinite dimensional Heisenberg algebra which is closely related to affine Lie algebras (see e.g. [K]). We shall construct a representation of the Heisenberg algebra on the homology group of the Hilbert scheme. In other words, the homology group will become a Fock space. The basic idea is to introduce certain in the product of the Hilbert scheme. Then they define operators on the homology group by a well-known procedure. They give generators of the Heisenberg algebra, and the only thing we must check is that they satisfy the defining relation. Here we remark that the components of the Hilbert scheme are parameterized by numbers of points and our representation will be constructed on the direct sum of homology groups of all components. Our correspondences live in the product of the different components. Thus it is quite essential to study all components together. Our construction has the same spirit as the author's construction [Nal], [Na4] of representations of affine Lie algebras on homology groups of moduli spaces of instantons1 on ALE spaces which are minimal resolutions of simple singularities. Certain correspondences, called Hecke correspondences, were used to define operators. These twist instantons along curves (irreducible components of the exceptional set), while ours twist ideals around points. In fact, the Hilbert scheme of points can be considered as the moduli space of rank 1 vector bundles, or more precisely torsion free sheaves. Our construction should be considered as a first step to extend [Nal], [Na4] to general 4-manifolds. The same program was also proposed by Ginzburg, Kapranov, and Vasserot [GKV].

Vertex operator algebras and representations of affine Lie algebras

We announce the construction of an explicit basis for all integrable highest weight modules over the Lie algebra A 1 (1). The construction uses representations of vertex operator algebras and leads to combinatorial identities of Rogers-Ramanujan-type.

An Application of Crystal Bases to Representations of Affine Lie Algebras

It has been shown that up to degree shifts any integrable highest weight (or standard) module of level k for an affine Lie algebra g can be imbedded in the tenser product of k copies of level one integrable highest weight modules. When the affine Lie algebra g is of classical type the path realizations of the crystal bases for the level one U q ( g )-modules have been used to obtain these results.

Representations of affine Lie algebras, parabolic differential equations, and Lamé functions

We consider correlation functions for the Wess-Zumino-Witten model on the torus with the insertion of a Cartan element; mathematically this means that we consider the function of the form $F=\Tr (\Phi_1 (z_1)\ldots \Phi_n (z_n)q^{-\d}e^{h})$ where $\Phi_i$ are intertwiners between Verma modules and evaluation modules over an affine Lie algebra $\ghat$, $\d$ is the grading operator in a Verma module and $h$ is in the Cartan subalgebra of $\g$. We derive a system of differential equations satisfied by such a function. In particular, the calculation of $q\frac{\d} {\d q} F$ yields a parabolic second order PDE closely related to the heat equation on the compact Lie group corresponding to $\g$. We consider in detail the case $n=1$, $\g = \sltwo$. In this case we get the following differential equation ($q=e^{\pi \i \tau}$): $ \left( -2\pi\i (K+2)\frac{\d}{\d\tau} +\frac{\d^2}{\d x^2}\right) F = (m(m+1)\wp(x+\frac{\tau}{2}) +c)F$, which for $K=-2$ (critical level) becomes Lam\'e equation. For the case $m\in\Z$ we derive integral formulas for $F$ and find their asymptotics as $K\to -2$, thus recovering classical Lam\'e functions.

W-strings from affine Lie algebras

The most disappointing aspect of W-strings is probably the fact that at least for the known models one does not revover a physical spectrum that differs much from that of ordinary string theory. It is hoped that this is not an intrinsic shortcoming or W-string theory, but rather that it is a consequence of the W-algebra realizations that have been chosen. In this note we point out a whole new class of possible W-strings built from representations of affine Lie algebras via (quantum) Drinfel'd-Sokolov reduction. Explicitly, we construct a BRST operator in terms of sl 3 currents which computes the physical spectrum of a W 3-string. As a special case of this construction, if we take a free-field realization of sl 3 , we recover the 2-scalar W 3-string. These results generalize to any W-algebra which can be obtained via quantum Drinfel'd-Sokolov reduction from an affine Lie algebra.

Toward Proof of Lusztig′s Conjecture Concerning Negative Level Representations of Affine Lie Algebras

Toward Proof of Lusztig′s Conjecture Concerning Negative Level Representations of Affine Lie Algebras

Representations of affine Lie algebras, ellipticr-matrix systems, and special functions

The author considers an elliptic analogue of the Knizhnik-Zamolodchikov equations—the consistent system of linear differential equations arising from the elliptic solution of the classical Yang-Baxter equation for the Lie algebra $$\mathfrak{s}\mathfrak{l}_N $$ . The solutions of this system are interpreted as traces of products of intertwining operators between certain representations of the affine Lie algebra . A new differential equation for such traces characterizing their behavior under the variation of the modulus of the underlying elliptic curve is deduced. This equation is consistent with the original system. It is shown that the system extended by the new equation is modular invariant, and the corresponding monodromy representations of the modular group are defined. Some elementary examples in which the system can be solved explicitly (in terms of elliptic and modular functions) are considered. The monodromy of the system is explicitly computed with the help of the trace interpretation of solutions. Projective representations of the braid group of the torus and representations of the double affine Hecke algebra are obtained.

Toward Proof of Lusztig′s Conjecture Concerning Negative Level Representations of Affine Lie Algebras

Toward Proof of Lusztig′s Conjecture Concerning Negative Level Representations of Affine Lie Algebras

Weight diagram construction of Lax operators

It is demonstrated and proven that cyclic weight diagrams corresponding to representations of affine Lie algebras allow one to construct the associated Lax operator. The resultant Lax operator is in the Miura-like form and generates the modified KdV equations. The algorithm is extended to the supersymmetric case.

Vertex operator algebras associated to representations of affine and Virasoro algebras

The first construction of the integrable highest-weight representations of affine Lie algebras or loop algebras by Kac i-K] was greatly inspired by the generalization of the Weyl denominator formula for affine roots systems discovered earlier by Macdonald [M]. Though the Macdonald identity found its natural context in representation theory, its mysterious modular invariance was not understood until the work of Witten [W-I on the geometric realization of representations of the loop groups corresponding to loop algebras. The work of Witten clearly indicated that the representations of loop groups possess a very rich structure of conformal field theory which appeared in physics literature in the work of Belavin, Polyakov, and Zamolodchikov [BPZ-I. Independently (though two years later), Borcherds, in an attempt to find a conceptual understanding of a certain algebra of vertex operators invariant under the Monster [FLM1], introduced in [B-I a new algebraic structure. We call vertex operator algebras a slightly modified version of Borcherd’s new algebras [FLM2].

Fock space representations of affine lie algebras and integral representations in the Wess-Zumino-Witten models

Fock space representations of affine Lie algebras are studied. Explicit forms of correction terms adding to the currentsF i (z) are determined. It is proved that the Sugawara energy-momentum tensor on the Fock spaces is quadratic in free bosons. Furthermore, screening operators are constructed. This implies the existence of generalized hypergeometric integrals satisfying the Knizhnik-Zamolodchikov equation.

Superalgebras, symplectic bosons and the Sugawara construction

New representations of affine Lie algebras are constructed using symplectic bosons of the sort that occur naturally in the BRST treatment of fermionic string theories. These representations are shown to have analogous properties to the current algebra representations in terms of free fermion fields, though they do not act in a positive space. In particular, the condition for the Sugawara construction of the Virasoro algebra to equal the free one is the existence of a superalgebra with a quadratic Casimir operator, paralleling the symmetric space theorem for fermionic field constructions. Both results are seen to be particular cases of a more general super-symmetric space theorem, which arises from considering an affinisation of the superalgebras. These algebras are realised in terms of free fermions and symplectic bosons and lead to a super-Sugawara construction of the Virasoro algebra. The conditions for this to equal a Virasoro algebra obtained from the free fields are provided by the super-symmetric space theorem.

Rademacher-type formulas for the multiplicities of irreducible highest-weight representations of affine Lie algebras

Rademacher-type formulas for the multiplicities of irreducible highest-weight representations of affine Lie algebras

Integrable representations of affine Lie-algebras

Integrable representations of affine Lie-algebras

Representations of affine Lie algebras and soliton equations

A few years ago the 'hidden symmetries’ of the soliton equations had been identified as affine Lie groups, also known as loop groups. The first extensive use of the representation theory of affine Lie algebras for the soliton equations have been developed in a series of works by mathematicians of the Kyoto school. We will review some of their results and develop them further on the basis of the representation theory. Thus an orbit of the simplest affine Lie group SL(2, C)^ in the fundamental representation V will provide the solutions of the Korteweg-de Vries equation, and similarly the solutions of the sine-Gordon equation will come from an orbit of the group (SL(2, C) x SL(2, C)) ^ in V x V*.

A natural representation of the Fischer-Griess Monster with the modular function J as character

We announce the construction of an irreducible graded module V for an "affine" commutative nonassociative algebra [unk]. This algebra is an "affinization" of a slight variant [unk] of the commutative nonassociative algebra B defined by Griess in his construction of the Monster sporadic group F(1). The character of V is given by the modular function J(q) = q(-1) + 0 + 196884q +.... We obtain a natural action of the Monster on V compatible with the action of [unk], thus conceptually explaining a major part of the numerical observations known as Monstrous Moonshine. Our construction starts from ideas in the theory of the basic representations of affine Lie algebras and develops further the calculus of vertex operators. In particular, the homogeneous and principal representations of the simplest affine Lie algebra A(1) ((l)) and the relation between them play an important role in our construction. As a corollary we deduce Griess's results, obtained previously by direct calculation, about the algebra structure of B and the action of F(1) on it. In this work, the Monster, a finite group, is defined and studied by means of a canonical infinite-dimensional representation.

Two constructions of affine Lie algebra representations and boson-fermion correspondence in quantum field theory

We establish an isomorphism between the vertex and spinor representations of affine Lie algebras for types D l (1) and D l + 1 (2) . We also study decomposition of spinor representations using the infinite family of Casimir operators and prove that they are either irreducible or have two irreducible components. We show that the vertex and spinor constructions of the representations can be reformulated in the language of two-dimensional quantum field theory. In this physical context, the two constructions yield the generalized sine-Gordon and Thirring models, respectively, already in renormalized form. The isomorphism of representations implies an equivalence of these two models which is known in quantum field theory as the boson-fermion correspondence

Spinor representations of affine Lie algebras

Let [unk] be an infinite-dimensional Kac-Moody Lie algebra of one of the types D(l+1) ((2)), B(l) ((1)), or D(l) ((1)). These algebras are characterized by the property that an elimination of any endpoint of their Dynkin diagrams gives diagrams of types B(l) or D(l) of classical orthogonal Lie algebras. We construct two representations of a Lie algebra [unk], which we call spinor representations, following the analogy with the classical case. We obtain that every spinor representation is either irreducible or has two irreducible components. This provides us with an explicit construction of fundamental representations of [unk], two for the type D(l+1) ((2)), three for B(l) ((1)), and four for D(l) ((1)). We note the profound connection of our construction with quantum field theory-in particular, with fermion fields. Comparing the character formulas of our representations with another construction of the fundamental representations of Kac-Moody Lie algebras of types A(l) ((1)), D(l) ((1)), E(l) ((1)), we obtain classical Jacobi identities and addition formulas for elliptic theta-functions.

Basic representations of affine Lie algebras and dual resonance models

Basic representations of affine Lie algebras and dual resonance models