One-dimensional configuration sums in vertex models and affine Lie algebra characters

We study the local state probabilities of the vertex models in the face formulation associated with the simple Lie algebras X _n =A _n, B _n, C _n, D _n. The corner transfer matrix method expresses them in terms of one-dimensional configuration sums. We show that the latter are the string functions of X _n ⁽¹⁾ modules. We also present similar results for the restricted face models of types B _n ⁽¹⁾, C _n ⁽¹⁾, D _n ⁽¹⁾.     

Fusion algebra at a rational level and cohomology of nilpotent subalgebras of\widehat{\mathfrak{s}\mathfrak{l}}(2)

We define and calculate the fusion algebra of a WZW model at a rational level using cohomological methods. As a byproduct, we obtain a cohomological characterization of admissible representations of ₂.     

The Kazhdan-Lusztig conjecture for finiteW-algebras

We study the representation theory of finiteW-algebras. After introducing parabolic subalgebras to describe the structure ofW-algebras, we define the Verma modules and give a conjecture for the Kac determinant. This allows us to find the completely degenerate representations of the finiteW-algebras. To extract the irreducible representations we analyse the structure of singular and subsingular vectors and find that, forW-algebras, in general the maximal submodule of a Verma module is not generated by singular vectors only. Surprisingly, the role of the (sub)singular vectors can be encapsulated in terms of a dual analogue of the Kazhdan-Lusztig theorem for simple Lie algebras. These involve dual relative Kazhdan-Lusztig polynomials. We support our conjectures with some examples, and briefly discuss applications and the generalisation to infiniteW-algebras.     

Remarks on the structure constants of the Verlinde algebra associated to sl3

We present a new formula for the structure constants of the Verlinde algebra associated to sl₃. We show that after an affine change of variables the structure constants, considered as a function of highest weights, become the weight function of a suitable sl₃ highest weight representation.Supported in part by NFS grants DMS-9400841 and DMS-9203929.     

Affine Yang-Mills theory and affine self-dual Chern-Simons Solitons

A four-dimensional affine Yang-Mills theory, i.e. Yang—Mills gauge theory with values in an affine Kac-Moody algebra, is constructed which can give rise to a spontaneous breakdown of the affine symmetry. The affine self-dual Yang-Mills equation (which is a special kind of affine YM theory) in four dimensions is dimensionally reduced to the affine self-dual Chem-Simons equation in two dimensions. The latter is shown to have soliton solutions which satisfy the conformal affine Toda equations.K. C. Wong Research Award Winner; address after 28 October 1992: Department of Mathematics, University of Queensland, Brisbane, Queensland 4072, Australia.     

On The Structure and Representations of the Insertion–Elimination Lie Algebra

We examine the structure of the insertion–elimination Lie algebra on rooted trees introduced in Connes and Kreimer (Ann. Henri Poincar 3(3):411–433, 2002). It possesses a triangular structure , like the Heisenberg, Virasoro, and affine algebras. We show in particular that it is simple, which in turn implies that it has no finite-dimensional representations. We consider a category of lowest-weight representations, and show that irreducible representations are uniquely determined by a “lowest weight” . We show that each irreducible representation is a quotient of a Verma-type object, which is generically irreducible.      

On quantum multi-hamiltonian systems

On studying various examples with one degree of freedom, we discuss the possibility of extending the classical concept of multi-Hamiltonian systems to the quantum domain. We show that an adaptation is possible in each case generally leading to distinguish classes of systems characterized globally by their observable algebras.     

The quantum super-Yangian and Casimir operators of Uq(gl(M |N))

TheZ ₂ graded Yangian Yq(gl(M |N)) associated with the Perk-SchultzR matrix is introduced. Its structural properties, the central algebra in particular, are studied. AZ ₂-graded associative algebra epimorphism Yq(gl(M |N)) Uq (gl(M |N)) is obtained in explicit form. Images of central elements of the quantum super-Yangian under this epimorphism yield the Casimir operators of the quantum supergroup Uq(gl(M |N)) constructed in an earlier publication.     

Quantum fields in 1+1-dimension carrying a true ray representation of the Poincaré group

In 1+1 spacetime dimensions there are genuine ray representations of the Poincaré group P ₊ ^sup . We shall construct a free relativistic quantum field theory, such that the fields _d are covariant under P ₊ ^sup and transform with a nontrivial infinitesimal exponent d.     

An example of cancellation of infinities in the star-quantization of fields

Within the *-quantization framework, it is shown how to remove some of the divergences occurring in theø ₂ ⁴ -theory by introducing a-dependent *-product cohomologically equivalent to the normal *-product.     

The central extension of Kac-Moody-Malcev algebras

We introduce a class of infinite-dimensional Kac-Moody-Malcev algebras. These algebras are the generalization of Lie algebras of the Kac-Moody type to Malcev algebras. We demonstrate that the central extensions of the Kac-Moody-Malcev algebras are given by the same cocycles as in the case of Lie algebras. Analogues of Kac-Moody-Malcev algebras may be also introduced in the case of an arbitrary Riemann surface.     

Loop Group-Valued CS and WZNW models,integrable systems,and self-dual Yang-Mills theory

Affine CS and WZNW theories with values in infinite-dimensional (loop) groups are proposed. It appears that the affine CS theory naturally introduces a spectral parameter into a CS theory. The Sinh-Gordon, KdV, and nonlinear Schrödinger equations are obtained, via Hamiltonian reductions, from the affine WZNW. It is shown that the self-dual Yang-Mills (SDYM) equation is related to the equation of motion of the affine WZNW and, thus, symmetry algebra underlying the SDYM can be identified with the affine two-loop Kac-Moody algebra of the affine WZNW.K. C. Wong Research Award Winner, address after 28 October 1992: Department of Mathematics, University of Queensland, Brisbane, Queensland 4072, Australia.     

Vertex operator representations of the quantum affine algebra U q(B r (1) )

We construct the level one vertex operator representations of the q-deformation U _q(B _r ⁽¹⁾ ) of the affine Kac-Moody algebra B _r ⁽¹⁾ . Beside the q-deformed vertex operators introduced by Frenkel and Jing, this construction involves a q-deformation of free fermionic fields.     

Hirota polynomials for the KP and BKP hierarchies

Using symmetric function techniques, we derive closed-form expressions for the Hirota polynomials for thepth modified KP and BKP hierarchies in terms of Schur and SchurQ-polynomials, respectively. The Hirota polynomials for the BKP hierarchy can also be expressed as Pfaffians while those for thepth modified KP hierarchies can, under certain conditions, be expressed as determinants.     

Loop algebras and their relation to the conformal structure of integrable systems

We use the theorem of Kostant, Adler and Symes to construct an infinite set of local polynomials in involution with respect to the Poisson bracket realisation of the Neveu-Schwartz sector of the N=1 superconformal algebra.     

The embedding structure and the shift operator of the U(1) lattice current algebra

The structure of block-spin embeddings of the U(1) lattice current algebra is described and the inner realizations of the shift automorphism are classified for an odd number of lattice sites. We present a particular inner shift operator which admits a factorization involving quantum dilogarithms analogous to the results of Faddeev and Volkov.On leave of absence from Steklov Mathematical Institute, St. Petersburg, Russia.     

R-matrix method for Heisenberg quantum groups

TheR-matrix method is systematically applied to get several Heisenberg quantum groups depending on two or three parameters. It turns out that the associatedR-matrices have to verify a weaker form of the QYBE. Only for particular cases of quantum groups, we can imposeR to be a solution of the QYBE. The corresponding quantum Heisenberg Lie algebras are obtained by duality.     

On the hopf structure of U p,q (gl(1∣1)) and the universal T-matrix of fun p,q (GL(1∣1))

The dually conjugate Hopf superalgebras Fun _p,q (GL(11)) and U _p,q (gl(11)) are studied using the Frønsdal-Galindo approach and the full Hopf structure of U _p,q (gl(11)) is extracted. A finite expression for the universal T-matrix, identified with the dual form and expressing the generalization of the exponential map of the classical groups, is obtained for Fun _p,q (GL(11)). In a representation with a colour index, the T-matrix assumes a form that satisfies a coloured graded Yang-Baxter equation.     

A new realization of the basic representation of A n (1)

The basic representation of A _n ⁽¹⁾ is constructed on a space of paths in (n + 1). A q-analogue of this is also presented. Our construction gives a Lie theoretical proof of a combinatorial identity used in the evaluation of the local state probabilities for a class of solvable lattice models.     

Fock space resolutions of the Virasoro highest weight modules withc ≤ 1

We extend Felder's construction of Fock space resolutions for the Virasoro minimal models to all irreducible modules withc1. In particular, we provide resolutions for the representations corresponding to the boundary and exterior of the Kac table.Supported by the NSF Grant #PHY-88-04561.Supported in part by the Department of Energy Contract #DE-FG03-84ER-40168 and by the USC Faculty Research and Innovation Fund.