Solutions to Knizhnik-Zamolodchikov equations with coefficients in non-bounded modules

We explicitly write dowm integral formulas for solutions to Knizhnik-Zamolodchikov equations with coefficients in non-bounded -- neither highest nor lowest weight -- \gtsl_{n+1}-modules. The formulas are closely related to WZNW model at a rational level.Comment: 13 page

Maurice Janet's algorithms on systems of linear partial differential equations

This article presents the emergence of formal methods in theory of partial differential equations (PDE) in the french school of mathematics through Janet's work in the period 1913-1930. In his thesis and in a series of articles published during this period, M. Janet introduced an original formal approach to deal with the solvability of the problem of initial conditions for finite linear PDE systems. His constructions implicitly used an interpretation of a monomial PDE system as a generating family of a multiplicative set of monomials. He introduced an algorithmic method on multiplicative sets to compute compatibility conditions, and to study the problem of the existence and the unicity of a solution to a linear PDE system with given initial conditions. The compatibility conditions are formulated using a refinement of the division operation on monomials defined with respect to a partition of the set of variables into multiplicative and non-multiplicative variables. M. Janet was a pioneer in the development of these algorithmic methods, and the completion procedure that he introduced on polynomials was the first one in a long and rich series of works on completion methods which appeared independently throughout the 20th century in various algebraic contexts

Representation theory of Neveu–Schwarz and Ramond algebras I: Verma modules

AbstractIn this article, we study the structure of Verma modules of N=1 super Virasoro algebras. As applications, we construct Bernstein-Gel'fand Gel'fand type resolutions. This article is the detailed and expanded version of Iohara and Koga (C. R. Acad. Sci. Paris Ser. I 328 (1999) 381)

Classification of Simple Lie Algebras on a Lattice

Let $\Lambda$ be a lattice of rank $n$. A Lie algebra on the lattice $\Lambda$ is a Lie algebra ${\cal L}=\oplus_{\lambda\in\Lambda}\,{\cal L}_{\lambda}$ such that $\dim\,{\cal L}_\lambda=1$ for all $\lambda$. In this article, we classify all simple graded Lie algebras on a lattice

Quantum affine symmetry in vertex models

We study the higher spin anologs of the six vertex model on the basis of its symmetry under the quantum affine algebra U_q(\slth). Using the method developed recently for the XXZ spin chain, we formulate the space of states, transfer matrix, vacuum, creation/annihilation operators of particles, and local operators, purely in the language of representation theory. We find that, regardless of the level of the representation involved, the particles have spin $1/2$, and that the $n$-particle space has an RSOS-type structure rather than a simple tensor product of the $1$-particle space. This agrees with the picture proposed earlier by Reshetikhin.Comment: 29 page