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The general solution to the complex Bateman equation in a space of arbitrary dimensions is constructed.

The quantum mechanical transition between a free particle Lagrangian and the Klein–Gordon field description of a free particle (particle-wave duality) is conjectured to extend to an analogous construction of relativistically invariant wave equations associated with strings and branes. Electromagnetic interactions in the two systems are discussed. It is emphasized that all integrable free field theories, including those of Dirac–Born–Infeld type, are associated with Lagrangians equivalent to divergences on the space of solutions of the equations of motion.

In this paper all deformations of the general linear group, subject to certain restrictions which in particular ensure a smooth passage to the Lie group limit, are obtained. Representations are given in terms of certains sets of creation and annihilation operators. These creation and annihilation operators may belong to a generalisation of the $q$-quark type or $q$-hadronic type, of $q$-boson or $q$-fermion type. We are also led to a natural definition of $q$-direct sums of q-algebras., Comment: 15 pages

Finite Euler hierarchies of field theory Lagrangians leading to universal equations of motion for new types of string and membrane theories and for {\it classical} topological field theories are constructed. The analysis uses two main ingredients. On the one hand, there exists a generic finite Euler hierarchy for one field leading to a universal equation which generalises the Plebanski equation of self-dual four dimensional gravity. On the other hand, specific maps are introduced between field theories which provide a ``triangular duality'' between certain classes of arbitrary field theories, classical topological field theories and generalised string and membrane theories. The universal equations, which derive from an infinity of inequivalent Lagrangians, are generalisations of certain reductions of the Plebanski and KdV equations, and could possibly define new integrable systems, thus in particular integrable membrane theories. Some classes of solutions are constructed in the general case. The general solution to some of the universal equations is given in the simplest cases., 42 p., Plain TeX, replaces previous unprintable version corrupted by mailer

This paper explores features of the infinite‐dimensional algebras that have been previously introduced. In particular, it is shown that the classical simple Lie algebras (AN, BN, CN, DN) may be expressed in an ‘‘egalitarian’’ basis with trigonometric structure constants. The transformation to the standard Cartan–Weyl basis, and the particularly transparent N→∞ limit that this formulation allows is provided.

Some features of Manin’s construction of quantum groups are developed and extended to supergroups.

The Yang equations for all self dual solutions of SU(r,s) gauge theory are exhibited in simple form. Algebraic and Backlund transformations of the solutions of these equations are derived. The Backlund transformations change an SU(r,s) solution into an SU(r−1, s+1) solution.

Nontrivial examples of Krichever–Novikov algebras are constructed. The construction involves parameters that satisfy eigenvalue conditions. In the linear case, for an r‐term algebra, the eigenvalues are 0,1,...,r−3.

Some properties of crossing matrices are deduced which are independent of the particular symmetry group from which the crossing matrices are derived. In particular, a, factorization of any elastic crossing matrix, analogous to the factorization of a rotation matrix in n dimensions is found, and the connection between, crossing and unitary matrices elucidated. In the special case of the crossing matrix for elastic scattering of particles which transform as representations of SU2, as well as the usual consistency requirement that each row should sum to unity, a new consistency requirement on the elements in a given column is proved. As a byproduct of this work, a possibly new quadratic identity for Racah coefficients is exhibited.

A new class of singularities associated with Landau‐Cutkosky diagrams is investigated. They correspond to solutions of the Landau equations with infinite internal momenta. Some of their properties depend on the spins of the participating particles. It is shown that a subset of these singularities, called pure second‐type singularities, does not appear on the physical sheet.

Generalized self‐duality type relations for gauge fields in higher‐dimensional spaces are discussed from the point of view of orbits, strata, and stability groups. It is explained how to obtain them as integrability conditions for first‐order gauge‐invariant linear differential equations.