A methodology for constructing conserved quantities with Lie symmetry infinitesimals in an Itô integral context is pursued. The basis of this construction relies on Lie bracket relations on both the instantaneous drift and diffusion of an Itô stochastic ordinary differential equation (SODE).

Various aspects of spaces of chiral blocks are discussed. In particular, conjectures about the dimensions of irreducible sub-bundles are reviewed and their relation to symmetry breaking conformal boundary conditions is outlined. —————— Invited talk by Jürgen Fuchs at the Workshop on Lie Theory and Its Applications in Physics -Lie III (Clausthal, July 1999); slightly extended version of contribu...

In a recent paper Carot et al. considered carefully the definition of cylindrical symmetry as a specialisation of the case of axial symmetry. One of their propositions states that if there is a second Killing vector, which together with the one generating the axial symmetry, forms the basis of a two-dimensional Lie algebra, then the two Killing vectors must commute, thus generating an Abelian g...

We study a modified version of an equation of the continuous Toda type in 1+1 dimensions. This equation contains a friction-like term which can be switched off by annihilating a free parameter ǫ. We apply the prolongation method, the symmetry and the approximate symmetry approach. This strategy allows us to get insight into both the equations for ǫ = 0 and ǫ 6= 0, whose properties arising in th...

The precise relationship between the arbitrary monodromy dependent 2-form appearing in the chiral WZNW symplectic form and the ‘exchange r-matrix’ that governs the corresponding Poisson brackets is established. Generalizing earlier results related to diagonal monodromy, the exchange r-matrices are shown to satisfy a new dynamical generalization of the classical modified Yang-Baxter equation, wh...

The need of classification of Lie bialgebras [1] comes from their close relation with q-deformations of universal enveloping algebras in the Drinfeld sense. To each such deformation there corresponds a Lie bialgebra which may be recovered from the first order of the deformation of the coproduct. It has also been shown [4] that each Lie bialgebra admits quantization. So the classification of Lie...

A class of new nonabelian gauge theories for vector fields on three manifolds is presented. The theories describe a generalization of three-dimensional YangMills theory featuring a novel nonlinear gauge symmetry and field equations for Lie-algebra valued vector potential fields. The nonlinear form of the gauge symmetry and field equations relies on the vector cross-product and vector curl opera...