ROOT SYSTEMS AND SYMMETRIES OF TORUS MANIFOLDS

The Symmetries of Equivalent Lagrangian Systems and Constants of Motion

In this paper Mathematical structure of time-dependent Lagrangian systems and their symmetries are extended and the explicit relation between constants of motion and infinitesimal symmetries of time-dependent Lagrangian systems are considered. Starting point is time-independent Lagrangian systems ,then we extend mathematical concepts of these systems such as equivalent lagrangian systems to th...

Symmetries of Hamiltonian systems on symplectic and Poisson manifolds

Symmetries and Motions in Manifolds

In these lectures the relations between symmetries, Lie algebras, Killing vectors and Noether’s theorem are reviewed. A generalisation of the basic ideas to include velocity-dependend co-ordinate transformations naturally leads to the concept of Killing tensors. Via their Poisson brackets these tensors generate an a priori infinitedimensional Lie algebra. The nature of such infinite algebras is...

Hidden symmetries and arithmetic manifolds

Let M be a closed, locally symmetric Riemannian manifold of nonpositive curvature with no local torus factors; for example take M to be a hyperbolic manifold. Equivalently, M = K\G/Γ where G is a semisimple Lie group and Γ is a cocompact lattice in G. For simplicity, we will always assume that Γ is irreducible, or equivalently that M is not finitely covered by a smooth product; we will also ass...

Frobenius manifolds for elliptic root systems

In this paper, we show that the quotient space of the domain by the reflection group for an elliptic root system has a structure of Frobenius manifold for the case of codimension 1. We also give a characterization of this Frobenius manifold structure under some suitable condition.