Regularization of Toda lattices by Hamiltonian reduction

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The Toda lattice defined by the Hamiltonian H=1 2 ∑ᵢ₌₁ⁿ pᵢ² + ∑ᵢ₌₁ⁿ⁻¹ νᵢ eqⁱ⁻qᵢ₊₁ with νᵢ∈ { ± 1}, which exhibits singular (blowing up) solutions if some of the νᵢ=-1, can be viewed as the reduced system following from a symmetry reduction of a subsystem of the free particle moving on the group G=SL(n,). The subsystem is T*Gₑ, where Gₑ=N+ A N- consists of the determinant one matrices with positive principal minors, and the reduction is based on the maximal nilpotent group N+ × N-. Using the Bruhat decomposition we show that the full reduced system obtained from T*G, which is perfectly regular, contains 2ⁿ⁻¹ Toda lattices. More precisely, if n is odd the reduced system contains all the possible Toda lattices having different signs for the νᵢ. If n is even, there exist two non-isomorphic reduced systems with different constituent Toda lattices. The Toda lattices occupy non-intersecting open submanifolds in the reduced phase space, wherein they are regularized by being glued together. We find a model of the reduced phase space as a hypersurface in ²ⁿ⁻¹. If νᵢ=1 for all i, we prove for n=2,3,4 that the Toda phase space associated with T*Gₑ is a connected component of this hypersurface. The generalization of the construction for the other simple Lie groups is also presented.