Dual moment maps into loop algebras

Abstract

Moment maps are defined from the space of rank-r deformations of a fixedn xn matrixA to the duals $$(\widetilde{g1}(r)^ + )^* , (\widetilde{g1}(n)^ + )^* $$ of the positive half of the loop algebras $$\widetilde{g1}(r),\widetilde{g1}(n)$$ . These maps are shown to give rise to the same invariant manifolds under Hamiltonian flow obtained through the Adler-Kostant-Symes theorem from the rings $$I(\widetilde{g1}(r)^* ),I(\widetilde{g1}(n)^* )$$ of invariant functions. This gives a dual characterization of integrable Hamiltonian systems as isospectral flow in the two loop algebras.