Abstract
We generalize the usual Lax equationd/dt L=[M, L] byd/dt L=−ϱ(M)L, where ϱ is an arbitrary representation of a Lie algebra g (the values ofM) in a representation spaceV (the values ofL). The usual classicalr-matrix programme for Hamiltonian integrable systems is generalized tor-matrices taking values in g⊗V. Ther-matrices are then considered as left invariant torsion-free covariant derivatives on a Lie groupK (with Lie algebraV *). The Classical Yang-Baxter Equation (CYBE) is equivalent to the flatness ofK whereas the Modified CYBE implies thatK is an affine locally symmetric space. An example is discussed.
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Communicated by N. Ya. Reshetikhin
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Bordemann, M. Generalized Lax pairs, the modified classical Yang-Baxter equation, and affine geometry of Lie groups. Commun.Math. Phys. 135, 201–216 (1990). https://doi.org/10.1007/BF02097662
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DOI: https://doi.org/10.1007/BF02097662