Outer automorphisms ofsl(2), integrable systems, and mappings

Outer automorphisms of infinite-dimensional representations of the Lie algebra sl(2) are used to construct Lax matrices for integrable Hamiltonian systems and discrete integrable mappings. The known results are reproduced, and new integrable systems are constructed. Classical r-matrices, corresponding to the Lax representation with the spectral parameter are dynamic. This scheme is advantageous because quantum systems naturally arise in the framework of the classical r-matrix Lax representation and the corresponding quantum mechanical problem admits a variable separation. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 118, No. 2, pp. 205–216, February, 1999.