# Homogeneous Manifolds, Factorisation Problems and Modified KdV Equations

## Abstract

Since the work [1] of G. Wilson the study of commuting flows in Lie groups has become an important tool in the theory of integrable systems. Our treatment of that subject can be summarised as follows. In a given Lie group *G* we can construct a set of commuting flows defined as the left translations by the elements of an abelian subgroup of *G*. Then, if one is able to define a subgroup *G* _{+} of the group *G* for wich the homogeneous manifold *G/G+* is locally isomorphic to a subgroup *G*_ ⊂ *G*, the differential description of the projection of the above mentioned flows in the Lie algebra of *G*_ is given by the integrable equations. The use of the classical r-matrix formalism[2] furnishes with a systematic approach to that construction in the context of the associated Lie algebra. See [3] for details and consequences of this construction for Banach-Lie groups of continous linear operators in Hilbert space. For loop groups[4], infinite dimensional Lie groups defined as the set of smooth maps on the circle with values in some simple group of finite dimension, one can consider two types of canonically defined abelian subgroups, principal and homogeneous subgroups[5], and deduce from them equations of the KdV and AKNS type respectively (see [6] for the AKNS case).

## Keywords

Abelian Subgroup Loop Group Factorisation Problem Homogeneous Manifold Loop Algebra## Preview

Unable to display preview. Download preview PDF.

## References

- 1.G. Wilson, C. R. Acad. Sci. Paris I 299, 587 (1984)MATHGoogle Scholar
- 2.M. A. Semenov-Tyan-Shanskii, Publ. RIMS, Kyoto Univ. 21,1237 (1985)MathSciNetCrossRefGoogle Scholar
- 3.F. Guil, Inverse Problems 5, 559 (1989)MathSciNetADSMATHCrossRefGoogle Scholar
- 4.A. Pressley and G. Segal,
*Loop Groups*(Oxford: Oxford University Press 1986)MATHGoogle Scholar - 5.V. Kac,
*Infinite Dimensional Lie Algebras*(Cambridge: Cambridge University Press 1985)MATHGoogle Scholar - 6.F. Guil and M. Mañas, Lett. Math. Phys. 19, 89 (1990)MathSciNetADSMATHCrossRefGoogle Scholar
- 7.V. G. Drinfeld and V. V. Sokolov, Sov. Math. Dokl. 32, 361 (1985)Google Scholar
- 8.F. Calogero and A. Degasparis, J. Math. Phys. 22, 23 (1981)MathSciNetADSMATHCrossRefGoogle Scholar
- 9.S. I. Svinolupov, V. V. Sokolov and R. I. Yamilov, Sov. Math. Dokl. 28, 165 (1983)MATHGoogle Scholar
- 10.I. M. Krichever and S. P. Novikov, Russ. Math. Surv. 35, 53 (1981)CrossRefGoogle Scholar
- 11.F. Guil and M. Mañas,
*Homogeneous Manifolds and Modified KdV Equations*submitted to J. Math. Phys.Google Scholar - 12.F. Guil and M. Mañas,
*Factorisation Problems related to Elliptic Curves, Landau-Lifshitz and Krichever-Novikov Equations*submitted to Phys. Lett. AGoogle Scholar