ON THE STRUCTURE OF THE OPERATOR COADJOINT ACTION FOR THE CURRENT ALGEBRA ON THE THREE-DIMENSIONAL TORUS

Abstract

For the current Lie algebra on the three-dimensional torus with non-standard Lie bracket some properties, in the case when the sum of adjoint and coadjoint operators on infinite-dimensional Lie algebra with scalar product has a finite norm are established. For the Landau-Lifshitz equation in the three-dimensional torus it is established that the operatorm mS = (ad+ ad* ) / 2mhas a finite norm, though it is not true the operators of the adjoint action adm and coadjoint ac-mtion ad ∗ . It follows that the coefficients of expansion of the solution in an orthonormal basis of eigenvectors of the La- place operator satisfy Lipschitz conditions. Thus, for the Landau-Lifshitz equation on the three-dimensional torus situationis similar to the equations of an ideal fluid and Korteweg de Vries. On the other hand, if for the equations of fluid dynamicsand Korteweg de Vries, this fact has been established in a general way, for the Landau-Lifshitz equation in the three- dimensional torus it is obtained specifically through the calculation of structural constants and the matrix of the coadjoint action for the current algebra with non-standard Lie bracket.

Keywords

• current algebra
• lie bracket
• operator of the adjoint action
• operator of the coadjoint action
• three- torus
• the landau-lifshitz equation
• compact operator
• lipschitz condition