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On some geometric characteristics of the orbit foliations of the co-adjoint action of some 5-dimensional solvable Lie groups

In this paper, discribe some geometric charateristics of the so-called MD(5,3C)-foliations and MD(5,4)- foliations, i.e., the foliations formed by the generic orbits of co-adjoint action of MD(5,3C)-groups and MD(5,4)-groups. | SCIENCE & TECHNOLOGY DEVELOPMENT, Vol 18, No.K4- 2015 On some geometric characteristics of the orbit foliations of the co-adjoint action of some 5-dimensional solvable Lie groups Le Anh Vu1 Nguyen Anh Tuan2 Duong Quang Hoa3 1 University of Economics and Law, VNU-HCM University of Physical Education and Sports, Ho Chi Minh city 3 Hoa Sen University, Ho Chi Minh city 2 (Manuscript Received on August 01st, 2015, Manuscript Revised August 27th, 2015) ABSTRACT: In this paper, we discribe some geometric charateristics of the so-called MD(5,3C)-foliations and MD(5,4)- foliations, i.e., the foliations formed by the generic orbits of co-adjoint action of MD(5,3C)-groups and MD(5,4)-groups. Key words: K-representation, K-orbits, MD-groups, MD-algebras, foliations. 1. INTRODUCTION It is well-known that Lie algebras are interesting objects with many applications not only in mathematics but also in physics. However, the problem of classifying all Lie algebras is still open, up to date. By the LeviMaltsev Theorem [5] in 1945, it reduces the task of classifying all finite-dimensional Lie algebras to obtaining the classification of solvable Lie algebras. There are two ways of proceeding in the classification of solvable Lie algebras: by dimension or by structure. It seems to be very difficult to proceed by dimension in the classification of Lie algebras of dimension greater than 6. However, it is possible to proceed by structure, i.e., to classify solvable Lie algebras with a specific given property. We start with the second way, i.e, the structure approach. More precisely, by Kirillov's Orbit Method [4], we consider Lie algebras whose correponding connected and simply Page 114 connected Lie groups have co-adjoint orbits (Korbits) which are orbits of dimension zero or maximal dimension. Such Lie algebras and Lie groups are called MD-algebras and MD-groups, respectively, in term of Diep [2]. The problem of classifying general MD-algebras (and corresponding .