Maximal subalgebras of general lie Algebra of rank two.

摘要

In the Lie theory, the classication problem for maximal subalgebras is a classical one. A number of problems in geometry and algebra lead to this one. This problem has been the focus of much research which produced very beautiful results. We first recall the famous papers of E. Dynkin, from the middle of 50's of past century, where the classication of semisimple subalgebras of complex semisimple Lie algebras has been obtained. Next, while studying the maximal subalgebras in nonclassical simple modular Lie algebras, new series of exceptional simple Lie algebras were discovered by H. Melikyan, nowadays it known as Melikyan algebras. Our goal is the characterization of maximal graded subalgebras in General Lie algebra of Cartan Wn over the algebraically closed eld of characteristic zero. The notion of an R-subalgebra and that of an S-subalgebra are introduced for maximal subalgebras. All maximal R-subalgebras are described completely. The number of conjugacy classes, representatives of conjugacy classes of all R- subalgebras are found. An invariant characterization of graded S-subalgebras is also obtained. The complete list of representatives from the conjugacy classes of graded maximal subalgebras is obtained for rank two case.