Maximal Graded Subalgebras For The Even Part Of The Restricted Witt Superalgebra

Since V. G. Kac classified the finite dimensional simple Lie superalgebras overalgebraically closed fields of characteristic zero, the theory of Lie superalgebras hasundergone a significant development. Over a field of prime characteristic, however,the classification problem is still open for the finite dimensional simple Lie super-algebras, in which those of Cartan type are of the most importance. In general,the study of the maximal subsystems of an algebraic system, such as finite groups,Lie groups Lie(super)algebras, is an essential part of structural characterization ofthe system. In classical Lie theory, the classification of maximal subalgebras ofsimple Lie algebras over the field of complex numbers is one of the beautiful resultsof that theory. In this paper, we study the structure of the even part g of Wittsuperalgebra g over a field of prime characteristic and characterize three families ofmaximal graded subalgebras of g by the method of Lie algebra. The work in thisaspect will be helpful to further understand the intrinsic properties of the modularLie superalgebras of Cartan type. The author expects the main results of this paperwill provide a useful reference for a further investigation.At first, we study the structure of g over a field of prime characteristic andconstruct some series of submodules of g1with respect to g0. Moreover, we discussthe relation and the irreducibility of these submodules by the special elements―(m+1)-leaders and (m+1)-defective elements. Then all the decomposable maximalgraded subalgebras of g are characterized. Motivated by the work on modular Liealgebra, we study deeply the special maximal graded subalgebras and the reduciblemaximal graded subalgebras of g by means of the Z-graded structure, the propertyof generators and the irreducibility of g. Furthermore, we give the classification upto isomorphism, determine the number of isomorphic classes and give the formulasof dimension of the special maximal graded subalgebras and the reducible maximal graded subalgebras of g, respectively.