Research On Some Problems Of Restricted Lie Superalgebras And Hom-lie Superalgebras

Lie algebras play an important role in the leading edge of modern mathematics. Motivated by the super symmetry in physics, Lie algebras were generalized to Lie su-peralgebras, and then became an active research field. Lie superalgebras over a field of characteristic prime number are called modular ones. The classification of simple mod-ular Lie superalgebras is an open problem still. The studying of Lie algebras shows that the restrictedness theories and cohomology are helpful for solving this problem. So, in modular Lie superalgebras, one work of this thesis is to build a series of basic theories in restricted Lie superalgebras. The other work is to compute the low-dimensional cohomol-ogy of the classical Lie superalgebra sIm|n with coefficients in restricted Lie superalgebras W, S and H of Cartan type. In the Lie superalgebras over s field of characteristic zero, we mainly study the structures of Hom-Lie superalgebras, including basic structural theo-ries and the Hom-structures on infinite-dimensional Z-graded simlpe linearly compact Lie superalgebras of vector fields. The main contents in this thesis are formulated as follows:Firstly, we generalize a series of definitions and properties from Lie algebras to super-case, and then establish some basic theories of restricted Lie superalgebras, includ-ing the structures and properties of the restricted envelops. In this part, the main focus is the toral rank of a restricted Lie superalgebra. Some important results of toral rank are obtained, for example, the sufficient and necessary conditions of the toral rank vanishing and the sufficient conditions of a torus having a maximal toral rank. As an application, the absolute toral ranks of the special linear Lie superalgebra sIm|n, the finite-dimensional restricted Lie superalgebras W and S of Cartan type and the toral rank of S in W are com-puted.Secondly, for the purpose of computing the cohomology of sIm|n with coefficients in W, we embed sIm|n in the null of W, and then W is viewed as a sIm|n-module by means of the adjoint representation. Next, we decompose W into some submodules and some weight spaces relative to the standard Cartan subalgebra of sIm|n. By a new method, the work is reduced to consider certain submodules and the weight-derivations, which preserve the weights. Afterwards, since the Special superalgebra S contains sIm|n as subalgebra and W contains S as sIm|n-submodule, we use the results obtained for W to compute the low-dimensional cohomology groups of sIm|n with coefficients in the Special superalgebra S. By the same way, we obtain the low-dimensional cohomology groups of sI2|1with coefficients in H. It is worthwhile to indicate that the new method we adopt for computing the cohomology are different from the simple computations as before and the results we obtained show certain differences for the cohomology of sIm|n in the field of characteristics zero and prime number p.Finally, over a field of characteristic zero, we study Hom-Lie superalgebras. In par-ticular, we proved that the simple Hom-Lie superalgebras have no any non trivial ideals, including right and left ones. Next, some properties of multiplicative Hom-Lie superal-gebras are obtained. In the final, we proved that the multiplicative Hom-structures on infinite-dimensional simple Lie superalgebras of vector fields are trivial by means of s-tudying the multiplicative Hom-structures on the O-components and-1-components of these algebras.We expected that the results of restricted Lie superalgebras in this thesis will be useful for the classification of simple modular Lie superalgebras and the work in Hom-Lie superalgebras will enrich the theories of Hom-algebras and provide references for the relative research of quantum group and physics.