Structures And Representations Of Modular Lie Superalgebras And Hom-type Algebras

The present thesis is devoted to studying structures for simple modular Lie superalge-bra M and simple modular Lie superalgebra N, as well as representations, deformations, extensions for δ-Hom-Jordan Lie superalgebras and d-Hom-Jordan Lie triple systems.Firstly, we study simple modular Lie superalgebra M. Finite-dimensional simple modular Lie superalgebra M and infinite-dimensional simple modular Lie superalge-bra M are constructed respectively. To begin with, we discuss the even part of finite-dimensional simple modular Lie superalgebra M, which is denoted by Mo. The generator set of Mo is found, moreover, the derivation algebra of Mo is determined. Then infinite-dimensional Lie superalgebra M is studied and its simplicity is proved. The natural fitration of the infinite-dimensional simple modular Lie superalgebra M is proved to be invariant under automorphisms by ad-nilpotent elements, so the infinite-dimensional simple modular Lie superalgebras M are classified up to isomorphisms.Secondly, we study simple modular Lie superalgebra N. Infinite-dimensional modu-lar Lie superalgebra N is constructed by generalizing Z2-graded to Z2m-graded, m ∈Z+, we prove the simplicity and obtain generators of N, then determine its derivation super-algebra.Thirdly,δ-Hom-Jordan Lie superalgebras are considered.δ-Hom-Jordan Lie superal-gebras are generalizations of Lie superalgebras and Hom-Lie superalgebras. By the direct sum of the space of a-derivations, any α-derivation gives rise to a derivation extension of a δ-Hom-Jordan Lie superalgebra. We discuss representation, apply it and semidirect prod-uct to construct new multiplicative δ-Hom-Jordan Lie superalgebras. Then, we study the trivial representations, adjoint representations and as-adjoint representations of δ-Hom-Jordan Lie superalgebras, respectively. Hom-Nijenhuis operators are introduced, more-over, we obtain that the deformation generated by a Hom-Nijenhuis operator is trivial. The notion of T*-extensions of δ-Hom-Jordan Lie superalgebras is given. It is determined that any finite-dimensional nilpotent metric δ-Hom-Jordan Lie superalgebra over an al-gebraically closed field of characteristic different from 2 is isometric to (a nondegenerate ideal of codimension 1 of) a T*-extension of a δ-Hom-Jordan Lie superalgebra.Fourthly, it is dedicated to δ-Hom-Jordan Lie triple systems, which are generaliza-tions of Lie triple systems and Hom-Lie triple systems. To start with, low order coho-mology spaces on J-Hom-Jordan Lie triple systems are defined, and it is determined that there is a one-to-one correspondence between equivalent classes of central extensions of multiplicative δ-Hom-Jordan Lie triple systems and the third cohomology space. Then, we discuss 1-parameter formal deformation of δ-Hom-Jordan Lie triple systems, and it shows that two equivalent 1-parameter formal deformations belong to the same third cohomology space. We also study Hom-Nijenhuis operators of δ-Hom-Jordan Lie triple systems, and give corresponding deformation. Finally, we discuss a special class of δ-Hom-Jordan Lie triple systems-δ-Jordan Lie triple systems. Using cohomology theory of δ-Hom-Jordan Lie triple systems, the abelian extension of δ-Jordan Lie triple systems is studied, and it is determined that two equivalent abelian extensions give the same representation. Besides, we obtain some results on T*-extensions of δ-Jordan Lie triple systems, and prove that sufficient and necessary conditions that two T*-extensions are equivalent.