Cohomology Theories Of Two Types Of Superalgebras And Their Applications

This thesis mainly studies the cohomology theories of Hom-Lic superalgebras and 3-Lie superalgebras,as well as their applications in deformat.ions and extensions.It consists of six chapters.In Chapter 17 we introduce the backgrounds and recent developments of the research work in this thesis,and then describe the main results.In Chapter 2,we review Hom-Lie superalgebras,3-Lie superalgebras,graded Lie algebras,Lie 3-algebras and other related algebras,and mainly introduce representation and cohomology theories of Hom-Lie superalgebras as well as 3-Lie superalgebras.In Cha.pter 3,we study cohomology spaces and deforma.tions of Heisenberg Hom-Lie superalgebras.Firstly,we give the definition of Heisenberg Hom-Lie superalgebras,and give the classification of 3-dimensional ones,up to isomorphism.Next,we compute the first and second cohomology spaces of 3-dimensional Heisenberg Hom-Lie superalgebras with respect to adjoint representations,and then characterize the infinitesimal deformations.In Chapter 4,we study cohomologies and deformations of relative Rota-Baxter operators on Hom-Lie superalgebras.To begin with,we define abilinear bracket,which can characterize the structures,representations and coboundary operators of Hom-Lie superalgebras.Next,we prove that relative Rota-Baxter operators on a Hom-Lie superalgebra are precisely Maurer-Cartan elements of a certain Z-graded Lie algebra,and give the relations of relative Rota-Baxter operators and semi-direct product Hom-Lie super algebras,Nijienhuis operators.Finally,we develop cohomologies and deformations of relative Rota-Baxter operators on Hom-Lie superalgebras,and prove that the infinitesimal deformations and the extendability of finite order deformations are controlled by the first and second cohomology spaces respectively.In Chapter 5,we study generalized representations and the corresponding cohomology theory of 3-Lic supcralgcbras,as well as generalized one-parameter formal deformations,generalized T*-extensions and abelian extensions.Firstly,we define generalized representations of 3-Lie superalgebras,and give the Ma.urer-Cartan characterizations of both usual representations and generalized representations.Meanwhile,we show several examples of generalized representations that are not usual representations.Next,we develop the cohomology and deformation theories corresponding to generalized representations,and prove that the infinitesimal generalized deformations and the extendability of finite order generalized deformations are controlled by the low order cohomology spaces.Finally,we prove that generalized T*-extensions are controlled by Maurer-Cartan elements of a certain differential graded Lie algebra,and a metric generalized T*-extension reduces to a usual one.Moreover,split and non-split abelian extensions shall be described by generalized semidirect product 3-Lie superalgebras and Maurer-Cartan elements of a certain differential graded Lie algebra respectively.In Chapter 6,we study the cohomologies and deformations of relative Rota-Baxter operators on 3-Lie superalgebras.Firstly,by a higher derived bracket,we construct a Lie-3 algebra and prove that relative Rota-Baxter operators on a 3-Lie superalgebra are precisely its Maurer-Cartan elements.Next,we give deformations of relative Rota-Baxter operators on 3-Lie superalgebras and the cohomologies that controll the deformations.Finally,we give the relation of relative Rota-Baxter operators on Lie superalgebras and 3-Lie superalgebras,and prove that there is a correspondence between the cohomology spaces of relative RotaBaxter operators on Lie superalgebras and those on 3-Lie superalgebras.