Structures And O-operators Of (Bihom)-Lie Superalgebras

In this thesis,we mainly study(σ,τ)-derivations and sympathetic structures of Lie superalgebras,complete Bihom-Lie superalgebras and their derivation superalgebras,and O-operators on Bihom-Lie algebras.This thesis is divided into five chapters as follows:Chapter 1 is the introduction.Firstly,the development of Lie algebras and Lie superalgebras is briefly introduced.After that,we introduce the progress of Hom-Lie algebras,Bihom-Lie algebras and Bihom-Lie superalgebras.In particular,the development of complete Lie algebras and complete Lie superalgebras are introduced.Then,we introduce some related results of generalized derivation algebras.Finally,some results of Rota-Baxter operators,O-operators and deformations are introduced.Chapter 2 is committed to studying the(σ,τ)-derivations of Lie superalgebras.The definition of the(σ,τ)-derivations of Lie superalgebras is given.Next,we research the relationship with derivations of Lie superalgebras.Particularly,if a(σ,σ)-derivation of Lie superalgebras is invertible,we use the properties of left-symmetric superalgebras,obtain a leftsymmetric superalgebra structure induced by a left-multiplication structure concerned with(σ,τ)-derivations.Moreover,we refer to the relationships with quasiderivations,(α,β,γ)derivations of Lie superalgebras,and give some examples.Finally,we define the interior of G-derivations of Lie superalgebras.In Particular,when G is a cyclic group,we calculate the corresponding Hilbert series.Chapter 3 is concentrated on studying the structure of sympathetic Lie superalgebras.We give the concept of sympathetic Lie superalgebras.After that,we show that every sympathetic Lie superalgebra can be uniquely decomposed to a direct sum of irreducible sympathetic ideals.Also,we prove that each sympathetic Lie superalgebra has a decomposition of the direct sum of the maximal sympathetic ideals and characteristic ideals.In addition,simply sympathetic Lie superalgebras is defined and its properties are studied.And finally,we also show that the derivations of Lie superalgebras are sympathetic.Chapter 4 study the complete Bihom-Lie superalgebras and its derivation superalgebras.We define complete Bihom-Lie superalgebras.After that,a sufficient condition is obtained to make Bihom-Lie superalgebras complete.Finally,it is prove that the derivation superalgebras of centerless perfect Bihom-Lie superalgebras are complete.Chapter 5 is devote to studying the O-operators for Bihom-Lie algebras.The definition of the O-operators on Bihom-Lie algebra with representations is given.By using the deformation complexes on Bihom-Lie algebras,we get the graded Li algebras with representations,and its Maurer-Cartan elements are the O-operators on Bihom-Lie algebras.After that,we obtain a differential graded Lie algebra associated with the O-operators.In addition,the O-operators can induce a Bihom-pre-Lie algebra.Finally,we study linear and formal deformations of O-operators on Bihom-Lie algebras with representations.Meanwhile,we give two equivalent formal deformations of the O-operators have cohomologous infinitesimals.