The Representations And Structures Of Bihom-Lie Superalgebras And 3-Bihom-Lie Algebras

There are five parts in this thesis.Part 1 is concerned with representations and cohomologies of Bihom-Lie superalgebras.Firstly,we construct the direct sum of Bihom-Lie superalgebras,and obtain a sufficient and necessary condition for a homomorphism of Bihom-Lie superalgebras.Next,we introduceα~kβ~l-derivations of Bihom-Lie superalgebras and prove that there exists the derivation exten-sion by anα~0β~1-derivation of Bihom-Lie superalgebras.Moreover,we introduce the notion of a representation of Bihom-Lie superalgebras,and then we have coboundary operators,chain complexes,and cohomology spaces with the representation.Also,we construct the semidirect product of a Bihom-Lie superalgebra and its representation.Finally,we calculate the lower-order cohomology spaces of trivial representations and adjoint representations.Part 2 is devoted to Bihom-Nijenhuis operators and extensions of Bihom-Lie super-algebras.First of all,we give the definition of Bihom-Nijenhuis operators of Bihom-Lie superalgebras and get that a deformation constructed by Bihom-Nijenhuis operators is a trivial deformation.Then we define quadratic Bihom-Lie superalgebras and construct T~*-extensions of quadratic Bihom-Lie superalgebras using adjoint representations.In addition,we obtain a sufficient and necessary condition that a quadratic Bihom-Lie superalgebra is isometric to a T~*-extension,and a sufficient and necessary condition for the equivalence and the isometric equivalence of T~*-extensions.Part 3 is committed to representations and cohomologies of 3-Bihom-Lie algebras.First-ly,we have that the tensor product of a 3-totally Bihom-associative algebra and a 3-Bihom-Lie algebra is a 3-Bihom-Lie algebra and find that 3-Bihom-Lie algebras and Bihom-Lie algebras can be constructed with each other.Meanwhile,we get a sufficient and neces-sary condition for linear maps of 3-Bihom-Lie algebras to be homomorphisms.Then,we introduce the notion of generalized derivations of 3-Bihom-Lie algebras,construct derivation extensions of 3-Bihom-Lie algebras by generalized derivations and obtain a condition for the isomorphism of derivation extensions.Also,we construct the semidirect product via repre-sentations of 3-Bihom-Lie algebras.Finally,we get coboundary operators,chain complexes,and cohomology spaces using the representations.Part 4 is dedicated to extensions of 3-Bihom-Lie algebras.In the beginning,we define a 3-linear mapθby representations of 3-Bihom-Lie algebras,construct T_θ-extensions of 3-Bihom-Lie algebras usingθ,and obtain a condition for the isomorphism of T_θ-extensions.Next,we study quadratic 3-Bihom-Lie algebras,construct T~*-extensions of quadratic 3-Bihom-Lie algebras using coadjoint representations,and prove a sufficient and necessary condition for the isometry of quadratic 3-Bihom-Lie algebras and T~*-extensions.And finally,we define abelian extensions of 3-Bihom-Lie algebras,obtain a representation and a closed 2-Bihom-cochain from an abelian extension and show that there is a one-to-one correspondence between equivalent classes of abelian extensions of 3-Bihom-Lie algebras and the second cohomology spaces.Part 5 is concentrated on product structures and complex structures on 3-Bihom-Lie algebras.To begin with,we introduce the definition of product structures on 3-Bihom-Lie algebras by Nijenhuis operators and obtain a sufficient and necessary condition that 3-Bihom-Lie algebras have product structures.Then we find that there are four types special product structures,and also prove a sufficient and necessary condition for their existence.Mean-while,we introduce the notion of complex structures on 3-Bihom-Lie algebras,there are also four types special complex structures,and we obtain a sufficient and necessary condition for the existence of these complex structures.Further,we show the relation between complex structures and product structures.