Some Results On Lie Superalgebras And Solvable Lie Algebras

Up to now,the study of Lie superalgebras and related topics has been one of the most active fields in mathematics.They are closely related to Lie algebras, homology and physics.In this thesis,we mainly consider some interesting problems in this field.In addition,we will also study a particular solvoble Lie algebra and give a complete classification.Now,we give a brief account of the contents of this thesis.Chapter 1 is introduction,in which we recall the background needed in our study,give the problems we will focus on and list the main results in this thesis.In Chapter 2,we mainly introduce the basic definitions of Leibniz superalgebra and its(co)homology.In addition,some general theories of central extensions of Leibnia superalgebras are investigated.A Lie superalgebra g can also be viewed as a Leibniz superalgebra,so in Chapter 3 we study the second Leibniz cohomology groups with trivial cofficients (HL~2(g,F)) for general Lie superalgebras,quadratic Lie superalgebras,and looplike Lie superalgebras by different means,and then give the explicit expressions and some examples,respectively.Based on these results,for some Lie superalgebras, we can get the difference between the one dimensional central extensions of these Lie superalgebras in Lie category and those in Leibniz category without complicated computations.In Chapter 4,we study the operator forms of the classical Yang-Baxter equation in Lie superalgebras.We introduce the notion of super O-operators and give their relation with the(standard) tensor form of the classical Yang-Baxter equation. Our main result is stated as Theorem 4.1 in Section 4.3.According to this result,we can get the close relations between the classical Yang-Baxter equation in Lie superalgebras and left-symmetric superalgebras which can be interpreted through the super O-operators.In particular,there are natural and explicit solutions of the classical Yang-Baxter equation in Lie superalgebras constructed from left-symmetric superalgebras. In Chapter 5,all finite-dimensional indecomposable solvable Lie algebras, having the quasifiliform Lie algebra N as the nilradical,are constructed and classifted. It turns out that the dimension of this kind of solvable Lie algebra is at most dim N+3.