Cohomology And Deformations Of Some Classes Lie Superalgebras

Lie superlagebras,as a natural generalization of Lie algebras,are important research objects in Lie theory.Since Lie superalgebras have important applications in supersymmetry problems in physics,its research has become an important topic in modern mathematics.In study of Lie superalgebras,cohomology and deformations of Lie superalgebras are important research topics that have attracted many researchers in recent years.The aim of this paper is to study cohomology and deformations of some classes of Lie superalgebras.Given a module of a Lie superalgebra,we can define a cochain,cocycle,coboundary and cohomology of the Lie superalgebra with coefficients in this module.In particular,the dimension of the cohomology space is called Betti number.Since the cohomology theory of Lie superalgebras has many applications in modern mathematics and physics,its reasearch has attracted the attention of many reseachers.From an operation over a module,Musson introduced the definition of cup products,which induce Z-graded superalgebra structures on the cochain and cohomology space,respectively.Char-acterizations of cup products and Betti numbers are the two most important research questions in the study of Lie superalgebra cohomology.For a Lie superalgebra,trivial represenation and adjoint representation are the two most common classes of representations.The cohomology of a Lie su-peralgebra with coefficients in the trivial module and adjoint module is called trivial cohomology and adjoint cohomology,respectively.The research on these two classes cohomology is a research topic that many scholars have paid attention to in recent years.In the case of trivial cohomology,the products of a field induce graded super-communicative associative superalgebra structures with identity elements on its cochain space and cohomology space through the cup product.As what happens in Lie case,the associative superalgebra structure on the cochains with coefficients in the trivial modules for a Lie superalgebra,which is arising from the cup product,is isomorphic to the super-exterior algebra generated by its dual superspace.By this isomorphism,Leites introduced the definitions of divided power algebras and divided power cohomology over a field of prime characteristic.Different from the case of trivial cohomology,the dimensions of divided cochain space and divided cohomology space are always finite dimensional.In the case of adjoint cohomology,since there is a one-to-one correspondence between the equivalence classes of infinitesimal deformations of a Lie superalgebra and the even 2-cohomology classes,we can describe the one-parameter formal deformations of a Lie superalgebras by calculating the even part of 2-cohomology.It is worth noting that different from the case of Lie algebras,odd 2-cocycles can not be used to construct formal deformations.So we only consider the even case when we compute formal deformations of a Lie superalgebra.The aim of this paper is to study trivial cohomology,adjoint cohomology and divided power cohomology of some classes of Lie superalgebras,including filiform Lie superalgebras,two-step nilpotent Lie superalgebras and metric Lie superalgebras.Suppose that the ground field F is an algebraically closed field of characteristic not 2 or 3.Suppose that char F=0 when we study trivial cohomology and adjoint cohomology.Suppose that char F=p>3,when we study divided power cohomology.In 1970,when studying the reducibility of nilpotent Lie algebra varieties,Vergne introduced the concept of filiform Lie algebras and pointed out that every filiform Lie algebra can be obtained by an infinitesimal deformation of model filiform Lie algebra L_n.Then this concept was generalized to the case of Lie superalgebras,called filiform Lie superalgebras.When studying nilptent Lie superalgebras,Gilg introduced the concept of super-nilindex.A nilpotent Lie superalgebra with the maximum super-nilindex is called a filiform Lie superalgebra.As what happens in Lie case completely,every filiform Lie superalgebra can be obtained by an infinitesimal deformation of model filiform Lie superalgebra L_n,m.Therefore,L_n,mbecomes an important research object.In terms of classification,Gilg studied the classification of low-dimensional filiform Lie superalgebras over the complex field.In terms of cohomology,Navarro et al.calculated the even part of the second cohomology of model filiform Lie superalgebra L_n,m,moreover,all infinitesimal deformations have been described completely.However,there is no general results of trivial cohomology of filiform Lie superalgebras.In the chapter three of this paper,we will study the trivial cohomology and divided power cohomology of filiform Lie superalgebras,including model filiform Lie superalgebras and low-dimensional filiform Lie superalgebras.Given a symplectic form on a symplectic space,we can define its Heisenberg Lie algebra.It is a two-step nilpotent Lie algebra with a one-dimensional center.Due to its application in the com-mutative relations of quantum mechanics,Heisenberg Lie algebra becomes an important research object in modern mathematics.In 2011,Rodríguez-Vallarte,Salgado and Sánchez-Valenzuela gen-eralized the concept of Heisenberg algebras to Lie superalgebras by studying its supersymmetry,which is called Heisenberg Lie superalgebra.That is a two-step nilpotent Lie superalgebra with a one-dimensional center.According to the parity of the centers,Heisenberg Lie superalgebras can be divided into even-center Heisenberg Lie superalgebras h_2m,nand odd-center Heisenberg Lie superal-gebras ba_n.Among nilpotent Lie superalgebras,The one-step nilpotent Lie superalgebra is an Abel Lie superalgebra which only has trivial multiplications,and the results of its trivial cohomology and adjoint cohomology can be directly obtained from the triviality of the coboundary operator on the cochain.Therefore,the study of two-step nilpotent Lie superalgebras has important reference significance for the study of general nilpotent Lie superalgebras.The multiplication of Lie superal-gebras can induce a superalgebra structure over the adjoint cohomology by means of cup products.Unlike the case of trivial cohomology,this multiplication is not always associative.In the chapter four of this paper,we will study adjoint cohomology of two-step nilpotent Lie superalgebras in terms of cup products and Betti numbers.First,we give a criterion for the triviality of the cup product.As an application,we obtain that the cup product on the adjoint cohomology of a Heisenberg Lie superalgebra is trivial.Next,we describe the Betti numbers of the adjoint cohomology of two-step nilpotent Lie superalgebras by means of Hochschild-Serre spectral sequences.In particular,we ob-tain the Betti number formula of adjoint cohomology of Heisenberg Lie superalgebras by conclusions of the Betti numbers of the trivial cohomology of Heisenberg Lie superalgebras.Metric Lie superalgebras are referred to a class of Lie superalgebras with even,non-degenerate,supersymmetric,and invariant bilinear forms,which can be regarded as a generalization of semisim-ple Lie algebras in Lie superalgebra case.For a metric Lie superalgebra,starting from a metric infinitesimal deformation,we can construct a metric deformation.In the chapter five of this paper,we will use this method to study all metric deformations of metric Lie superalgebras with dimension not more than 6 over the complex field.