Infinite-Dimensional Lie Algebras And Leibniz Algebras

1.Central extensions play an important role in the theory of Lie algebras, and it is therefore not surprising that there are many researches on central extensions of various classes of Lie algebras. Recently there are many researches on central extensions of Leibniz algebras.In the chapter 2, some general theory of central extensions of Leibniz algebras are investigated. The universal central extension of g D for a unital associative dialgebra and a finite-dimensional simple Lie algebra 0 is obtained. All one-dimensional Leibniz central extensions on the algebras of differential operators over C[t, t-1] and C((t)), as well as on the quantum 2-torus, the Virasoro-like algebra and its q-analog are studied. We determine all nontrivial Leibniz 2-cocycles on these infinite dimensional Lie algebras.2. Recently there are several researches on the Steinberg algebras st(n, A), stl(n, A) and the Steinberg unitary algebras sin (n, A), stul(n, A) over a unital associative algebra A. In the chapter 3, we construct the Steinberg Leibniz algebra and Steinberg unitary Leibniz algebra and prove that they are universal central extensions of the special linear matrix Leibniz algebra si (n, D) and the elementary matrix Leibniz algebra eul(n, D, - ,r) over an associative dialgebra D respectively (n > 3). These investigations play important roles in the study of Leibniz algebras graded by finite root systems and Leibniz K-Theory.3. The concept of a Lie algebra graded by a finite root system was defined and investigated by Berman and Moody [BM] as an approach for studying various important classes of Lie algebras such as the intersection matrix Lie algebras of Slodowy [S], which arise in the study of singularities, or the extended affine Lie algebras of [AABGP]. The unifying theme is that these Lie algebras exhibit a grading by finite (possibly nonreduced) root system A. Lie algebras graded by finite root systems have been investigated in many papers.In the chapter 4, we introduce the definition of Leibniz algebras graded by finite root systems and give the structure of Leibniz algebras graded by finite root system of type A, D and E. This investigation gives an approach for studying more various classes of Leibniz algebras, which are not Lie algebras in general.4. Associative dialgebras are generalizations of associative algebras which give rise to Leibniz algebras instead of Lie algebras. So it is also interesting to study associative super dialgebras and Leibniz superalgebras. In the chapter 5, we study the general theory of Leibniz superalgebras and give some universal central extensions of some Leibniz superalgebras.5. Recently, Lie superalgebras graded by finite root systems are also studied in several papers. In the chapter 6, we mainly study Leibniz superalgebras graded by finite root systems. The structure of Leibniz superalgebras graded by finite root systems of type 6. Vertex operator representations for affine Lie algebras have a number of beautiful applications to the theories of modular forms, solitons, combinatoral identities and others. The toroidal Lie algebras are natural generalizations of affine Kac-Moody Lie algebras which were introduced and studied in [MRY] and [RM]. In [MRYjand [RM], the vertex representations in the homogeneous picture were constructed for toroidal Lie algebras of simply laced type on Fock space through the case of vertex operators. This construction is a generalization of the Frenkel-Kac and Segal lever-one construction of the affine Kac-Moody Lie algebras.In the chapter 7, the vertex operator representation for the toroidal Lie algebras of type G2 is constructed. The vertex operator representation for the affine Leibniz algebras is also constructed.