Left-Symmetric Algebraic Structures On Some Infinite Dimensional Lie Algebras

Left-symmetric algebras, an important class of non-associative algebras, were first introduced by Cayley as a special type of rooted tree algebras. These algebras are closely related to Lie algebra and have many applications in mathematical physics and other mathematical branches, such as Lie group, differential geometry and cohomology theory.The super-Virasoro algebras, also known as the superconformal algebras, are nontrivial graded extensions of the Virasoro algebra to Lie superalgebra version; both the Virasoro algebra and the super-Virasoro algebras play important roles in theoretical physics such as conformal field theory and string theory. In the second chapter of this dissertation, we classify the compatible left-symmetric superalgebra structures on the N =2 Ramond and Neveu-Schwarz superconformal algebras under certain conditions.In the paper [35], Prof. Chengming Bai and Xiaomin Tang classified a type of nongraded left-symmetric algebraic structures on the Witt algebra under certain rational condition. In this paper, we show that this rational condition is not necessary. This leads to a more elegant classification of the left-symmetric algebraic structures and Novikov algebraic structures on the Witt algebra.