Non-graded Left-symmetric Algebraic Structures Of General Form On Witt Algebras

Left-symmetric algebras,also known as pre-Lie algebras,were first proposed by A.Cayley at the end of the 19th century,and were also involved in the study of affine manifolds and affine structures of Lie groups.Since the 20th century,left-symmetric algebra has played an important role in many mathematical fields,such as the complex structures of Lie groups and Lie algebras,and the classical and quantum Yang-Baxter equations.However,since the left-symmetric algebra is a non-associative algebra,it does not have complete structure theories or representation theories like finite dimensional associative algebras,complex semisimple Lie algebras.In generally,we study its structures by studying its sub-adjacent Lie algebra.In this thesis,we study the compatible nongraded left-symmetric algebraic structures on the Witt algebra.Based on the work of Professors Hongjia Chen and Xiangqian Guo,some simplifications are made to the specific conditions required for the general forms of non-graded left-symmetric algebraic structures on the Witt algebra.In this thesis,we calculated the cases of p=3,4,5(p is the number of non-graded parts)in details,and obtained good properties.We expect that if there is a better way,the condition can be generalized to the general case.