The Compatible Left-symmetric Superalgebraic Structures On The W(2,2) Superalgebra

In this paper,we mainly use the known conclusions that the structures of compatible left-symmetric algebras on the W(2,2)Lie algebra to determine the structures of compatible left-symmetric superalgebras on the W(2,2)superalgebra.The W(2,2)superalgebra referred to in this paper is an infinite dimensional Lie superalgebra obtained by affine construction of a 2|2 dimensional Balinsky-Novikov superalgebra and then through the central extension.Its even part is the W(2,2)Lie algebra.The main structure of this paper is as follows:In the introduction,we introduce the related research background and current situation.In the second section,some basic knowledge and conclusions are introduced,including the definition of the W(2,2)superalgebra and left-symmetric superalgebra,and the structures of compatible left-symmetric algebra on the W(2,2)Lie algebra.In the third section,by using the compatible left-symmetric algebraic structures on the centreless W(2,2)Lie algebra,the compatible left-symmetric superalgebraic structures on the centreless W(2,2)superalgebra are determined.In the fourth section,we determine the structures of the compatible left-symmetric superalgebras on the W(2,2)superalgebras.