Classification Of Finite-dimensional Simple Balinsky-Novikov Superalgebras Over Fields Of Characteristic Zero

In the late 19th century,mathematician S.Lie introduced a class of important non-associative algebras-Lie algebras when studying continuous transformation groups,and then developed many algebras which are closely related to Lie algebras.Novikov algebra is one of them.It is a special kind of pre-Lie algebra or it can be called left symmetric algebra.This is because its algebraic structure satisfies that the right multiplication operator is commutative,and the left multiplication operator is symmetric,which means left multiplication operator forming a Lie algebra.Novikov algebra was originally produced by mathematicians I.Gel’fand and Dorfman when studying Hamiton operators of formal variational operators.Balinsky and Novikov gave the definition in 1985.Finally,Novikov algebra was named by mathematician Osborne.Novikov algebras are closely related to Virasoro algebras in physics.Virasolo algebra and its extension can be realized by affine expansion of Novikov algebra,and Novikov algebra is also closely related to fluid dynamics.Therefore,the research on Novikov algebra has attracted a lot of attention,which has made some progress and achieved some important results.In fact,there are two different types of supersymmetric extensions of Novikov algebra in essence.One is the Novikov superalgebra defined by Xu Xiaoping,and the other is Balinsky-Novikov superalgebra which will be studied in this article.The concept of Balinsky-Novikov superalgebra was defined by Balinsky at first and was introduced by Balinsky in order to construct super-Virasoro type Lie superalgebra in 1987.It is essentially a Z₂-graded nonassociative superalgebra,which can be regarded as a kind of super analog of Novikov algebra.In addition,Balinsky-Novikov superalgebras can also be used to construct local translation invariant Lie superalgebras of vector-valued functions on lines.They can be associated with a class of important infinite dimensional Lie superalgebras in physics,including super-Virasoro algebras with N=1,some infinite dimensional Lie superalgebras and vertex superalgebras.They can be naturally realized through Balinsky-Novikov superalgebras with invariant bilinear forms.These Lie superalgebras have important connections with many branches of mathematics and physics.Therefore,the research on Balinsky-Novikov superalgebras has both theoretical significance and application value,which will help us to solve some problems between mathematics and physics.In this article,we will focus on the study of finite dimensional simple Balinsky-Novikov superalgebras over fields of characteristic zero,and mainly discuss the classification of finite dimensional simple Balinsky-Novikov superalgebras over general characteristic zero fields and some relevant questions.We have introduced some research backgrounds of Novikov algebra and Balinsky-Novikov superalgebra at beginning,as well as some existing research results,and described some definitions related to Novikov algebra and Balinsky-Novikov superalgebra,and proved some required properties.Then we have carried out further research on Balinsky-Novikov superalgebra.Firstly,we have probed into the classification of Balinsky-Novikov superalgebras in real number field.Through the discussion about the classification of odd and even parts of Balinsky-Novikov superalgebra in real number field,we have given the complete classification of finite dimensional simple Balinsky-Novikov superalgebra in real number field.Secondly,a super analog of Zel’manov theorem of Novikov algebra is proved over fields of characteristic zero.On this basis,we analyzed the even part of Balinsky-Novikov superalgebra in combination with Zel’manov theorem,and discussd the odd part of Balinsky-Novikov superalgebra on a finite extension of general characteristic zero field.Finally,we gave the complete classification of Balinsky-Novikov superalgebra over fields of characteristic zero.At the end of the article,it is also proved that the basic Lie superalgebra of the finite dimensional Balinsky-Novikov superalgebra is solvable.