Infinite Dimensional Lie Superalgebra And Balinsky-Novikov Superalgebra

The Balinsky-Novikov（BN）superalgebras were introduced by Balinsky in[2].These nonas-sociative Z₂-graded superalgebras can be regarded as certain supersymmetric generalization of the Novikov algebras.By certian affinalizations of BN superalgebras,we can obtain some infinite di-mensional Lie superalgebras,which play important roles in mathematics and physics.In this paper,we study classification and constructions of BN algebras,and discuss their re-lations with infinite dimensional Lie superalgebras.First,we classify 3-dimensional complex BN superalgebras;Second,we give two canonical constructions of BN superalgebra via differential algebras and double constructions,and then we describle a conceptual approach to Lie superalge-bras via BN superalgebras;Finally,we give a notion of BNL superalgebras,which can be regarded as certian supersymmetric generalization of that of the Gel’fand-Dorfman algebras.We study con-structions of BNL superalgebras and relations with infinite dimensional Lie superalgebras.