Star Presentation Poisson Algebras And Bialgebras

A Poisson algebra is an algebra with a Lie algebra structure and a commutative associative algebra structure which are entwined by the Leibniz rule.There is an equiv-alent presentation with only one binary operation.We call it a star presentation Poisson algebra(or briefly,an sp-Poisson algebra)in case of confusion with the classical notion of a Poisson algebra.This thesis is devoted to the study of the bialgebra theory of Poisson algebras in terms of sp-Poisson algebras.The notion of sp-Poisson bialgebras,which is equiva-lent to the notion of Poisson bialgebras,is first proposed.Many properties of Poisson bialgebras,including the equivalences between Poisson bialgebras,matched pairs and Manin triples,could be described via the star presentation.In particular,we study a special case of sp-Poisson bialgebras as an analogue of the known coboundary Poisson bialgebra,and this construction leads to an algebraic equation on an sp-Poisson algebra as an analogue of CYBE on a Lie algebra.We call it star presentation Poisson classical Yang-Baxter equation(or briefly,sp-Poisson CYBE).We would like to point out that the known coboundary Poisson bialgebra is just a special case of this construction.This means although a star presentation Poisson algebra is equivalent to a Poisson algebra in the usual presentation,the former might reveal more nontrivial intrinsic implications for some applications.That is one of the main aims of this thesis.At last,we introduce the notions of O-operators of an sp-Poisson algebra and pre-sp-Poisson algebras,and then discuss their relations with sp-Poisson CYBE.In particu-lar,each pre-sp-Poisson algebra gives a skew-symmetric solution of sp-Poisson CYBE,and hence constructs an sp-Poisson bialgebra.Moreover,although the presentation of pre-sp-Poisson algebras given in this thesis is different from the notion of pre-Poisson algebras given by Aguiar,both notions are in fact equivalent.