Extending Structures for Gel'fand-Dorfman Bialgebras  

作　　者：Jia Jia WEN Yan Yong HONG 

机构地区：[1]School of Mathematics,Hangzhou Normal University,Hangzhou 311121,P.R.China

出　　处：《Acta Mathematica Sinica,English Series》2024年第2期619-638,共20页数学学报（英文版）

基　　金：Supported by the National Natural Science Foundation of China(Grant Nos.12171129,11871421);the Zhejiang Provincial Natural Science Foundation of China(Grant No.LY20A010022);the Scientific Research Foundation of Hangzhou Normal University(Grant No.2019QDL012)。

摘　　要：Gel'fand-Dorfman bialgebra,which is both a Lie algebra and a Novikov algebra with some compatibility condition,appeared in the study of Hamiltonian pairs in completely integrable systems.They also emerged in the description of a class special Lie conformal algebras called quadratic Lie conformal algebras.In this paper,we investigate the extending structures problem for Gel'fand-Dorfman bialgebras,which is equivalent to some extending structures problem of quadratic Lie conformal algebras.Explicitly,given a Gel'fand-Dorfman bialgebra(A,o,[.,.]),this problem is how to describe and classify all Gel'fand-Dorfman bialgebra structures on a vector space E(A⊂E)such that(A,o,[.,.])is a subalgebra of E up to an isomorphism whose restriction on A is the identity map.Motivated by the theories of extending structures for Lie algebras and Novikov algebras,we construct an object gH2(V,A)to answer the extending structures problem by introducing the notion of a unified product for Gel'fand-Dorfman bialgebras,where V is a complement of A in E.In particular,we investigate the special case when dim(V)=1 in detail.

关 键 词：Gel'fand-Dorfman bialgebra Lie conformal algebra Extending structures problem Novikovalgebra 

分 类 号：O152.5[理学—数学]