论文标题
扩展gel'fand-dorfman bialgebras的结构
Extending structures for Gel'fand-Dorfman bialgebras
论文作者
论文摘要
Gel'Fand-Dorfman Bialgebra既是Lie代数，又是具有一定兼容条件的Novikov代数，出现在完全可以整合的系统中的Hamiltonian Pairs和一类称为Quadratic lie Conformal Algebras的特殊谎言组合代数中出现。在本文中，我们研究了Gel'fand-dorfman bialgebras的扩展结构问题，该问题等同于某些扩展结构的二次结构结构式代数。明确地给出了一个gel'fand-dorfman bialgebra $（a，\ circ，[\ cdot，\ cdot]）$，此问题要求如何描述和分类所有gel'fand-dorfman bialgerbraic bialgerbraic在矢量空间$ e $ $ $ $ $（a \ subset e $）上的$（$ is $ cd cd cd cd cd cd cd cd cd cd cd cd） $ e $ $ e $的子代数为同构，其对$ a $的限制是身份图。 Motivated by the theories of extending structures for Lie algebras and Novikov algebras, we construct an object $\mathcal{GH}^2(V,A)$ to answer the extending structures problem by introducing a definition of unified product for Gel'fand-Dorfman bialgebras, where $V$ is a complement of $A$ in $E$.特别是，当$ \ text {dim}（v）= 1 $详细时，我们研究了特殊情况。
Gel'fand-Dorfman bialgebra, which is both a Lie algebra and a Novikov algebra with some compatibility condition, appears in the study of Hamiltonian pairs in completely integrable systems and a class of 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 $(A, \circ, [\cdot,\cdot])$, this problem asks that how to describe and classify all Gel'fand-Dorfman bialgebraic structures on a vector space $E$ $(A\subset E$) such that $(A, \circ, [\cdot,\cdot])$ 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 $\mathcal{GH}^2(V,A)$ to answer the extending structures problem by introducing a definition of unified product for Gel'fand-Dorfman bialgebras, where $V$ is a complement of $A$ in $E$. In particular, we investigate the special case when $\text{dim}(V)=1$ in detail.
