论文标题
差异的反对称无限双齿，相干衍生和泊松式双齿
Differential Antisymmetric Infinitesimal Bialgebras, Coherent Derivations and Poisson Bialgebras
论文作者
论文摘要
我们通过将ASI Bialgebras的研究推广到差分代数的背景下，为差异代数差异代数建立了一种差异代数的双重理论，称为差异。它们的特征是差异代数的双重结构以及匹配的差分代数对。差分代数中关联阳式方程的类似物的反对称解提供了差异性ASI双gebras，而$ \ Mathcal {O} $的概念也是差异代数和不同代数的操作员，也引入了不同的树枝状代数代数来生产前者。另一方面，在ASI Bialgebra上的相干推导的概念被引入为差分ASI Bialgebra的等效结构。它们包括对ASI Bialgebras的派生，以及ASI Bialgebra上的一组相干推导组成了一个Lie代数，该代数是谎言组的Lie代数，该代数由该ASI Bialgebra上的相干自动化组成。最后，我们将差分ASI Bialgebras的研究应用于泊松式bialgebras，将泊松代数的构造从交换性差分代数的构造延伸到具有两个通勤派生到双齿的背景，这与poisson bialgebras的构建理论一致。特别是，我们构建了来自差异Zinbiel代数的泊松双子。
We establish a bialgebra theory for differential algebras, called differential antisymmetric infinitesimal (ASI) bialgebras by generalizing the study of ASI bialgebras to the context of differential algebras, in which the derivations play an important role. They are characterized by double constructions of differential Frobenius algebras as well as matched pairs of differential algebras. Antisymmetric solutions of an analogue of associative Yang-Baxter equation in differential algebras provide differential ASI bialgebras, whereas in turn the notions of $\mathcal{O}$-operators of differential algebras and differential dendriform algebras are also introduced to produce the former. On the other hand, the notion of a coherent derivation on an ASI bialgebra is introduced as an equivalent structure of a differential ASI bialgebra. They include derivations on ASI bialgebras and the set of coherent derivations on an ASI bialgebra composes a Lie algebra which is the Lie algebra of the Lie group consisting of coherent automorphisms on this ASI bialgebra. Finally, we apply the study of differential ASI bialgebras to Poisson bialgebras, extending the construction of Poisson algebras from commutative differential algebras with two commuting derivations to the context of bialgebras, which is consistent with the well constructed theory of Poisson bialgebras. In particular, we construct Poisson bialgebras from differential Zinbiel algebras.
