We give two kinds of reduction relating Poisson manifolds and Jacobi manifolds by classifying reducible Dirac and Jacobi-Dirac structures into two classes, respectively. The proof needs only some direct calculation, without using the existence of momentum mappings or the introducing of admissible functions, etc.. Meanwhile we present some examples and applications.

Trangular Jacobi bialgebroids are a generalization of triangular Lie bialge- broids in the sense of Mackenzie and Xu.In this paper,we will discuss the properties of this type.Meanwhile,we will show the deformed triangular Jacobi bialgebroids,as a consequence,we get a strict Jacobi-Nijenhuis structure.

The notion of the dual characteristic pair of Dirac structures is introduced,using which, the authors give the conditions for maximally isotropic sub-bundles being integrable. From this result they obtain a condition for inducing natural Dirac structures on the sub-manifolds of Dirac manifolds, which generalizes Courant's result. Moreover, the conditions for Poisson manifolds inducing Poisson structures on its sub-manifolds are obtained,which improves those given by Weinstein and Courant. Finally, they prove that there is a 1-1 correspondence between the reducible Dirac structures of presymplectic forms and the symplectic structures of the reductive manifolds.

This paper discusses some properties of DN manifolds, such as the submanifolds of DN manifolds, the reduction of DN manifolds and the fundamental vector fields on DN manifolds.

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Protobialgebroids include several kinds of algebroid structures such as Lie algebroid, Lie bialgebroid, Lie quasi-bialgebroid, etc. In this paper, the Dirac theories are generalized from Lie bialgebroid to protobialgebroid. We give the integrable conditions for a maximally isotropic sub-bundle being a Dirac structure for a protobialgebroid by the notion of a characteristic pair. From the integrable conditions, we found out that the Dirac structure has closed relations with the twisting of a protobialgebroid. At last, some special cases of the Dirac structures for protobialgebroids are discussed.

We study polynomial representations of finite dimensional (R or C) Lie algebras. As a total classification, we show that there are altogether three types of such nontrivial representations and give their subtle structures.

This paper is a continuation of the work done in [1]. We define Dirac-Nijenhuis structures (DN structures) on Lie bialgebroids, a generalization of the notion of Dirac-Nijenhuis structures on manifolds given in [1]. However, the problems we are going to deal with are more complex than the ones in [1]. The necessary and sufficient conditions for a structure to be a DN structure on a Lie bialgebroid as well as several examples of these structures are given. At the end of the paper we discuss the fundamental sections of DN structures.

This paper gives the definition of Dirac-Nijenhuis manifolds (DN-manifolds). It discusses their properties and the relations among DN-manifolds, Poisson-Nijenhuis manifolds (PN-manifolds) and presymplectic-Nijenhuis manifolds (ΩN-manifolds).

In this paper, some properties of reduction for symplectic Γ-spaces are discussed. The properties of stable subgroups are discussed. We find that the symplectic action of a symplectic groupoid on a symplectic manifold can induce a symplectic map between reduced symplectic manifolds. This symplectic action can be characterized by the action of its induced symplectic groupoid on a symplectic manifold. Lastly, we shall discuss Poisson reduction and give a Poisson reduction theorem.

In this paper, we use the notion of morphisms of Lie bialgebroids to discuss thePoisson groupoids action, we obtain some properties of the Poisson action of a Poisson groupoid on aPoisson manifold in the sense of the morphisms of Lie bialgebroids.

A symplectic reduction method for symplectic G-spaces is given in this paper without using the existence of momentum mappings. By a method similar to the above one, the arthors give a symplectic reduction method for the Poisson action of Poisson Lie groups on symplectic manifolds, also without using the existence of momentum mappings. The symplectic reduction method for momentum mappings is thus a special case of the above results.

Let (Γ P,α,β) be a Poisson groupoid. Weprove first a existence theorem for coisotropic bisections in Γ. Thenwe prove that (Γ K,φ,ψ) is also a Poisson groupoid if K isa coisotropic bisection in Γ. And it is isomorphism with (Γ P,α,β). Furthermore, using the results above we obtain someproperties and a necessary and sufficient condition of coisotropicbisections.

令(┏ (→→)P,a,β)是Poisson群胚.如果它的每个α-纤维与β-纤维至多交于一点,则┏在任一点x的特征分布有直和分解△(x) = △α(x)+ △β(x),其中△α(x)CTxα-1(u), △β(x)CTxβ-1(v)且它们都是(x)的辛子空间.由此得到辛叶Sx的辛子流形Sαx和Sβx,使在映射之α下,Sβx辛微分同胚于P中辛叶Su,在映射之β下,Sαx反辛微分同胚于P中辛叶Sv (定理4和5).对于一般的Poisson群胚,也可得到类似的Sαx和Sβx,它们差一局部辛微分同胚是唯一确定的(定理6).把以上结果用于辛群胚,还可得到一些更具体的性质(定理7及其推论).