Courant Algebroids and Strongly and Strongly Homotopy Lie Algebras

Courant algebroids are structures which include as examples the doubles of Lie bialgebras and the bundle TM T^*M with the bracket introduced by T. Courant for the study of Dirac structures. Within the category of Courant algebroids one can construct the doubles of Lie bialgebroids, the infinitesimal objects for Poisson groupoids. We show that Courant algebroids can be considered as strongly homotopy Lie algebras.     

Homogeneous generalized holomorphic structures

Homogeneous generalized holomorphic structures in the context of homogeneous principal fiber bundles are investigated. They are characterized in terms of Lie algebra data, and the generalized Dolbeault cohomology groups associated to a homogeneous generalized holomorphic vector bundle are identified with certain relative Lie algebra cohomology groups. We also provide some examples, using generalized flag manifolds.     

Lie Algebras Associated with Group U（n）

Starting from the subgroups of the group U（n）, the corresponding Lie algebras of the Lie algebra Al are presented, from which two well-known simple equivalent matrix Lie algebras are given. It follows that a few expanding Lie algebras are obtained by enlarging matrices. Some of them can be devoted to producing double integrable couplings of the soliton hierarchies of nonlinear evolution equations. Others can be used to generate integrable couplings involving more potential functions. The above Lie algebras are classified into two types. Only one type can generate the integrable couplings, whose Hamiltonian structure could be obtained by use of the quadratic-form identity. In addition, one condition on searching for integrable couplings is improved such that more useful Lie algebras are enlightened to engender. Then two explicit examples are shown to illustrate the applications of the Lie algebras. Finally, with the help of closed cycling operation relations, another way of producing higher-dimensional Lie algebras is given.     

Lie Algebras Associated with Group U(n)

Starting from the subgroups of the group U(n), the corresponding Lie algebras of the Lie algebra A1 are presented, from which two well-known simple equivalent matrix Lie algebras are given. It follows that a few expanding Lie algebras are obtained by enlarging matrices. Some of them can be devoted to producing double integrable couplings of the soliton hierarchies of nonlinear evolution equations. Others can be used to generate integrable couplings involving more potential functions. The above Lie algebras are classified into two types. Only one type can generate the integrable couplings, whose Hamiltonian structure could be obtained by use of the quadratic-form identity. In addition, one condition on searching for integrable couplings is improved such that more useful Lie algebras are enlightened to engender. Then two explicit examples are shown to illustrate the applications of the Lie algebras. Finally, with the help of closed cycling operation relations, another way of producing higher-dimensional Lie algebras is given.     

Manageability of Multiplicative Unitaries Associated with Double Lie Groups

We prove that unitary multiplicative operators associated with double Lie groups are manageable.     

Su (2) Poisson–Lie T Duality

Poisson–Lie target space duality is a framework where duality transformations are properly defined. In this Letter, we investigate the dual pair of -models defined by the double SO(3,1) in the Iwasawa decomposition.     

A characterization of strict Jacobi-Nijenhuis manifolds through the theory of Lie algebroids

We obtain a characterization of strict Jacobi-Nijenhuis structures using the equivalent notions of generalized Lie bialgebroid and Jacobi bialgebroid.     

Some compatible Poisson structures and integrable bi-Hamiltonian systems on four dimensional and nilpotent six dimensional symplectic real Lie groups

We provide an alternative method for obtaining of compatible Poisson structures on Lie groups by means of the adjoint representations of Lie algebras. In this way we calculate some compatible Poisson structures on four dimensional and nilpotent six dimensional symplectic real Lie groups. Then using Magri-Morosi’s theorem we obtain new bi-Hamiltonian systems with four dimensional and nilpotent six dimensional symplectic real Lie groups as phase spaces.     

Nambu–Dirac Structures for Lie Algebroids

The theory of Nambu–Poisson structures on manifolds is extended to the context of Lie algebroids in a natural way based on the derived bracket associated with the Lie algebroid differential. A new way of combining Nambu–Poisson structures and triangular Lie bialgebroids is described in this work. Also, we introduce the concept of a higher order Dirac structure on a Lie algebroid. This allows to describe both Nambu–Poisson structures and Dirac structures on manifolds in the same setting.     

Continual Lie algebra bicomplexes and integrable models

We introduce bicomplex structures associated with Saveliev-Vershik continual Lie algebras, and derive non-linear dynamical systems resulting from the bicomplex conditions. Examples related to classes of continual Lie algebras, including contact Lie, Poisson bracket, and Hilbert-Cartan ones are discussed. Using the bicomplex linearization problem, we derive corresponding conservation laws. Presented at the International Colloquium “Integrable Systems and Quantum Symmetries”, Prague, 16–18 June 2005.     

<Emphasis Type="Italic">Q</Emphasis>-Manifolds and Mackenzie Theory

Double Lie algebroids were discovered by Kirill Mackenzie from the study of double Lie groupoids and were defined in terms of rather complicated conditions making use of duality theory for Lie algebroids and double vector bundles. In this paper we establish a simple alternative characterization of double Lie algebroids in a supermanifold language. Namely, we show that a double Lie algebroid in Mackenzie’s sense is equivalent to a double vector bundle endowed with a pair of commuting homological vector fields of appropriate weights. Our approach helps to simplify and elucidate Mackenzie’s original definition; we show how it fits into a bigger picture of equivalent structures on ‘neighbor’ double vector bundles. It also opens ways for extending the theory to multiple Lie algebroids, which we introduce here.     

Linearization Problems for Lie Algebroids and Lie Groupoids

In this Letter, we discuss a series of linearization problems – for Poisson structures, Lie algebroids, and Lie groupoids. The last problem involves a conjecture on the structure of proper groupoids. Attempting to prove this by the method of averaging leads to problems concerning almost actions of compact groups and almost invariant submanifolds for compact group actions. The Letter ends with a discussion of possible extensions of the convexity theorems for momentum maps of hamiltonian actions of compact groups.     

Compatible Lie Bialgebras

A compatible Lie algebra is a pair of Lie algebras such that any linear combination of the two Lie brackets is a Lie bracket.We construct a bialgebra theory of compatible Lie algebras as an analogue of a Lie bialgebra.They can also be regarded as a "compatible version" of Lie bialgebras,that is,a pair of Lie bialgebras such that any linear combination of the two Lie bialgebras is still a Lie bialgebra.Many properties of compatible Lie bialgebras as the "compatible version" of the corresponding properties of Lie bialgebras are presented.In particular,there is a coboundary compatible Lie bialgebra theory with a construction from the classical Yang-Baxter equation in compatible Lie algebras as a combination of two classical Yang-Baxter equations in Lie algebras.Furthermore,a notion of compatible pre-Lie algebra is introduced with an interpretation of its close relation with the classical Yang-Baxter equation in compatible Lie algebras which leads to a construction of the solutions of the latter.As a byproduct,the compatible Lie bialgebras St into the framework to construct non-constant solutions of the classical Yang-Baxter equation given by Golubchik and Sokolov.     

Two Integrable Couplings of Modified Tu Hierarchy

Two different integrable couplings of the modified Tu hierarchy are obtained under the zero curvature equation by using two higher dimension Lie algebras. Furthermore, a complex Hamiltonian structures of the second integrable couplings is presented by taking use of the variational identity.     

Composition of Lie Group Elements from Basis Lie Algebra Elements

It is shown explicitly how one can obtain elements of Lie groups as compositions of products of other elements based on the commutator properties of associated Lie algebras. Problems of this kind can arise naturally in control theory. Suppose an apparatus has mechanisms for moving in a limited number of ways with other movements generated by compositions of allowed motions. Two concrete examples are: (1) the restricted parallel parking problem where the commutator of translations in y and rotations in the xy-plane yields translations in x. Here the control problem involves a vehicle that can only perform a series of translations in y and rotations with the aim of efficiently obtaining a pure translation in x; (2) involves an apparatus that can only perform rotations about two axes with the aim of performing rotations about a third axis. Both examples involve three-dimensional Lie algebras. In particular, the composition problem is solved for the nine three- and four-dimensional Lie algebras with non-trivial solutions. Three different solution methods are presented. Two of these methods depend on operator and matrix representations of a Lie algebra. The other method is a differential equation method that depends solely on the commutator properties of a Lie algebra. Remarkably, for these distinguished Lie algebras the solutions involve arbitrary functions and can be expressed in terms of elementary functions.     

The Application of Discrete Variational Identity on Semi-Direct Sums of Lie Algebras

With non-semisimple Lie algebras, the trace identity was generalized to discrete spectral problems. Then the corresponding discrete variational identity was used to a class of semi-direct sums of Lie algebras in a lattice hierarchy case and obtained Hamiltonian structures for the associated integrable couplings of the lattice hierarchy. It is a powerful tool for exploring Hamiltonian structures of discrete soliton equations.     

The 4th structure

According to conclusions made by F. Montaner and E. Zelmanov (unpublished), there exist four Lie bialgebra structures on g[u] up to classical twisting. Therefore we investigated four possible cases for the classical double and the corresponding Lie co-bracket on g[u]. Among them, a new case, where the Lie bialgebra structures are in anone-to-one correspondence with the first introduced herequasi-rational solutions of the CYBE. We also consideredquasi-rational r-matrices in sl(n), particularly, we calculated all the quasi-rationalr-matrices in sl(2).