Double Lie Group Research Articles

Double principal bundles

We define double principal bundles (DPBs), for which the frame bundle of a double vector bundle, double Lie groups and double homogeneous spaces are basic examples. It is shown that a double vector bundle can be realized as the associated bundle of its frame bundle. Also dual structures, gauge transformations and connections in DPBs are investigated.

Flat Affine and Symplectic Geometries on Lie Groups

In this paper we exhibit a family of flat left invariant affine structures on the double Lie group of the oscillator Lie group of dimension 4, associated to each solution of classical Yang-Baxter equation given by Boucetta and Medina. On the other hand, using Koszul's method, we prove the existence of an immersion of Lie groups between the group of affine transformations of a flat affine and simply connected manifold and the classical group of affine transformations of $\mathbb{R}^n$. In the last section, for each flat left invariant affine symplectic connection on the group of affine transformations of the real line, describe for Medina-Saldarriaga-Giraldo, we determine the affine symplectomorphisms. Finally we exhibit the Hess connection, associated to a Lagrangian bi-foliation, which is flat left invariant affine.

Dirac approach to constrained submanifolds in a double loop group: From Wess-Zumino-Novikov-Witten to Poisson-Lie σ-model

We study the restriction to a family of second class constrained submanifolds in the cotangent bundle of a double Lie group equipped with a 2-cocycle extended symplectic form to build the corresponding Dirac brackets. It is shown that, for 2-cocycle vanishing on each isotropic subspace of the associated Manin triple, the Dirac bracket contains no traces of the cocycle. We also investigate the restriction of the left translation action of the double Lie group on its cotangent bundle, where it fails to be a canonical transformation. However, the Hamiltonian symmetry is restored on some special submanifolds. The main application is to loop groups, showing that a WZNW-type model on the double Lie group with a quadratic Hamilton function in the momentum maps associated with the left translation action on the cotangent bundle with the canonical symplectic form, restricts to a collective system on some special submanifolds. There, the Lagrangian version coincides with the so-called Poisson-Lie σ -model.

Lattices in some symplectic or affine nilpotent Lie groups

The main aim of this paper is the description of a large class of lattices in some nilpotent Lie groups, sometimes filiforms, carrying a flat left invariant linear connection and often a left invariant symplectic form. As a consequence we obtain an infinity of, non homeomorphic, compact affine or symplectic nilmanifolds.We review some new facts about the geometry of compact symplectic nilmanifolds and we describe symplectic reduction for these manifolds. For the Heisenberg–Lie group, defined over a local associative and commutative finite dimensional real algebra, a necessary and sufficient condition for the existence of a left invariant symplectic form, is given. Finally in the symplectic case we show that a lattice in the group determines naturally lattices in the double Lie group corresponding to any solution of the classical Yang–Baxter equation.

The lie group and integrability of the Fisher type travelling wave equation

For the Fisher-type wave equation, which has two stable states and one unstable state, it is proved that only in two particular cases, the corresponding travelling wave equation admits a double parameter Lie group, and based on a method different to the traditional one, its two independent first integrals are given. It is proved further that in the two integrable cases, the different bounded and non-trivial travelling wave solutions, which are corresponding the invariant manifolds of the corresponding equation under the Lie transformation, can be expressed with elementary functions although they cannot be obtained directly from the two independent first integrals.

Poisson–Lie [formula omitted]-duality and non-trivial monodromies

We describe a general framework for studying duality among different phase spaces which share the same symmetry group H . Solutions corresponding to collective dynamics become dual in the sense that they are generated by the same curve in H . Explicit examples of phase spaces which are dual with respect to a common non-trivial coadjoint orbit O c , 0 ( α , 1 ) ⊂ h ∗ are constructed on the cotangent bundles of the factors of a double Lie group H = N ⋈ N ∗ . In the case H = L D , the loop group of a Drinfeld double Lie group D , we built up a hamiltonian description of Poisson–Lie T -duality for non-trivial monodromies and its relation with non-trivial coadjoint orbits is obtained.

Double Products and Hypersymplectic Structures on ℝ4 n

In this paper we give a procedure to construct hypersymplectic structures on $R^{4n}$ beginning with affine-symplectic data on $R^{2n}$. These structures are shown to be invariant by a 3-step nilpotent double Lie group and the resulting metrics are complete and not necessarily flat. Explicit examples of this construction are exhibited.

From double Lie groups to quantum groups

It is shown, using geometric methods, that there is a $C^*$-algebraic quantum group related to any double Lie group (also known as a matched pair of Lie groups or a bicrossproduct Lie group). An algebra underlying this quantum group is the algebra of a di

Classical Yang-Baxter equation and left invariant affine geometry on lie groups

Let G be a Lie group with Lie algebra $ \Cal G: = T_\epsilon G$ and $T^*G = \Cal G^* \rtimes G$ its cotangent bundle considered as a Lie group, where G acts on $\Cal G^*$ via the coadjoint action. We show that there is a 1-1 correspondance between the skew-symmetric solutions $r\in \wedge^2 \Cal G$ of the Classical Yang-Baxter Equation in G, and the set of connected Lie subgroups of $T^*G$ which carry a left invariant affine structure and whose Lie algebras are lagrangian graphs in $ \Cal G \oplus \Cal G^*$. An invertible solution r endows G with a left invariant symplectic structure and hence a left invariant affine structure. In this case we prove that the Poisson Lie tensor $\pi := r^+ - r^-$ is polynomial of degree at most 2 and the double Lie groups of $(G,\pi)$ also carry a canonical left invariant affine structure. In the general case of (non necessarly invertible) solutions r, we supply a necessary and suffisant condition to the geodesic completness of the associated affine structure

Manageability of Multiplicative Unitaries Associated with Double Lie Groups

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

Classical dynamics, alternative carrier spaces and group-valued constants of motion

We show that a second order differential equation, when represented as a vector field on TQ or T ∗Q , may admit alternative Lagrangian and Hamiltonian descriptions. When Q is in addition a Lie group, the bundle may be a double Lie group; in this case we find group-valued constants of motion.

Poisson structures on double Lie groups

Lie bialgebra structures are reviewed and investigated in terms of the double Lie algebra, of Manin- and Gauss-decompositions. The standard R-matrix in a Manin decomposition then gives rise to several Poisson structures on the correponding double group, which is investigated in great detail.

Completely Integrable Systems — A Generalization

We present a slight generalization of the notion of completely integrable systems such that they can be integrated by quadratures. We use this generalization to integrate dynamical systems on double Lie groups.