On simple real Lie bialgebras (original) (raw)
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Lie bialgebras of complex type and associated Poisson Lie groups
Journal of Geometry and Physics, 2008
In this work we study a particular class of Lie bialgebras arising from Hermitian structures on Lie algebras such that the metric is ad-invariant. We will refer to them as Lie bialgebras of complex type. These give rise to Poisson Lie groups G whose corresponding duals G * are complex Lie groups. We also prove that a Hermitian structure on g with ad-invariant metric induces a structure of the same type on the double Lie algebra Dg = g ⊕ g * , with respect to the canonical ad-invariant metric of neutral signature on Dg. We show how to construct a 2n-dimensional Lie bialgebra of complex type starting with one of dimension 2(n − 2), n ≥ 2. This allows us to determine all solvable Lie algebras of dimension ≤ 6 admitting a Hermitian structure with ad-invariant metric. We exhibit some examples in dimension 4 and 6, including two one-parameter families, where we identify the Lie-Poisson structures on the associated simply connected Lie groups, obtaining also their symplectic foliations.
Some remarks on the classification of Poisson Lie groups
Contemporary Mathematics, 1994
W e describe some results in the problem of classifying the bialgebra structures on a given nite dimensional Lie algebra. We consider two aspects of this problem. One is to see which Lie algebras arise (up to isomorphism) as the big algebra in a Manin triple, and the other is to try and determine all the exact Poisson structures for a given semisimple Lie algebra. We follow here the presentation of the talk that one of us gave at the Yokohama Symposium; in particular, we recall many w ell known properties so that it is essentially self-contained.
Classification of real three-dimensional Lie bialgebras and their Poisson–Lie groups
Journal of Physics A-mathematical and General, 2005
Classical r-matrices of the three-dimensional real Lie bialgebras are obtained. In this way all three-dimensional real coboundary Lie bialgebras and their types (triangular, quasitriangular or factorizable) are classified. Then, by using the Sklyanin bracket, the Poisson structures on the related Poisson-Lie groups are obtained.
Poisson structures on double Lie groups
Journal of Geometry and Physics, 1998
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. 1991 Mathematics Subject Classification. 22E30, 58F05, 70H99.
Mixed product Poisson structures associated to Poisson Lie groups and Lie bialgebras
We introduce and study some mixed product Poisson structures on product manifolds associated to Poisson Lie groups and Lie bialgebras. When the Lie bialgebras are quasitriangular, our construction is equivalent to that of fusion products of quasi-Poisson G-manifolds introduced by Alekseev, Kosmann-Schwarzbach, and Meinrenken. Our main examples include mixed product Poisson structures on products of flag varieties defined by Belavin-Drinfeld rmatrices.
On Lie Algebroids and Poisson Algebras
We introduce and study a class of Lie algebroids associated to faithful modules which is motivated by the notion of cotangent Lie algebroids of Poisson manifolds. We also give a classification of transitive Lie algebroids and describe Poisson algebras by using the notions of algebroid and Lie connections.
1+1) Schrodinger Lie bialgebras and their Poisson-Lie groups
1999
All Lie bialgebra structures for the (1+1)-dimensional centrally extended Schrodinger algebra are explicitly derived and proved to be of the coboundary type. Therefore, since all of them come from a classical r-matrix, the complete family of Schrodinger Poisson-Lie groups can be deduced by means of the Sklyanin bracket. All possible embeddings of the harmonic oscillator, extended Galilei and gl(2) Lie bialgebras within the Schrodinger classification are studied. As an application, new quantum (Hopf algebra) deformations of the Schrodinger algebra, including their corresponding quantum universal R-matrices, are constructed.
Classification of real three-dimensional Poisson–Lie groups
Journal of Physics A: Mathematical and Theoretical, 2012
All real three dimensional Poisson-Lie groups are explicitly constructed and fully classified under group automorphisms by making use of their one-to-one correspondence with the complete classification of real three-dimensional Lie bialgebras given in [1]. Many of these 3D Poisson-Lie groups are non-coboundary structures, whose Poisson brackets are given here for the first time. Casimir functions for all three-dimensional PL groups are given, and some features of several PL structures are commented.
On the geometry of Lie algebras and Poisson tensors
Journal of Physics A: Mathematical and General, 1994
A geometric programme to analyze the structure of Lie algebras is presented with special emphasis on the geometry of linear Poisson tensors. The notion of decomposable Poisson tensors is introduced and an algorithm to construct all solvable Lie algebras is presented. Poisson-Liouville structures are also introduced to discuss a new class of Lie algebras that include as a subclass semisimple Lie algebras. A decomposition theorem for Poisson tensors is proved for a class of Poisson manifolds including linear ones. Simple Lie algebras are also discussed from this viewpoint and lower dimensional real Lie algebras are analyzed.
2012
We prove a 2-categorical analogue of a classical result of Drinfeld: there is a one-to-one correspondence between connected, simply-connected Poisson Lie 2-groups and Lie 2-bialgebras. In fact, we also prove that there is a one-to-one correspondence between connected, simply connected quasi-Poisson 2-groups and quasi-Lie 2-bialgebras. Our approach relies on a "universal lifting theorem" for Lie 2-groups: an isomorphism between the graded Lie algebras of multiplicative polyvector fields on the Lie 2-group on one hand and of polydifferentials on the corresponding Lie 2algebra on the other hand.