Lie Bialgebra Structures Research Articles

Quantization of continuum Kac–Moody algebras

Continuum Kac-Moody algebras have been recently introduced by the authors and O. Schiffmann. These are Lie algebras governed by a continuum root system, which can be realized as uncountable colimits of Borcherds-Kac-Moody algebras. In this paper, we prove that any continuum Kac-Moody algebra is canonically endowed with a non-degenerate invariant bilinear form. The positive and negative Borel subalgebras form a Manin triple with respect to this pairing, inducing on the continuum Kac-Moody algebra a topological quasi-triangular Lie bialgebra structure. We then construct an explicit quantization, which we refer to as a continuum quantum group, and we show that the latter is similarly realized as an uncountable colimit of Drinfeld-Jimbo quantum groups.

Dual Lie Bialgebra Structures of W-algebra W(2, 2) Type

In the present paper, we investigate the dual Lie coalgebras of the centerless W(2, 2) algebra by studying the maximal good subspaces. Based on this, we construct the dual Lie bialgebra structures of the centerless W(2, 2) Lie bialgebra. As by-products, four new infinite dimensional Lie algebras are obtained.

Drinfel’d double structures for Poincaré and Euclidean groups

All non-isomorphic three-dimensional Poisson homogeneous Euclidean spaces are constructed and analyzed, based on the classification of coboundary Lie bialgebra structures of the Euclidean group in 3-dimensions, and the only Drinfel’d double structure for this group is explicitly given. The similar construction for the Poincaré case is reviewed and the striking differences between the Lorentzian and Euclidean cases are underlined. Finally, the contraction scheme starting from Drinfel’d double structures of the Lie algebra is presented.

Lie bialgebra structures on generalized loop Schrödinger-Virasoro algebras

We give a classification of Lie bialgebra structures on generalized loop Schrodinger-Virasoro algebras $$\mathfrak{sv}$$ . Then we find out that not all Lie bialgebra structures on generalized loop Schrodinger-Virasoro algebras $$\mathfrak{sv}$$ are triangular coboundary.

On the Classification of Lie Bialgebras by Cohomological Means

We approach the classification of Lie bialgebra structures on simple Lie algebras from the viewpoint of descent and non-abelian cohomology. We achieve a description of the problem in terms of faithfully flat cohomology over an arbitrary ring over \mathbb{Q} , and solve it for Drinfeld-Jimbo Lie bialgebras over fields of characteristic zero. We consider the classification up to isomorphism, as opposed to equivalence, and treat split and non-split Lie algebras alike. We moreover give a new interpretation of scalar multiples of Lie bialgebras hitherto studied using twisted Belavin-Drinfeld cohomology.

The Poincaré group as a Drinfel’d double

The eight nonisomorphic Drinfel’d double (DD) structures for the Poincaré Lie group in (2 + 1) dimensions are explicitly constructed in the kinematical basis. Also, the two existing DD structures for a non-trivial central extension of the (1 + 1) Poincaré group are also identified and constructed, while in (3 + 1) dimensions no Poincaré DD structure does exist. Each of the DD structures here presented has an associated canonical quasitriangular Poincaré -matrix whose properties are analysed. Some of these -matrices give rise to coisotropic Poisson homogeneous spaces with respect to the Lorentz subgroup, and their associated Poisson Minkowski spacetimes are constructed. Two of these (2 + 1) noncommutative DD Minkowski spacetimes turn out to be quotients by a Lorentz Poisson subgroup: the first one corresponds to the double of with trivial Lie bialgebra structure, and the second one gives rise to a quadratic noncommutative Poisson Minkowski spacetime. With these results, the explicit construction of DD structures for all Lorentzian kinematical groups in (1 + 1) and (2 + 1) dimensions is completed, and the connection between (anti-)de Sitter and Poincaré -matrices through the vanishing cosmological constant limit is also analysed.

Dual Lie Bialgebra Structures of Twisted Schrödinger-Virasoro Type

In this paper, the structures of dual Lie bialgebras of twisted Schrödinger-Virasoro type are investigated. By studying the maximal good subspaces, we determine the dual Lie coalgebras of the twisted Schrödinger-Virasoro algebras. Then based on this, we construct the dual Lie bialgebra structures of this type. As by-products, four new infinite dimensional Lie algebras are obtained.

Extended noncommutative Minkowski spacetimes and hybrid gauge symmetries

We study the Lie bialgebra structures that can be built on the one-dimensional central extension of the Poincaré and (A)dS algebras in (1 + 1) dimensions. These central extensions admit more than one interpretation, but the simplest one is that they describe the symmetries of (the noncommutative deformation of) an Abelian gauge theory, U(1) or SO(2) on Minkowski or (A)dS spacetime. We show that this highlights the possibility that the algebra of functions on the gauge bundle becomes noncommutative. This is a new way in which the Coleman–Mandula theorem could be circumvented by noncommutative structures, and it is related to a mixing of spacetime and gauge symmetry generators when they act on tensor-product states. We obtain all Lie bialgebra structures on centrally-extended Poincaré and (A)dS which are coisotropic w.r.t. the Lorentz algebra, and therefore admit the construction of a noncommutative principal gauge bundle on a quantum homogeneous spacetime. It is shown that several different types of hybrid noncommutativity between the spacetime and gauge coordinates are allowed. In one of these cases, an alternative interpretation of the central extension leads to a new description of the well-known canonical noncommutative spacetime as the quantum homogeneous space of a quantum Poincaré algebra.

Lie bialgebra structures on 2-step nilpotent graph algebras

We generalize a result on the Heisenberg Lie algebra that gives restrictions to possible Lie bialgebra cobrackets on 2-step nilpotent algebras with some additional properties. For the class of 2-step nilpotent Lie algebras coming from graphs, we describe these extra properties in a very easy graph-combinatorial way. We exhibit applications for fn, the free 2-step nilpotent Lie algebra.

The Goldman–Turaev Lie bialgebra in genus zero and the Kashiwara–Vergne problem

In this paper, we describe a surprising link between the theory of the Goldman–Turaev Lie bialgebra on surfaces of genus zero and the Kashiwara–Vergne (KV) problem in Lie theory. Let Σ be an oriented 2-dimensional manifold with non-empty boundary and K a field of characteristic zero. The Goldman–Turaev Lie bialgebra is defined by the Goldman bracket {−,−} and Turaev cobracket δ on the K-span of homotopy classes of free loops on Σ.Applying an expansion θ:Kπ→K〈x1,…,xn〉 yields an algebraic description of the operations {−,−} and δ in terms of non-commutative variables x1,…,xn. If Σ is a surface of genus g=0 the lowest degree parts {−,−}−1 and δ−1 are canonically defined (and independent of θ). They define a Lie bialgebra structure on the space of cyclic words which was introduced and studied by Schedler [31]. It was conjectured by the second and the third authors that one can define an expansion θ such that {−,−}={−,−}−1 and δ=δ−1. The main result of this paper states that for surfaces of genus zero constructing such an expansion is essentially equivalent to the KV problem. In [24], Massuyeau constructed such expansions using the Kontsevich integral.In order to prove this result, we show that the Turaev cobracket δ can be constructed in terms of the double bracket (upgrading the Goldman bracket) and the non-commutative divergence cocycle which plays the central role in the KV theory. Among other things, this observation gives a new topological interpretation of the KV problem and allows to extend it to surfaces with arbitrary number of boundary components (and of arbitrary genus, see [2]).

Goldman–Turaev formality from the Knizhnik–Zamolodchikov connection

For an oriented 2-dimensional manifold Σ of genus g with n boundary components, the space Cπ1(Σ)/[Cπ1(Σ),Cπ1(Σ)] carries the Goldman–Turaev Lie bialgebra structure defined in terms of intersections and self-intersections of curves. Its associated graded Lie bialgebra (under the natural filtration) is described by cyclic words in H1(Σ) and carries the structure of a necklace Schedler Lie bialgebra. The isomorphism between these two structures in genus zero has been established in [13] using Kontsevich integrals and in [2] using solutions of the Kashiwara–Vergne problem.In this note, we give an elementary proof of this isomorphism over C. It uses the Knizhnik–Zamolodchikov connection on C\\{z1,…zn}. We show that the isomorphism naturally depends on the complex structure on the surface. The proof of the isomorphism for Lie brackets is a version of the classical result by Hitchin [9]. Surprisingly, it turns out that a similar proof applies to cobrackets.Furthermore, we show that the formality isomorphism constructed in this note coincides with the one defined in [2] if one uses the solution of the Kashiwara–Vergne problem corresponding to the Knizhnik–Zamolodchikov associator.

AdS Poisson homogeneous spaces and Drinfel’d doubles

The correspondence between Poisson homogeneous spaces over a Poisson–Lie group G and Lagrangian Lie subalgebras of the classical double is revisited and explored in detail for the case in which is a classical double itself. We apply these results to give an explicit description of some coisotropic 2d Poisson homogeneous spaces over the group , namely 2d anti de Sitter space, 2d hyperbolic space and the lightcone in 3d Minkowski space. We show how each of these spaces is obtained as a quotient with respect to a Poisson-subgroup for one of the three inequivalent Lie bialgebra structures on and as a coisotropic one for the others.We then construct families of coisotropic Poisson homogeneous structures for 3d anti de Sitter space and show that the ones that are quotients by a Poisson subgroup are determined by a three-parameter family of classical r-matrices for , while the non Poisson-subgroup cases are much more numerous. In particular, we present the two Poisson homogeneous structures on that arise from two Drinfel’d double structures on . The first one realises as a quotient of by the Poisson-subgroup , while the second one, the non-commutative spacetime of the twisted κ-AdS deformation, realises as a coisotropic Poisson homogeneous space.

Dual Lie bialgebra structures of loop and current Virasoro\\ algebras

The loop and current Virasoro algebras are respectively the tensor product Lie algebras of the Virasoro algebra with the polynomial and Laurent polynomial algebra. In the present paper, the structures of dual Lie bialgebras of loop and current Virasoro algebras are investigated. As a result, a series of new infinite dimensional Lie algebras are obtained.

Lie bialgebra structures on derivation Lie algebra over quantum tori

We investigate Lie bialgebra structures on the derivation Lie algebra over the quantum torus. It is proved that, for the derivation Lie algebra W over a rank 2 quantum torus, all Lie bialgebra structures on W are the coboundary triangular Lie bialgebras. As a by-product, it is also proved that the first cohomology group H1(W, W ⊗ W) is trivial.

Lie bialgebras, fields of cohomological dimension at most 2 and Hilbert's Seventeenth Problem

We investigate Lie bialgebra structures on simple Lie algebras of non-split type A. It turns out that there are several classes of such Lie bialgebra structures, and it is possible to classify some of them. The classification is obtained using Belavin–Drinfeld cohomology sets, which are introduced in the paper. Our description is particularly detailed over fields of cohomological dimension at most two, and is related to quaternion algebras and the Brauer group. We then extend the results to certain rational function fields over real closed fields via Pfister's theory of quadratic forms and his solution to Hilbert's Seventeenth Problem.

The 2-plectic structures induced by the Lie bialgebras

In this paper we show that if the Lie algebra $\mathfrak{g}$ admits a Lie bialgebra structure and $\mathcal{D}$ is a Lie group with Lie algebra $\mathfrak{d}$, the double of $\mathfrak{g}$, then $\mathcal{D}$ or its quotient by a suitable Lie subgroup admits a $2$-plectic structure. In particular it is shown that the imaginary part of the Killing form on $\mathfrak{sl}(n, \mathbb{C})$ (as a real Lie algebra) induces a $2$-plectic structure on $SL(n, \mathbb{C})$.

The Lie bialgebra structure of the vector space of cyclic words

This paper presents a combinatorial proof of the existence of a Lie bialgebra structure over the vector space of reduced cyclic words. Any surface with non-empty boundary has an associated vector space determined by the corresponding surface symbol, this space is known as the space of reduced cyclic words. The Lie bialgebra structure over this space was introduced by Chas in the article Combinatorial Lie bialgebras of curves on surfaces, where a proof of the existence of this structure is given. This proof is based on the construction of an isomorphism between the space of reduced cyclic words and the space of curves on a surface.

Classification of Quantum Groups and Belavin–Drinfeld Cohomologies

In the present article we discuss the classification of quantum groups whose quasi-classical limit is a given simple complex Lie algebra g. This problem is reduced to the classification of all Lie bialgebra structures on g(K) , where K= C((ħ)). The associated classical double is of the form g(K) ⊗ KA, where A is one of the following: K[ e] , where e2= 0 , K⊕ K or K[ j] , where j2= ħ. The first case is related to quasi-Frobenius Lie algebras. In the second and third cases we introduce a theory of Belavin–Drinfeld cohomology associated to any non-skewsymmetric r-matrix on the Belavin–Drinfeld list (Belavin and Drinfeld in Soviet Sci Rev Sect C: Math Phys Rev 4:93–165, 1984). We prove a one-to-one correspondence between gauge equivalence classes of Lie bialgebra structures on g(K) and cohomology classes (in case II) and twisted cohomology classes (in case III) associated to any non-skewsymmetric r-matrix.

Integrable deformations of Rössler and Lorenz systems from Poisson–Lie groups

A method to construct integrable deformations of Hamiltonian systems of ODEs endowed with Lie–Poisson symmetries is proposed by considering Poisson–Lie groups as deformations of Lie–Poisson (co)algebras. Moreover, the underlying Lie–Poisson symmetry of the initial system of ODEs is used to construct integrable coupled systems, whose integrable deformations can be obtained through the construction of the appropriate Poisson–Lie groups that deform the initial symmetry. The approach is applied in order to construct integrable deformations of both uncoupled and coupled versions of certain integrable types of Rössler and Lorenz systems. It is worth stressing that such deformations are of non-polynomial type since they are obtained through an exponentiation process that gives rise to the Poisson–Lie group from its infinitesimal Lie bialgebra structure. The full deformation procedure is essentially algorithmic and can be computerized to a large extent.

Classification of quantum groups and Lie bialgebra structures on [formula omitted]. Relations with Brauer group

Given an arbitrary field F of characteristic 0, we study Lie bialgebra structures on sl(n,F), based on the description of the corresponding classical double. For any Lie bialgebra structure δ, the classical double D(sl(n,F),δ) is isomorphic to sl(n,F)⊗FA, where A is either F[ε], with ε2=0, or F⊕F or a quadratic field extension of F. In the first case, the classification leads to quasi-Frobenius Lie subalgebras of sl(n,F). In the second and third cases, a Belavin–Drinfeld cohomology can be introduced which enables one to classify Lie bialgebras on sl(n,F), up to gauge equivalence. The Belavin–Drinfeld untwisted and twisted cohomology sets associated to an r-matrix are computed. For the Cremmer–Gervais r-matrix in sl(3), we also construct a natural map of sets between the total Belavin–Drinfeld twisted cohomology set and the Brauer group of the field F.