Lie Bialgebra Research Articles

Jacobi–Lie symmetry and Jacobi–Lie T-dual sigma models on group manifolds

Using the concept of Jacobi–Lie group and Jacobi–Lie bialgebra, we generalize the definition of Poisson–Lie symmetry to Jacobi–Lie symmetry. In this regard, we generalize the concept of Poisson–Lie T-duality to Jacobi–Lie T-duality and present Jacobi–Lie T-dual sigma models on Lie groups, which have Jacobi–Lie symmetry. Using this symmetry, new cases of duality appear and some examples are given. This generalization may provide insights to understand the quantum features of Poisson–Lie T-duality, in a more satisfactory way.

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.

3-Leibniz bialgebras (3-Lie bialgebras)

The aim of this paper is to extend the notion of Leibniz bialgebra (and Lie bialgebra) to [Formula: see text]-Leibniz bialgebra (and [Formula: see text]-Lie bialgebra) by use of the cohomology complex of [Formula: see text]-Leibniz algebras. Some theorems about Leibniz bialgebras are extended and proved in the case of [Formula: see text]-Leibniz bialgebras ([Formula: see text]-Lie bialgebras). Moreover, a new theorem on the correspondence between [Formula: see text]-Leibniz bialgebra and its associated Leibniz bialgebra is proved. Finally, some examples are discussed in detail.

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.

3-Lie bialgebras and 3-Lie classical Yang–Baxter equations in low dimensions

In this paper, we give low-dimensional examples of local cocycle 3-Lie bialgebras and double construction 3-Lie bialgebras which were introduced in the study of the classical Yang–Baxter equation and Manin triples for 3-Lie algebras. We give an explicit and practical formula to compute the skew-symmetric solutions of the 3-Lie classical Yang–Baxter equation (CYBE). As an illustration, we obtain all skew-symmetric solutions of the 3-Lie CYBE in complex 3-Lie algebras of dimensions 3 and 4 and then the induced local cocycle 3-Lie bialgebras. On the other hand, we classify the double construction 3-Lie bialgebras for complex 3-Lie algebras in dimensions 3 and 4 and then give the corresponding eight-dimensional pseudo-metric 3-Lie algebras.

Irregular Riemann–Hilbert Correspondence, Alekseev–Meinrenken Dynamical r-Matrices, and Drinfeld Twists

In 2004, Enriquez-Etingof-Marshall suggested a new approach to the Ginzburg-Weinstein linearization theorem for a quasitriangular Lie bialgebra $(\g,r)$. This approach is based on solving a system of PDEs for a gauge transformation between the classical r-matrix $r$ and the Alekseev-Meinrenken dynamical r-matrix. They proved that the semiclassical limit of an admissible Drinfeld twist gives rise to a solution of the PDEs. In this paper, we explain that preferred gauge transformations can be constructed as connection maps for a certain irregular Riemann-Hilbert problem (provided $r$ is the standard classical r-matrix). Along the way, we give a symplectic geometric interpretation of their PDEs, as a symplectic neighborhood version of the Ginzburg-Weinstein linearization theorem. Our construction is based on earlier works by Boalch. We then prove that for semisimple Lie algebra $\g$, any solution of the PDEs for the gauge transformation is the semiclassical limit of an admissible Drinfeld twist. As byproducts, we find a surprising relation between the connection maps and Drinfeld twists as well as a new description of the Lu-Weinstein symplectic double.

N-Lie bialgebras

We define n-Lie coalgebras with rank r and discuss their structures. We also introduce n-Lie bialgebras and investigate their structures. A triple is an n-Lie bialgebra if and only if is a conformal 1-cocycle on the n-Lie algebra L associated to L-modules , . Furthermore we study two-dimensional extensions of finite dimensional n-Lie bialgebras, and construct an -dimensional -Lie bialgebra associated to an m dimensional n-Lie bialgebra . Finally we discuss the bialgebra structure on the finite dimensional simple n-Lie algebra , and prove that the only bialgebra structures on the simple n-Lie algebra are of rank zero and rank two.

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.

Lie Groups in Quasi-Poisson Geometry and Braided Hopf Algebras

We extend the notion of Poisson-Lie groups and Lie bialgebras from Poisson to \frak{g} -quasi-Poisson geometry and provide a quantization to braided Hopf algebras in the corresponding Drinfeld category. The basic examples of these \frak{g} -quasi-Poisson Lie groups are nilpotent radicals of parabolic subgroups. We also provide examples of moment maps in this new context coming from moduli spaces of flat connections on surfaces.

From Braided Infinitesimal Bialgebras to Braided Lie Bialgebras

The present paper is a continuation of [1], where we considered braided infinitesimal Hopf algebras (i.e., infinitesimal Hopf algebras in the Yetter-Drin feld category for any Hopf algebra H), and constructed their Drinfeld double as a generalization of Aguiar’s result. In this paper we mainly investigate the necessary and sufficient condition for a braided infinitesimal bialgebra to be a braided Lie bialgebra (i.e., a Lie bialgebra in the category ).

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 $SL(n, \mathbb{C})$.

Classification of four-dimensional real Lie bialgebras of symplectic type and their Poisson–Lie groups

We classify all four-dimensional real Lie bialgebras of symplectic type and obtain the classical r-matrices for these Lie bialgebras and Poisson structures on all the associated four-dimensional Poisson–Lie groups. We obtain some new integrable models where a Poisson–Lie group plays the role of the phase space and its dual Lie group plays the role of the symmetry group of the system.

Higher genus Kashiwara–Vergne problems and the Goldman–Turaev Lie bialgebra

We define a family KV(g,n+1) of Kashiwara–Vergne problems associated with compact connected oriented 2-manifolds of genus g with n+1 boundary components. The problem KV(0,3) is the classical Kashiwara–Vergne problem from Lie theory. We show the existence of solutions to KV(g,n+1) for arbitrary g and n. The key point is the solution to KV(1,1) based on the results by B. Enriquez on elliptic associators. Our construction is motivated by applications to the formality problem for the Goldman–Turaev Lie bialgebra g(g,n+1). In more detail, we show that every solution to KV(g,n+1) induces a Lie bialgebra isomorphism between g(g,n+1) and its associated graded grg(g,n+1). For g=0, a similar result was obtained by G. Massuyeau using the Kontsevich integral. For g≥1, n=0, our results imply that the obstruction to surjectivity of the Johnson homomorphism provided by the Turaev cobracket is equivalent to the Enomoto–Satoh obstruction.

Classification of real low-dimensional Jacobi (generalized)–Lie bialgebras

We describe the definition of Jacobi (generalized)–Lie bialgebras [Formula: see text] in terms of structure constants of the Lie algebras [Formula: see text] and [Formula: see text] and components of their 1-cocycles [Formula: see text] and [Formula: see text] in the basis of the Lie algebras. Then, using adjoint representations and automorphism Lie groups of Lie algebras, we give a method for classification of real low-dimensional Jacobi–Lie bialgebras. In this way, we obtain and classify real two- and three-dimensional Jacobi–Lie bialgebras.

Belavin\u2013Drinfeld solutions of the Yang\u2013Baxter equation: Galois cohomology considerations

We relate the Belavin–Drinfeld cohomologies (twisted and untwisted) that have been introduced in the literature to study certain families of quantum groups and Lie bialgebras over a non algebraically closed field mathbb {K} of characteristic 0 to the standard non-abelian Galois cohomology H^1(mathbb {K}, mathbf{H}) for a suitable algebraic mathbb {K}-group mathbf{H}. The approach presented allows us to establish in full generality certain conjectures that were known to hold for the classical types of the split simple Lie algebras.

LIE SUPER-BIALGEBRAS ON GENERALIZED LOOP SUPER-VIRASORO ALGEBRAS

In this article we consider Lie super-bialgebra structures on the generalized loop super-Virasoro algebra <TEX>${\mathcal{G}}$</TEX>. By proving that the first cohomology group <TEX>$H^1({\mathcal{G}},{\mathcal{G}}{\otimes}{\mathcal{G}})$</TEX> is trivial, we obtain that all such Lie bialgebras are triangular coboundary.

On the T-leaves of some Poisson structures related to products of flag varieties

For a connected abelian Lie group T acting on a Poisson manifold (Y,π) by Poisson isomorphisms, the T-leaves of π in Y are the orbits of the symplectic leaves of π under T, and the leaf stabilizer of a T-leaf is the subspace of the Lie algebra of T that is everywhere tangent to all the symplectic leaves in the T-leaf. In this paper, we first develop a general theory on T-leaves and leaf stabilizers for a class of Poisson structures defined by Lie bialgebra actions and quasitriangular r-matrices. We then apply the general theory to four series of holomorphic Poisson structures on products of flag varieties and related spaces for a complex semi-simple Lie group G. We describe their T-leaf decompositions, where T is a maximal torus of G, in terms of (open) generalized Richardson varieties and generalized double Bruhat cells associated to conjugacy classes of G, and we compute their leaf stabilizers and the dimension of the symplectic leaves in each T-leaf.

M2 to D2 and vice versa by 3-Lie and Lie bialgebra

Using the concept of 3-Lie bialgebra, which has recently been defined in arXiv:1604.04475, we construct Bagger-Lambert-Gustavson (BLG) model for M2-brane on Manin triple of a special 3-Lie bialgebra. Then by using the correspondence and relation between those 3-Lie bialgebra with Lie bialgebra, we reduce this model to an $N=(4,4)$ WZW model (D2-brane), such that, its algebraic structure is a Lie bialgebra with one 2-cocycle. In this manner by using correspondence of 3-Lie bialgebra and Lie bialgebra (for this special 3-Lie algebra) one can construct M2-brane from a D2-brane and vice versa.