Lie Bialgebra Structures Research Articles

The 4th structure

In this paper we describe all Lie bialgebra structures on the polynomial Lie algebra $\mathbf{g}[u]$, where $\mathbf{g}$ is a simple, finite dimensional, complex Lie algebra. The results are based on an unpublished paper Montaner and Zelmanov. Further, we introduce quasi-rational solutions of the CYBE and describe all quasi-rational $r$-matrices for $\mathbf{sl}(2)$.

The Drinfeld doublegl(n) ⊕tn

The two isomorphic Borel subalgebras of gl(n), realized on upper and lower triangular matrices, allow us to consider the gl(n) \opus t_n algebra as a self-dual Drinfeld double. Compatibility conditions impose the choice of an orthonormal basis in the Cartan subalgebra and fix the basis of gl(n). A natural Lie bialgebra structure on gl(n) is obtained, that offers a new perspective for its standard quantum deformation.

Lie Bialgebras of Generalized Virasoro–like Type

In this paper, Lie bialgebra structures on generalized Virasoro–like algebras are studied. It is proved that all such Lie bialgebras are triangular coboundary.

Lie bialgebras of generalized Witt type

In a paper by Michaelis a class of infinite-dimensional Lie bialgebras containing the Virasoro algebra was presented. This type of Lie bialgebras was classified by Ng and Taft. In this paper, all Lie bialgebra structures on the Lie algebras of generalized Witt type are classified. It is proved that, for any Lie algebra $W$ of generalized Witt type, all Lie bialgebras on $W$ are coboundary triangular Lie bialgebras. As a by-product, it is also proved that the first cohomology group $H^1(W,W \otimes W)$ is trivial.

Classical quasi-trigonometric r−matrices of Cremmer-Gervais type and their quantization

We propose a method of quantization of certain Lie bialgebra structures on the polynomial Lie algebras related to quasi-trigonometric solutions of the classical Yang-Baxter equation. The method is based on so-called affinization of certain seaweed algebras and their quantum analogues.

Classical quasi-trigonometric r-matrices of Cremmer-Gervais type and their quantization

We propose a method of quantization of certain Lie bialgebra structures on the polynomial Lie algebras related to quasi-trigonometric solutions of the classical Yang-Baxter equation. The method is based on so-called affinization of certain seaweed algebras and their quantum analogues.

Cut vertices in commutative graphs

The homology of Kontsevich's commutative graph complex parameterizes finite type invariants of odd dimensional manifolds. This {\it graph homology} is also the twisted homology of Outer Space modulo its boundary, so gives a nice point of contact between geometric group theory and quantum topology. In this paper we give two different proofs (one algebraic, one geometric) that the commutative graph complex is quasi-isomorphic to the quotient complex obtained by modding out by graphs with cut vertices. This quotient complex has the advantage of being smaller and hence more practical for computations. In addition, it supports a Lie bialgebra structure coming from a bracket and cobracket we defined in a previous paper. As an application, we compute the rational homology groups of the commutative graph complex up to rank 7.

Trigonometric Dynamical r-Matrices Over Poisson Lie Base

Let $\g$ be a finite dimensional complex Lie algebra and $\l\subset \g$ a Lie subalgebra equipped with the structure of a factorizable quasitriangular Lie bialgebra. Consider the Lie group $\Exp \l$ with the Semenov-Tjan-Shansky Poisson bracket as a Poisson Lie manifold for the double Lie bialgebra $\D\l$. Let $\Nc_\l(0)\subset \l$ be an open domain parameterizing a neighborhood of the identity in $\Exp \l$ by the exponential map. We present dynamical $r$-matrices with values in $\g\wedge \g$ over the Poisson Lie base manifold $\Nc_\l(0)$.

A Hopf algebra quantizing a necklace Lie algebra canonically associated to a quiver

Ginzburg, Bocklandt, and Le Bruyn defined an infinite-dimensional Lie algebra canonically associated to any quiver. Following suggestions of V. Turaev, P. Etingof, and Ginzburg, we define a cobracket and prove that it defines a Lie bialgebra structure. We then present a Hopf algebra quantizing this Lie bialgebra and prove that it is a Hopf algebra satisfying the Poincaré-Birkhoff-Witt (PBW) property. We also define representations to differential operators quantizing the trace representations of the Lie algebra defined by Ginzburg.

On the Classical Double of Parabolic Subalgebras#

Given a complex simple finite-dimensional Lie algebra g with fixed root system, there exists a so-called classical Drinfeld–Jimbo r-matrix, r. Consider any parabolic subalgebra P S ⊆ g defined by a subset S of the set of simple roots. We prove that the Lie bialgebra structure on g defined by r can be restricted to P S . Moreover, it turns out that the corresponding classical double D(P S ) is isomorphic to g ⊕ Red(P S ), where Red(P S ) denotes the reductive part of P S .

Existence of triangular Lie bialgebra structures II

We characterize finite-dimensional Lie algebras over an arbitrary field of characteristic zero which admit a non-trivial (quasi-) triangular Lie bialgebra structure.

Quantizations of the Witt algebra and of simple Lie algebras in characteristic p

We explicitly quantize the Witt algebra in characteristic 0 equipped with its Lie bialgebra structures discovered by Taft. Then, we study the reduction modulo p of our formulas. This gives p − 1 families of polynomial noncocommutative deformations of a restricted enveloping algebra of a simple Lie algebra in characteristic p (of Cartan type). In particular, this yields new families of noncommutative and noncocommutative Hopf algebras of dimension p p in char p.

Three-dimensional quantum algebras: a Cartan-like point of view

A perturbative quantization procedure for Lie bialgebras is introduced and used to classify all three dimensional complex quantum algebras compatible with a given coproduct. The role of elements of the quantum universal enveloping algebra that, analogously to generators in Lie algebras, have a distinguished type of coproduct is discussed, and the relevance of a symmetrical basis in the universal enveloping algebra stressed. New quantizations of three dimensional solvable algebras, relevant for possible physical applications for their simplicity, are obtained and all already known related results recovered. Our results give a quantization of all existing three dimensional Lie algebras and reproduce, in the classical limit, the most relevant sector of the complete classification for real three dimensional Lie bialgebra structures given by X. Gomez in J. Math. Phys. Vol. 41. (2000) 4939.

On simple real lie bialgebras

The explicit list of all almost factorizable Lie bialgebra structures on real absolutely simple Lie algebras is given. By “absolutely simple” we mean a real Lie algebra whose complexification is simple.

Combinatorial Lie bialgebras of curves on surfaces

Goldman (Invent. Math. 85(2) (1986) 263) and Turaev (Ann. Sci. Ecole Norm. Sup. (4) 24 (6) (1991) 635) found a Lie bialgebra structure on the vector space generated by non-trivial free homotopy classes of curves on a surface. When the surface has non-empty boundary, this vector space has a basis of cyclic reduced words in the generators of the fundamental group and their inverses. We give a combinatorial algorithm to compute this Lie bialgebra on this vector space of cyclic words. Using this presentation, we prove a variant of Goldman's result relating the bracket to disjointness of curve representatives when one of the classes is simple. We exhibit some examples we found by programming the algorithm which answer negatively Turaev's question about the characterization of simple curves in terms of the cobracket. Further computations suggest an alternative characterization of simple curves in terms of the bracket of a curve and its inverse. Turaev's question is still open in genus zero.

Infinitesimal operations on complexes of graphs

In two seminal papers Kontsevich used a construction called graph homology as a bridge between certain infinite dimensional Lie algebras and various topological objects, includ- ing moduli spaces of curves, the group of outer automorphisms of a free group, and invariants of odd dimensional manifolds. In this paper, we show that Kontsevich's graph complexes, which include graph complexes studied earlier by Culler and Vogtmann and by Penner, have a rich algebraic structure. We define a Lie bracket and cobracket on graph complexes, and in fact show that they are Batalin-Vilkovisky algebras, and therefore Gerstenhaber algebras. We also find natural subcomplexes on which the bracket and cobracket are compatible as a Lie bialgebra. Kontsevich's graph complex construction was generalized to the context of operads by Ginzburg and Kapranov, with later generalizations by Getzler-Kapranov and Markl. In (CoV), we show that Kontsevich's results in fact extend to general cyclic operads. For some operads, including the examples associated to moduli space and outer automorphism groups of free groups, the subcomplex on which we have a Lie bi-algebra structure is quasi-isomorphic to the entire con- nected graph complex. In the present paper we show that all of the new algebraic operations canonically vanish when the homology functor is applied, and we expect that the resulting con- straints will be useful in studying the homology of the mapping class group, finite type manifold invariants and the homology of Out(F n).

Symplectic leaves of complex reductive Poisson-Lie groups

All factorizable Lie bialgebra structures on complex reductive Lie algebras were described by Belavin and Drinfeld. We classify the symplectic leaves of all related Poisson-Lie groups. A formula for their dimensions is also proved.

Classification des triples de Manin pour les algèbres de Lie réductives complexes: Avec un appendice de Guillaume Macey

We study real and complex Manin triples for a complex reductive Lie algebra g . First, we generalize results of E. Karolinsky (1996, Math. Phys. Anal. Geom 3 , 545–563; 1999, Preprint math.QA.9901073) on the classification of Lagrangian subalgebras. Then we show that, if g is noncommutative, one can attach to each Manin triple in g another one for a strictly smaller reductive complex Lie subalgebra of g . This gives a powerful tool for induction. Then we classify complex Manin triples in terms of what we call generalized Belavin–Drinfeld data. This generalizes, by other methods, the classification of A. Belavin and V. G. Drinfeld of certain r -matrices, i.e., the solutions of modified triangle equations for constants (cf. A. Belavin and V. G. Drinfeld, “Triangle Equations and Simple Lie Algebras,” Mathematical Physics Reviews, Vol. 4, pp. 93–165, Harwood Academic, Chur, 1984, Theorem 6.1). We get also results for real Manin triples. In passing, we retrieve a result of A. Panov (1999, Preprint math.QA.9904156) which classifies certain Lie bialgebra structures on a real simple Lie algebra.

Existence of solutions of the classical yang- baxter equation for a real lie Algebra

We characterize finite-dimensional Lie algebras over the real numbers for which the classical Yang-Baxter equation has a non-trivial skew-symmetric solution (resp. a non-trivial solution with invariant symmetric part). Equivalently, we obtain a characterization of those finite-dimensional real Lie algebras which admit a non-trivial (quasi-) triangular Lie bialgebra structure.

Quantization of inhomogeneous Lie bialgebras

A self-dual class of Lie bialgebra structures ( g, g ∗) on inhomogeneous Lie algebras g describing kinematical symmetries is considered. In that class, both g and g ∗ split into the semi-direct sums g= h▷ v and g ∗= h ∗◁ v ∗ with abelian ideals of translations v and h ∗ . We build the explicit quantization of the universal enveloping algebra U( g) , including the coproduct, commutation relations among generators, and, in case of coboundary g , the universal R-matrix. This class of Lie bialgebras forms a self-dual category stable under the classical double procedure. The quantization turns out to be a functor to the category of Hopf algebras which commutes with operations of dualization and double.