arXiv research

We study generalized Lie bialgebroids over a single point, that is, generalized Lie bialgebras. Lie bialgebras are examples of generalized Lie bialgebras. Moreover, we prove that the last ones can be considered as the infinitesimal invariants of Lie groups endowed with a certain type of Jacobi structures. We also propo…

The paper connects Lie bialgebras, Rota-Baxter Lie algebras, and their properties.

Study exact Lie bialgebras from flat Lie groups, classifying them.

New algebraic tools solve Poisson and Lie bialgebra problems.

Introduces NL bialgebras combining Lie and Nijenhuis structures.

Decomposes Goldman-Turaev Lie bialgebra via cutting a surface.

The paper defines projective structures for Lie bialgebras and Poisson-Lie groups.

Extends distribution algebra concept to Lie groupoids.

Develops a bialgebra theory for post-Lie algebras using geometric interpretations and bilinear forms.

Introduces Poisson double algebroids and their relation to Lie 2-bialgebras.

We introduce and study some mixed product Poisson structures on product manifolds associated to Poisson Lie groups and Lie bialgebras. For quasitriangular Lie bialgebras, our construction is equivalent to that of fusion products of quasi-Poisson G-manifolds introduced by Alekseev, Kosmann- Schwarzbach, and Meinrenken. …

We survey a geometric approach to the Johnson homomorphisms using the Goldman-Turaev Lie bialgebra.

Poisson sigma models represent an interesting use of Poisson manifolds for the construction of a classical field theory. Their definition in the language of fibre bundles is shown and the corresponding field equations are derived using a coordinate independent variational principle. The elegant form of equations of mot…

The Morse complex is shown to be an infinite functor.

New combinatorial approach to Goldman-Turaev Lie bialgebra using cyclic word partitions.

New bialgebra structures for relative Poisson algebras are introduced.

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 He…

Study formalities on closed surfaces using connections.

We discuss natural operations on loops in a quasi-surface and show that these operations define a structure of a quasi-Lie bialgebra in the module generated by the set of free homotopy classes of non-contractible loops.

Classifies Lie bialgebras using Darboux families.

We review the extent to which the universal enveloping algebra of a Lie-Rinehart algebra resembles a Hopf algebra, and refer to this structure as a Rinehart bialgebra. We then prove a Cartier-Milnor-Moore type theorem for such Rinehart bialgebras.

Functorial approach connects operads to Lie bialgebras.

New manifolds help understand group actions on complex chains.

Study Lie bialgebra structures on flat metric Lie algebras, leading to explicit Poisson-Lie groups.

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-L…

We define \textit{graded manifolds} as a version of supermanifolds endowed with an additional $\mathbb Z$-grading in the structure sheaf, called \textit{weight} (not linked with parity). Examples are ordinary supermanifolds, vector bundles over supermanifolds, double vector bundles, iterated constructions like $TTM$, e…

By introducing a refinement of the Goldman-Turaev Lie bialgebra, we interpret the divergence cocycle in the Kashiwara-Vergne problem and the Enomoto-Satoh obstructions for the surjectivity of the Johnson homomorphisms as some part of a regular homotopy version of the Turaev cobracket.

Using technique of wheeled props we establish a correspondence between the homotopy theory of unimodular Lie 1-bialgebras and the famous Batalin-Vilkovisky formalism. Solutions of the so called quantum master equation satisfying certain boundary conditions are proven to be in 1-1 correspondence with representations of …

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 loops on an orientable surface. Chas (Combinatorial Lie bialgebras of curves on surfaces, Topology 43 (2004) 543), b…

Goldman and Turaev 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 combinator…

We define a family ${\rm KV}^{(g,n)}$ of Kashiwara-Vergne problems associated with compact connected oriented 2-manifolds of genus $g$ with $n+1$ boundary components. The problem ${\rm KV}^{(0,3)}$ is the classical Kashiwara-Vergne problem from Lie theory. We show the existence of solutions of ${\rm KV}^{(g,n)}$ for ar…

We define an abstract notion of double Lie algebroid, which includes as particular cases: (1) the double Lie algebroid of a double Lie groupoid in the sense of the author, such as the iterated tangent bundle of an ordinary manifold, and various iterated tangent/cotangent constructions in symplectic and Poisson geometry…

Abstract: Extends Drinfeld correspondence to infinite-dimensional Lie groups.

For an oriented 2-dimensional manifold $Σ$ of genus $g$ with $n$ boundary components the space $\mathbb{C}π_1(Σ)/[\mathbb{C}π_1(Σ), \mathbb{C}π_1(Σ)]$ carries the Goldman-Turaev Lie bialgebra structure defined in terms of intersections and self-intersections of curves. Its associated graded (under the natural filtratio…

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.

We construct a Poisson isomorphism between the formal Poisson manifolds g^* and G^*, where g is a finite dimensional quasitriangular Lie bialgebra. Here g^* is equipped with its Lie-Poisson (or Kostant-Kirillov-Souriau) structure, and G^* with its Poisson-Lie structure. We also quantize Poisson-Lie dynamical r-matrices…

We introduce a simplified version of Drinfeld's equations that still solves Kashiwara-Vergne equations.

We remark some basic facts on homological aspects of involutive Lie bialgebras and their involutive bimodules, and present some problems on surface topology related to these facts.

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 $\mathbb{K}$ a field of characteristic zero. The Goldman-Turaev Lie…

This paper explores the relationship between Leibniz algebras and Nijenhuis operators.

Introduces Lie-Nijenhuis bialgebroids for Poisson-Nijenhuis groupoids.

The notions of \emph{Poisson Lie group} and \emph{Poisson homogeneous space} are extended to the Dirac category. The theorem of Drinfel$'$d (\cite{Drinfeld93}) on the one-to-one correspondence between Poisson homogeneous spaces of a Poisson Lie group and a special class of Lagrangian subalgebras of the Lie bialgebra as…

This work is devoted to the study of a class of Poisson-Lie groups endowed with left invariant metrics. The triples $(G,π,<,>)$ are considered, where $G$ is a simply connected Lie group, ?$π$ is a multiplicative Poisson tensor and $<,>$ is a left invariant riemannian metric such that Hawkins conditions are satisfied. H…

Courant algebroids are structures which include as examples the doubles of Lie bialgebras and the direct sum of tangent and cotangent bundles with the bracket introduced by T. Courant for the study of Dirac structures. Within the category of Courant algebroids one can construct the doubles of Lie bialgebroids, the infi…

A Lie version of Turaev's $\overline{G}$-Frobenius algebras from 2-dimensional homotopy quantum field theory is proposed. The foundation for this Lie version is a structure we call a \textit{$\frak{g}$-quasi-Frobenius Lie algebra} for $\frak{g}$ a finite dimensional Lie algebra. The latter consists of a quasi-Frobenius…

The notion of a generalized Lie bialgebroid (a generalization of the notion of a Lie bialgebroid) is introduced in such a way that a Jacobi manifold has associated a canonical generalized Lie bialgebroid. As a kind of converse, we prove that a Jacobi structure can be defined on the base space of a generalized Lie bialg…

Study cohomology of Lie algebroids over algebraic spaces using derived functors and Čech cohomology.

For a compact oriented surface $Σ$ of genus $g$ with $n+1$ boundary components, the space $\mathfrak{g}(Σ)$ spanned by free homotopy classes of loops in $Σ$ carries the structure of a Lie bialgebra equipped with a natural decreasing filtration, whose structure morphisms are called the Goldman bracket and the (framed) T…