arXiv research

New structure found in loops on quasi-surfaces.

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…

In this dissertation we study Courant algebroids, objects that first appeared in the work of T. Courant on Dirac structures; they were later studied by Liu, Weinstein and Xu who used Courant algebroids to generalize the notion of the Drinfeld double to Lie bialgebroids. As a first step towards understanding the complic…

This work analyses types of group actions on families of $t$-dependent vector fields of a particular class, the hereby called quasi-Lie families. We devise methods to obtain the defined here quasi-Lie invariants, namely a kind of functions constant along the orbits of the above-mentioned actions. Our techniques lead to…

Deforms Poisson quasi-Nijenhuis manifolds using closed 2-forms.

The theory of quasi-Lie systems, i.e. systems of first order ordinary differential equations which can be related via a generalised flow to Lie systems, is extended to systems of partial differential equations and its applications to obtaining $t$-dependent superposition rules and integrability conditions are analysed.…

We propose a definition of Poisson quasi-Nijenhuis Lie algebroids as a natural generalization of Poisson quasi-Nijenhuis manifolds and show that any such Lie algebroid has an associated quasi-Lie bialgebroid. Therefore, also an associated Courant algebroid is obtained. We introduce the notion of a morphism of quasi-Lie…

Introduces NL bialgebras combining Lie and Nijenhuis structures.

New bialgebra structures for relative Poisson algebras are introduced.

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…

New algebraic tools solve Poisson and Lie bialgebra problems.

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

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

A quasi-Lie scheme is a geometric structure that provides t-dependent changes of variables transforming members of an associated family of systems of first-order differential equations into members of the same family. In this note we introduce two quasi-Lie schemes for studying second-order Gambier equations in a geome…

Decomposes Goldman-Turaev Lie bialgebra via cutting a surface.

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

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

Extends distribution algebra concept to Lie groupoids.

The Morse complex is shown to be an infinite functor.

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 introduce the notion of pseudo-Poisson Nijenhuis manifolds. These manifolds are generalizations of Poisson Nijenhuis manifolds by Magri and Morosi \cite{MM}. We show that any pseudo-Poisson Nijenhuis manifold has an associated quasi-Lie bialgebroid as in the case of Poisson quasi-Nijenhuis manifolds by Sti$\acute{\m…

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

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

Study formalities on closed surfaces using connections.

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.

To each ribbon graph we assign a so-called L-space, which is a Lagrangian subspace in an even-dimensional vector space with the standard symplectic form. This invariant generalizes the notion of the intersection matrix of a chord diagram. Moreover, the actions of Morse perestroikas (or taking a partial dual) and Vassil…

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…

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

Functorial approach connects operads to Lie bialgebras.

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.

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…

New manifolds help understand group actions on complex chains.

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…

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 …

Classifies Lie bialgebras using Darboux families.

We introduce the notion of Hamiltonian spaces for Manin pairs over manifolds, using the so-called generalized Dirac structures. As an example, we describe Hamiltonian spaces of a quasi-Lie bialgebroid using this general framework. We also discuss reduction of Hamiltonian spaces of this general type.

In these lecture notes, we give a quick account of the theory of Poisson groupoids and Lie bialgebroids. In particular, we discuss the universal lifting theorem and its applications including integration of quasi-Lie bialgebroids, integration of Poisson Nijenhuis structures and Alekseev and Kosmann-Schwarzbach's theory…

We study quasi-Jacobi and Jacobi-quasi bialgebroids and their relationships with twisted Jacobi and quasi Jacobi manifolds. We show that we can construct quasi-Lie bialgebroids from quasi-Jacobi bialgebroids, and conversely, and also that the structures induced on their base manifolds are related via a quasi Poissoniza…

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…

In this paper, we introduce generalized almost para-contact manifolds and obtain normality conditions in terms of classical tensor fields. We show that such manifolds naturally carry certain Lie bialgebroid/quasi-Lie algebroid structures on them and we relate this new generalized manifolds with classical almost para-co…

New Lie 2-algebra structure for multiplicative forms on quasi-Poisson groupoids.

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

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…

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

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…

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.