A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
Extends distribution algebra concept to Lie groupoids.
New manifolds help understand group actions on complex chains.
The Morse complex is shown to be an infinite functor.
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.
Lie algebras with -action are studied.
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.
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 …
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.
Proof of Goldman-Turaev Lie bialgebra structure using Knizhnik-Zamolodchikov connection.
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. …
New structure found in loops on quasi-surfaces.
We survey a geometric approach to the Johnson homomorphisms using the Goldman-Turaev Lie bialgebra.
Solves Kashiwara-Vergne problems for higher genus surfaces, linking to Lie bialgebra formality.
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…
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…
The paper studies T-leaves and stabilizers in Poisson structures on flag varieties.
Study formalities on closed surfaces using connections.
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.
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 -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 , 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.
Paper connects Lie bialgebra and KV problem on genus zero 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 loops on an orientable surface. Chas (Combinatorial Lie bialgebras of curves on surfaces, Topology 43 (2004) 543), b…
New proof connects Kashiwara-Vergne equations to Goldman-Turaev Lie bialgebra.
Local Poisson groupoids over mixed product Poisson structures defined and applied.
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 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…
Paper surveys Johnson homomorphisms and related tools.
Abstract: Extends Drinfeld correspondence to infinite-dimensional Lie groups.
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.
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…
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.
The data of a "2D field theory with a closed string compactification" is an equivariant chain level action of a cell decomposition of the union of all moduli spaces of punctured Riemann surfaces with each component compactified as a pseudomanifold with boundary. The axioms on the data are contained in the following ass…
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.
The word `double' was used by Ehresmann to mean `an object X in the category of all X'. Double categories, double groupoids and double vector bundles are instances, but the notion of Lie algebroid cannot readily be doubled in the Ehresmann sense, since a Lie algebroid bracket cannot be defined diagrammatically. In this…
This paper explores the relationship between Leibniz algebras and Nijenhuis operators.