Related papers: On Lie bialgebras of loops on orientable surface
We show that there exists a Lie a bracket on the cohomology of any type of (bi)algebras over an operad or a PROP, induced by a strongly homotopy Lie structure on the defining cochain complex, such that the associated "quantum" master…
The first cohomology group of a generalized loop Virasoro algebra with coefficients in the tensor product of its adjoint module is shown to be trivial. The result is applied to prove that Lie bialgebra structures on generalized loop…
A double algebra is a linear space $V$ equipped with linear map $V\otimes V\to V\otimes V$. Additional conditions on this map lead to the notions of Lie and associative double algebras. We prove that simple finite-dimensional Lie double…
Classical contact Lie algebras are the fundamental algebraic structures on the manifolds of contact elements of configuration spaces in classical mechanics. In this paper, we determine the structure of the currently largest known category…
We investigate the group gradings on the algebra of upper triangular matrices over an arbitrary field, viewed as a Lie algebra. These results were obtained a few years early by the same authors. We provide streamlined proofs, and present a…
In this note, we give an explicit counterexample to the simple loop conjecture for representations of surface groups into PSL(2,R). Specifically, we show that for any surface with negative Euler characteristic and genus at least 1, there…
Some examples are given of finite dimensional Lie bialgebras whose brackets and cobrackets are determined by pairs of $r$-matrices.
We consider the Lie algebra of all vector fields on a contact manifold as a module over the Lie subalgebra of contact vector fields. This module is split into a direct sum of two submodules: the contact algebra itself and the space of…
We exhibit a natural Lie algebra structure on the graded space of cyclic coinvariants of a symplectic vector space.
We discuss a new approach to computing the standard algebraic operations on homotopy classes of loops in surfaces: the homological intersection number, Goldman's Lie bracket, and the author's Lie cobracket. Our approach uses fillings of the…
We derive the notions of BV unital infinitesimal bialgebra and BV Frobenius algebra from the topology of suitable compactifications of moduli spaces of decorated genus 0 curves. We construct these structures respectively on reduced…
We study zero cycles on rationally connected varieties defined over characteristic zero Laurent fields with algebraically closed residue fields. We show that the degree map induces an isomorphism for rationally connected threefolds defined…
The aim here is to sketch the development of ideas related to brackets and similar concepts: Some purely group theoretical combinatorics due to Ph. Hall led to a proof of the Jacobi identity for the Whitehead product in homotopy theory.…
We describe the general non-associative version of Lie theory that relates unital formal multiplications (formal loops), Sabinin algebras and non-associative bialgebras. Starting with a formal multiplication we construct a non-associative…
We give a natural monomorphism from the necklace Lie coalgebra, defined for any quiver, to Connes and Kreimer's Lie coalgebra of trees, and extend this to a map from a certain quiver-theoretic Hopf algebra to Connes and Kreimer's…
All coboundary Lie bialgebras and their corresponding Poisson--Lie structures are constructed for the oscillator algebra generated by $\{\aa,\ap,\am,\bb\}$. Quantum oscillator algebras are derived from these bialgebras by using the…
Lie bialgebra structures on the extended affine Lie algebra $\widetilde{sl_2(\mathbb{C}_q)}$ are investigated. In particular, all Lie bialgebra structures on $\widetilde{sl_2(\mathbb{C}_q)}$ are shown to be triangular coboundary. This…
This paper is devoted to a classification of topological Lie bialgebra structures on the Lie algebra $\mathfrak{g}[\![x]\!]$, where $ \mathfrak{g} $ is a finite-dimensional simple Lie algebra over an algebraically closed field $ F $ of…
We consider algebras over a field $k$ of characteristic zero. The article is concerned with the isomorphism of graded vectorspaces \[ H(\gl(A))\iso\wedge (HC(A)[-1]) \] between the Lie algebra homology of matrices and the free graded…
Lie bialgebra structures on $e(2)$ are classified. For two Lie bialgebra structures which are not coboundaries (i.e. which are not determined by a classical $r$-matrix) we solve the cocycle condition, find the Lie-Poisson brackets and…