Related papers: Euclidean E-models
We establish Ecalle's mould calculus in an abstract Lie-theoretic setting and use it to solve a normalization problem, which covers several formal normal form problems in the theory of dynamical systems. The mould formalism allows us to…
We study the class of Lorentzian symmetric polynomials and Lorentzian symmetric functions, which are defined to be symmetric functions for which every truncation of variables is Lorentzian. Similar to the space of Lorentzian polynomials, we…
Two-dimensional Yang-Mills models in a pseudo-euclidean space are considered from a point of view of a class of nonlinear Klein-Gordon-Fock equations. It is shown that the Nahm reduction does not work, another choice is proposed and…
Following V. Drinfeld and G. Olshansky, we construct Manin triples $(\fg, \fa, \fa^*)$ such that $\fg$ is different from Drinfeld's doubles of $\fa$ for several series of Lie superalgebras $\fa$ which have no even invariant bilinear form…
These are the notes of lectures delivered at Grenoble's summer school on \emph{Arakelov Geo\-me\-try and Diophantine Applications}, in June 2017. They constitute an introduction to the study of Euclidean lattices and of their invariants…
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…
We provide a coarse classification of all 8-dimensional Manin triples, that describe Poisson--Lie T-dualities between 4-dimensional group manifold solutions to supergravity equations. We find several such dualities and one Poisson--Lie…
These notes present an introduction to an analytic version of deformation quantization. The central point is to study algebras of physical observables and their irreducible representations. In classical mechanics one deals with real Poisson…
We show that the partition function of free Maxwell theory on a generic Euclidean four-manifold transforms in a non-trivial way under electric-magnetic duality. The classical part of the partition sum can be mapped onto the genus-one…
We consider two integrable deformations of 2d sigma models on supercosets associated with AdS_n x S^n. The first, the "eta-deformation" (based on the Yang-Baxter sigma model), is a one-parameter generalization of the standard superstring…
Given a Lorentzian spacetime $(M, g)$ and a non-vanishing timelike vector field $u(\lambda)$ with level surfaces $\Sigma$, one can construct on $M$ a Euclidean metric $g_E^{ab} = g^{ab} + 2 u^a u^b$. Motivated by this, we consider a class…
This paper is a survey article on bi-Hamiltonian systems on the dual of the Lie algebra of vector fields on the circle. We investigate the special case where one of the structures is the canonical Lie-Poisson structure and the second one is…
We determine a substantial part of the unitary representation theory of the Drinfeld double of a $q$-deformation of a compact Lie group in terms of the complexification of the compact Lie group. Using this, we show that the dual of every…
We construct two-parameter deformation of an universal enveloping algebra $U(g[u])$ of a polynomial loop algebra $g[u]$, where $g$ is a finite-dimensional complex simple Lie algebra (or superalgebra). This new quantum Hopf algebra called…
These pedagogical lectures given at the Corfu Summer Institute 2018 review two generalised notions of T-duality, non-Abelian T-duality and Poisson-Lie duality, and their applications. We explain how each of these has seen recent application…
We study a one-parameter family of self-adjoint normal operators for the X-ray transform on the closed Euclidean disk ${\mathbb D}$, obtained by considering specific singularly weighted $L^2$ topologies. We first recover the well-known…
Classical equations of motion for three-dimensional sigma-models in curved background are solved by a transformation that follows from the Poisson-Lie T-plurality and transform them into the equations in the flat background. Transformations…
We study the $T\overline{T}$ deformation of two dimensional quantum field theories from a Hamiltonian point of view, focusing on aspects of the theory in Lorentzian signature. Our starting point is a simple rewriting of the spatial integral…
A method to construct integrable deformations of Hamiltonian systems of ODEs endowed with Lie-Poisson symmetries is proposed by considering Poisson-Lie groups as deformations of Lie-Poisson (co)algebras. Moreover, the underlying Lie-Poisson…
In this paper, we establish a complete structural description of flat Lorentzian Lie groups, i.e., Lie groups endowed with a flat left invariant Lorentzian metric, thereby resolving a long-standing open problem in the theory of…