Affine Lie Group Research Articles

On k-polycosymplectic Marsden–Weinstein reductions

We review and slightly improve the known k-polysymplectic Marsden–Weinstein reduction theory by removing some technical conditions on k-polysymplectic momentum maps by developing a theory of affine Lie group actions for k-polysymplectic momentum maps, removing the necessity of their co-adjoint equivariance. Then, we focus on the analysis of a particular case of k-polysymplectic manifolds, the so-called fibred ones, and we study their k-polysymplectic Marsden–Weinstein reductions. Previous results allow us to devise a k-polycosymplectic Marsden–Weinstein reduction theory, which represents one of our main results. Our findings are applied to study coupled vibrating strings and, more generally, k-polycosymplectic Hamiltonian systems with field symmetries. We show that k-polycosymplectic geometry can be understood as a particular type of k-polysymplectic geometry. Finally, a k-cosymplectic to ℓ-cosymplectic geometric reduction theory is presented, which reduces, geometrically, the space-time variables in a k-cosymplectic framework. An application of this latter result to a vibrating membrane with symmetries is given.

Globally Optimal Fetoscopic Mosaicking Based on Pose Graph Optimisation With Affine Constraints

Fetoscopic laser ablation surgery could be guided using a high-quality panorama of the operating site, representing a map of the placental vasculature. This can be achieved during the initial inspection phase of the procedure using image mosaicking techniques. Due to the lack of camera calibration in the operating room, it has been mostly modelled as an affine registration problem. While previous work mostly focuses on image feature extraction for visual odometry, the challenges related to large-scale reconstruction (re-localisation, loop closure, drift correction) remain largely unaddressed in this context. This letter proposes using pose graph optimisation to produce globally optimal image mosaics of placental vessels. Our approach follows the SLAM framework with a front-end for visual odometry and a back-end for long-term refinement. Our front-end uses a recent state-of-the-art odometry approach based on vessel segmentation, which is then managed by a key-frame structure and the bag-of-words (BoW) scheme to retrieve loop closures. The back-end, which is our key contribution, models odometry and loop closure constraints as a pose graph with affine warpings between states. This problem in the special Euclidean space cannot be solved by existing pose graph algorithms and available libraries such as G2O. We model states on affine Lie group with local linearisation in its Lie algebra. The cost function is established using Mahalanobis distance with the vectorisation of the Lie algebra. Finally, an iterative optimisation algorithm is adopted to minimise the cost function. The proposed pose graph optimisation is first validated on simulation data with a synthetic trajectory that has different levels of noise and different numbers of loop closures. Then the whole system is validated using real fetoscopic data that has three sequences with different numbers of frames and loop closures. Experimental results validate the advantage of the proposed method compared with baselines.

Duflo-Moore Operator for The Square-Integrable Representation of 2-Dimensional Affine Lie Group

In this paper, we study the quasi-regular and the irreducible unitary representation of affine Lie group of dimension two. First, we prove a sharpening of Fuhr’s work of Fourier transform of quasi-regular representation of . The second, in such the representation of affine Lie group is square-integrable then we compute its Duflo-Moore operator instead of using Fourier transform as in F hr’s work.

A topological conjugacy of invariant flows on some class of Lie groups

The aim of this paper is to classify invariant flows on Lie group $G$ whose Lie algebra $\mathfrak{g}$ is associative or semisimple. Specifically, we present this classification from the hyperbolicity of the lift flows on $G \times \mathfrak{g}$. Then we apply this construction to some special cases as ${\rm Gl}(2,{\Bbb R})$ and affine Lie group.

Holomorphic Cartan geometries on complex tori

In [6], it was asked whether all flat holomorphic Cartan geometries (G,H) on a complex torus are translation invariant. We answer this affirmatively under the assumption that the complex Lie group G is affine. More precisely, we show that every holomorphic Cartan geometry of type (G,H), with G a complex affine Lie group, on any complex torus is translation invariant.

Flat affine or projective geometries on Lie groups

This paper deals essentially with affine or projective transformations of Lie groups endowed with a flat left invariant affine or projective structure. These groups are called flat affine or flat projective Lie groups. We give necessary and sufficient conditions for the existence of flat left invariant projective structures on Lie groups. We also determine Lie groups admitting flat bi-invariant affine or projective structures. These groups could play an essential role in the study of homogeneous spaces M=G/H having a flat affine or flat projective structures invariant under the natural action of G on M. A. Medina asked several years ago if the group of affine transformations of a flat affine Lie group is a flat projective Lie group. In this work we provide a partial positive answer to this question.

Trimmed strategy for affine registration of point sets

We propose a trimmed strategy for affine registration of point sets using the Lie group parameterization. All affine transformations form an affine Lie group, thus finding an optimal transformation in registration is reduced to finding an optimal element in the affine group. Given two point sets (with outliers) and an initial element in the transformation group, we seek the optimal group element iteratively by minimizing an energy functional. This is conducted by sequentially finding the closest correspondence of two point sets, estimating the overlap rate of two sets, and finding the optimal affine transformation via the exponential map of the affine group. This method improves the trimmed iterative closest point algorithm (TrICP) in two aspects: (1) We use the Lie group parameterization to implement TrICP. (2) We also extend TrICP to the case of affine transformations. The performance of the proposed algorithm is demonstrated by using the LiDAR data acquired in the Mount St. Helens area. Both visual inspections and evaluation index (root mean trimmed squared distance) indicate that our algorithm performs consistently better than TrICP and other related algorithms, especially in the presence of outliers and missing points.

Structure of symplectic Lie groups and momentum map

We describe the structure of the Lie groups endowed with a left-invariant symplectic form, called symplectic Lie groups, in terms of semi-direct products of Lie groups, symplectic reduction and principal bundles with affine fiber. This description is particularly nice if the group is Hamiltonian, that is, if the left canonical action of the group on itself is Hamiltonian. The principal tool used for our description is a canonical affine structure associated with the symplectic form. We also characterize the Hamiltonian symplectic Lie groups among the connected symplectic Lie groups. We specialize our principal results to the cases of simply connected Hamiltonian symplectic nilpotent Lie groups or Frobenius symplectic Lie groups. Finally we pursue the study of the classical affine Lie group as a symplectic Lie group.

Deformable Object Tracking Based on Lie Algebra

Dynamic deformation of object is a distinct problem in image-based tracking.A novel deformable object tracking method based on Lie algebra is presented.This method uses Gabor feature as target token,models deformation using affine Lie group,and optimizes parameters directly on manifold,which can be solved by exponential mapping between Lie group and its Lie algebra.We physically illustrate the essential of the geometric transformation in the tracking,then analyze the advantage of our method and give a direct proof of local quadratic convergence of the algorithm.The experimental results demonstrate that the Lie algebra based method makes significant improvements in convergence speed,tracking stability and accuracy in comparison to Euclidean based algorithms.

Covariance Tracking via Geometric Particle Filtering

Region covariance descriptor recently proposed has been approved robust and elegant to describe a region of interest, which has been applied to visual tracking. We develop a geometric method for visual tracking, in which region covariance is used to model objects appearance; then tracking is led by implementing the particle filter with the constraint that the system state lies in a low dimensional manifold: affine Lie group. The sequential Bayesian updating consists of drawing state samples while moving on the manifold geodesics; the region covariance is updated using a novel approach in a Riemannian space. Our main contribution is developing a general particle filtering-based racking algorithm that explicitly take the geometry of affine Lie groups into consideration in deriving the state equation on Lie groups. Theoretic analysis and experimental evaluations demonstrate the promise and effectiveness of the proposed tracking method.

On K(E<sub>9</sub>)

We study the maximal compact subgroup K(E9) of the affine Lie group E9(9) and its on-shell realization as an R symmetry of maximal N = 16 supergravity in two dimensions. We first give a rigorous definition of the group K(E9), which lives on the double cover of the spectral parameter plane, and show that the infinitesi mal action of K(E9) on the chiral components of the bosons and the fermions is determined in terms of an expansion of the Lie algebra of K(E9) about the two branch points of this cover; this implies in particular that the fermions of N = 16 supergravity transform in a spinor representation of K(E9). The fermionic equations of motion can be fitted into the lowe st components of a single K(E9) covariant ‘Dirac equation’, with the linear system of N = 16 supergravity as the gauge connection. These results suggest the existence of an ‘off-shell’ realization of K(E9) in terms of an infinite component spinor representation. We c onclude with some coments on ‘generalized holonomies’ of M theory.

Affine lie group approach to a derivative nonlinear Schrödinger equation and its similarity reduction

A generalization of the Drinfel'd-Sokolov hierarchies is proposed using the framework of Sato-Wilson dressing method. In the case of the A1(1)-type affine Lie group, we obtain the hierarchy that includes the derivative nonlinear Schrodinger equation. We give two types of affine Weyl group symmetry of the hierarchy based on the Gauss decomposition. The fourth Painleve equation and its Weyl group symmetry are obtained as a similarity reduction. We also discuss the connection between the reduced system and monodromy-preserving deformations.

Groupes de Lie pseudo-riemanniens plats

The determination of affine Lie groups (i.e., which carry a left-invariant affine structure) is an open problem. In this work we begin the study of Lie groups with a left-invariant, flat pseudo-Riemannian metric (flat pseudo-Riemannian groups). We show that in such groups the left-invariant affine structure defined by the Levi-Civita connection is geodesically complete if and only if the group is unimodular. We also show that the cotangent manifold of an affine Lie group is endowed with an affine Lie group structure and a left-invariant, flat hyperbolic metric. We describe a double extension process which allows us to construct all nilpotent, flat Lorentzian groups. We give examples and prove that the only Heisenberg group which carries a left invariant, flat pseudo-Riemannian metric is the three dimensional one.

Uniqueness of left invariant symplectic structures on the affine Lie group

We show the uniqueness of left invariant symplectic structures on the affine Lie group A ( n , R ) A(n,\mathbf {R}) under the adjoint action of A ( n , R ) A(n,\mathbf {R}) , by giving an explicit formula of the Pfaffian of the skew symmetric matrix naturally associated with A ( n , R ) A(n,\mathbf {R}) , and also by giving an unexpected identity on it which relates two left invariant symplectic structures. As an application of this result, we classify maximum rank left invariant Poisson structures on the simple Lie groups S L ( n , R ) SL(n,\mathbf {R}) and S L ( n , C ) SL(n, \mathbf {C}) . This result is a generalization of Stolin’s classification of constant solutions of the classical Yang-Baxter equation for s l ( 2 , C ) \mathfrak {sl}(2, \mathbf {C}) and s l ( 3 , C ) \mathfrak {sl}(3,\mathbf {C}) .

THREE-POINT FUNCTIONS IN CONFORMAL FIELD THEORY WITH AFFINE LIE GROUP SYMMETRY

In this paper we develop a general method for constructing three-point functions in conformal field theory with affine Lie group symmetry, continuing our recent work on two-point functions. The results are provided in terms of triangular coordinates used in a wave function description of vectors in highest weight modules. In this framework, complicated couplings translate into ordinary products of certain elementary polynomials. The discussions pertain to all simple Lie groups and arbitrary integrable representation. An interesting by-product is a general procedure for computing tensor product coefficients, essentially by counting integer solutions to certain inequalities. As an illustration of the construction, we consider in great detail the three cases SL(3), SL(4) and SO(5).

THE WZW MODEL AS A DYNAMICAL SYSTEM ON AFFINE LIE GROUPS

Working directly on affine Lie groups, we construct several new formulations of the WZW model. In one formulation WZW is expressed as a one-dimensional mechanical system whose variables are coordinates on the affine Lie group. When written in terms of the affine group element, this formulation exhibits a two-dimensional WZW term. In another formulation WZW is written as a two-dimensional field theory, with a three-dimensional WZW term, whose fields are coordinates on the affine group. On the basis of these equivalent formulations, we develop a translation dictionary in which the new formulations on the affine Lie group are understood as mode formulations of the conventional WZW formulation on the Lie group. Using this dictionary, we also express WZW as a three-dimensional field theory on the Lie group with a four-dimensional WZW term.

Spherical functions on affine Lie groups

We describe vector valued conjugacy equivariant functions on a group K in two cases -- K is a compact simple Lie group, and K is an affine Lie group. We construct such functions as weighted traces of certain intertwining operators between representations of K. For a compact group $K$, Peter-Weyl theorem implies that all equivariant functions can be written as linear combinations of such traces. Next, we compute the radial parts of the Laplace operators of $K$ acting on conjugacy equivariant functions and obtain a comple- tely integrable quantum system with matrix coefficients, which in a special case coincides with the trigonometric Calogero-Sutherland-Moser multi-particle system. In the affine Lie group case, we prove that the space of equivariant functions having a fixed homogeneity degree with respect to the action of the center of the group is finite-dimensional and spanned by weighted traces of intertwining operators. This space coincides with the space of Wess-Zumino-Witten conformal blocks on an elliptic curve. We compute the radial part of the second order Laplace operator on the affine Lie group acting on equivariant functions, and find that it is a certain parabolic partial differential operator, which degenerates to the elliptic Calogero-Sutherland-Moser hamiltonian as the central charge tends to minus the dual Coxeter number (the critical level). Quantum integrals of this hamiltonian are obtained as radial part of the higher Sugawara operators which are central at the critical level.

MAXWELL-CHERN-SIMONS ELECTRODYNAMICS ON A DISK

The Maxwell-Chern-Simons (MCS) Lagrangian is the Maxwell Lagrangian augmented by the Chern-Simons term. In this paper, we study the MCS and Maxwell Lagrangians on a disk D. They are of interest for the quantum Hall effect, and also when the disk and its exterior are composed of different media. We show that quantization is not unique, but depends on a nonnegative parameter λ. 1/λ is the penetration depth of the fields described by these Lagrangians into the medium in the exterior of D. For λ=0, there are edge observables and edge states localized at the boundary ∂D for the MCS system. They describe the affine Lie group [Formula: see text]. Their excitations carry zero energy, signifying an infinite degeneracy of all states of the theory. There is also an additional infinity of single particle excitations of exactly the same energy proportional to |k|, k being the strength of the Chern-Simons term. The MCS theory for λ=0 has the huge symmetry group [Formula: see text]. In the Maxwell theory, the last-mentioned excitations are absent while the edge observables, which exist for λ=0, commute. Also, these excitations are described by states which are not localized at ∂D and are characterized by a continuous and infinitely degenerate spectrum. All these degeneracies are lifted and edge observables and their states cease to exist for λ>0. The novel excitations discovered in this paper should be accessible to observations. We will discuss issues related to observations, as also the generalization of the present considerations to vortices, domain walls and monopoles, in a paper under preparation.

Semigroups of the basic affine lie group and their automorphisms

UDC 519.46 In recent years the theory of semigroups of Lie groups has been intensively studied. The problem of calculating the automorphisms of connected subsemigroups generates particular interest. For Abelian Lie groups this problem has been completely solved by A.D. Aleksandrov [1]. However, for noncom- mutative Lie groups, no notable results have been obtained yet (the basic affine Lie group [2, 3] and 3-dimensional Lie groups [4] being exceptions). In this article we find the automorphisms of connected subsemigroups with quasicontingency of the form L x K (see Definition 1 below) for the case of the basic anne Lie group. This completely solves the problem of computing the automorphisms of subsemigroups of the basic affine Lie group. The basic anne Lie group is a connected simply connected real Lie group whose Lie algebra is given, in a suitable basis X1,..., Xn, by nontrivial commutation relations

Representations of affine Lie algebras and soliton equations

A few years ago the 'hidden symmetries’ of the soliton equations had been identified as affine Lie groups, also known as loop groups. The first extensive use of the representation theory of affine Lie algebras for the soliton equations have been developed in a series of works by mathematicians of the Kyoto school. We will review some of their results and develop them further on the basis of the representation theory. Thus an orbit of the simplest affine Lie group SL(2, C)^ in the fundamental representation V will provide the solutions of the Korteweg-de Vries equation, and similarly the solutions of the sine-Gordon equation will come from an orbit of the group (SL(2, C) x SL(2, C)) ^ in V x V*.