scholarly journals Geometric Models for Lie–Hamilton Systems on ℝ2

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1053
Author(s):  
Julia Lange ◽  
Javier de Lucas
Keyword(s):  
Lie Algebras ◽  
Poisson Structure ◽  
Quotient Space ◽  
Geometric Description ◽  
Natural Manner ◽  
Geometric Models ◽  
Hamilton System ◽  
Higher Dimensional ◽  
Symplectic Leaves ◽  
Lie Systems

This paper provides a geometric description for Lie–Hamilton systems on R 2 with locally transitive Vessiot–Guldberg Lie algebras through two types of geometric models. The first one is the restriction of a class of Lie–Hamilton systems on the dual of a Lie algebra to even-dimensional symplectic leaves relative to the Kirillov-Kostant-Souriau bracket. The second is a projection onto a quotient space of an automorphic Lie–Hamilton system relative to a naturally defined Poisson structure or, more generally, an automorphic Lie system with a compatible bivector field. These models give a natural framework for the analysis of Lie–Hamilton systems on R 2 while retrieving known results in a natural manner. Our methods may be extended to study Lie–Hamilton systems on higher-dimensional manifolds and provide new approaches to Lie systems admitting compatible geometric structures.

scholarly journals Quiver Hecke Algebras and 2-Lie Algebras

2012 ◽  
Vol 19 (02) ◽  
pp. 359-410 ◽  
Author(s):  
Raphaël Rouquier
Keyword(s):  
Representation Theory ◽  
Lie Algebras ◽  
Hecke Algebras ◽  
Geometric Description ◽  
Monoidal Categories ◽  
Quiver Varieties ◽  
Basic Properties

We provide an introduction to the 2-representation theory of Kac-Moody algebras, starting with basic properties of nil Hecke algebras and quiver Hecke algebras, and continuing with the resulting monoidal categories, which have a geometric description via quiver varieties, in certain cases. We present basic properties of 2-representations and describe simple 2-representations, via cyclotomic quiver Hecke algebras, and through microlocalized quiver varieties.

scholarly journals DEFORMATIONS OF FOUR-DIMENSIONAL LIE ALGEBRAS

2007 ◽  
Vol 09 (01) ◽  
pp. 41-79 ◽  
Author(s):  
ALICE FIALOWSKI ◽  
MICHAEL PENKAVA
Keyword(s):  
Lie Algebras ◽  
Moduli Space ◽  
Geometric Description ◽  
Exceptional Points ◽  
Versal Deformations

We study the moduli space of four-dimensional ordinary Lie algebras, and their versal deformations. Their classification is well known; our focus in this paper is on the deformations, which yield a picture of how the moduli space is assembled. Surprisingly, we get a nice geometric description of this moduli space essentially as an orbifold, with just a few exceptional points.

scholarly journals Affine symmetric group

2021 ◽  
Vol 4 (1) ◽  
pp. 3
Author(s):  
Joel Brewster Lewis
Keyword(s):  
Representation Theory ◽  
Symmetric Group ◽  
Mathematical Structure ◽  
Number Line ◽  
Geometric Description ◽  
Generators And Relations ◽  
Higher Dimensional ◽  
Finite Set ◽  
Finite Symmetric Group

The affine symmetric group is a mathematical structure that describes the symmetries of the number line and the regular triangular tesselation of the plane, as well as related higher dimensional objects. It is an infinite extension of the symmetric group, which consists of all permutations (rearrangements) of a finite set. In additition to its geometric description, the affine symmetric group may be defined as the collection of permutations of the integers (..., −2, −1, 0, 1, 2, ...) that are periodic in a certain sense, or in purely algebraic terms as a group with certain generators and relations. These different definitions allow for the extension of many important properties of the finite symmetric group to the infinite setting, and are studied as part of the fields of combinatorics and representation theory.

Deformation of the Poisson Structure Related to Algebroid Bracket of Differential Forms and Application to Real Low Dimentional Lie Algebras

2019 ◽  
Vol 20 ◽  
pp. 122-130
Author(s):  
Alina Dobrogowska ◽  
◽  
Grzegorz Jakimowicz ◽  
Marzena Szajewska ◽  
Karolina Wojciechowicz
Keyword(s):  
Lie Algebras ◽  
Poisson Structure ◽  
Differential Forms

Assembly/Disassembly Simulation Early During a Design Process

2001 ◽  
Author(s):  
Jean-Claude Léon ◽  
Nicolas Rejneri ◽  
Gilles Debarbouillé
Keyword(s):  
Design Process ◽  
Algorithm Complexity ◽  
Geometric Description ◽  
Geometric Models ◽  
Number Of Components ◽  
Simulation Process ◽  
Technological Data ◽  
Selection Of ◽  
Sequencing Process

Abstract In this article, a method is presented which can automatically generate all feasible disassembly sequences of a product both from geometric and technological points of views. The components of the assembly are modelled with accurate B-Rep models and the mobility between them is modelled as effective translations, rotations or helical movements. In order to reduce the algorithm complexity inherited from such models, i.e. the number of geometrically feasible sequences and the complexity of the movements, this approach is based on technological data associated to the geometric description of components. The geometric and technological data are taken into account to create associations of components, i.e. the number of components is reduced, and to define criteria for the evaluation and selection of disassembly sequences in order to reduce the number of sequences effectively built and hence the algorithm complexity. Then, user criteria are incorporated to select the best component to mount/dismount for each operation and produce assembly/disassembly sequences. A path trajectory planner is coupled with the sequencing process to generate paths in 3D that avoid obstacles. This approach incorporates geometric models of components as well as technological data, which may be available early during a design process. All these data are not strictly required for the simulation process. Therefore, the simulation can be continuously enriched during the design process and can be initiated early in this process.

scholarly journals Witten Volume Formulas for Semi-Simple Lie Algebras

Integers ◽  
2011 ◽  
Vol 11 (4) ◽  
Author(s):  
Jianqiang Zhao
Keyword(s):  
Lie Algebras ◽  
Dedekind Sums ◽  
Combinatorial Method ◽  
Simple Lie Algebras ◽  
Higher Dimensional

AbstractIn this paper we provide an algebraic derivation of the explicit Witten volume formulas for a few semi-simple Lie algebras by combining a combinatorial method with the ideas used by Gunnells and Sczech in the computation of higher-dimensional Dedekind sums.

scholarly journals Darboux Families and the Classification of Real Four-Dimensional Indecomposable Coboundary Lie Bialgebras

Symmetry ◽  
2021 ◽  
Vol 13 (3) ◽  
pp. 465
Author(s):  
Javier de Lucas ◽  
Daniel Wysocki
Keyword(s):  
Vector Field ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Matrix Representations ◽  
Lie Bialgebras ◽  
Higher Dimensional ◽  
Trivial Center ◽  
Darboux Polynomial

This work introduces a new concept, the so-called Darboux family, which is employed to determine coboundary Lie bialgebras on real four-dimensional, indecomposable Lie algebras, as well as geometrically analysying, and classifying them up to Lie algebra automorphisms, in a relatively easy manner. The Darboux family notion can be considered as a generalisation of the Darboux polynomial for a vector field. The classification of r-matrices and solutions to classical Yang–Baxter equations for real four-dimensional indecomposable Lie algebras is also given in detail. Our methods can further be applied to general, even higher-dimensional, Lie algebras. As a byproduct, a method to obtain matrix representations of certain Lie algebras with a non-trivial center is developed.

EXPLICIT TRIGONOMETRIC YANG-BAXTERIZATION

1991 ◽  
Vol 06 (21) ◽  
pp. 3735-3779 ◽  
Author(s):  
MO-LIN GE ◽  
YONG-SHI WU ◽  
KANG XUE
Keyword(s):  
Quantum Group ◽  
Lie Algebras ◽  
Braid Group ◽  
Classical Limit ◽  
Group Representation ◽  
Direct Method ◽  
Explicit Solutions ◽  
Simple Lie Algebras ◽  
Higher Dimensional

We present an explicit prescription for trigonometric Yang-Baxterization. Given a braid group representation (BGR) in appropriate form, our prescription generates explicit solutions to the quantum Yang-Baxter equations (QYBE). We have proved the correctness of this prescription, together with a variation of it, for BGR’s having two or three unequal eigenvalues. All the known explicit results obtained by Jimbo and Bazhanov are reproduced. More YB solutions are obtained from “standard” BGR’s associated with fundamental and higher dimensional representations of simple Lie algebras. Our prescription also applies to the new “nonstandard” BGR’s, which are beyond the reach of present quantum group techniques but obtainable by our previous direct method. New explicit examples include the standard ones with 6 of SU(3), 10 of SU(5) and exotic ones with C2 and D2, etc. The classical limit of our YB solutions is also studied. It turns out that some of our exotic QYB solutions have an unusual classical limit. Our results suggest that the QYBE embodies a much richer structure than we thought.

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