Lie groups from a homotopy point of view

1974 ◽  
pp. 121-131 ◽  
Author(s):  
David Rector ◽  
James Stasheff
Keyword(s):  
Lie Groups ◽  
Point Of View

scholarly journals Twisted 6d (2, 0) SCFTs on a circle

2021 ◽  
Vol 2021 (7) ◽  
Author(s):  
Zhihao Duan ◽  
Kimyeong Lee ◽  
June Nahmgoong ◽  
Xin Wang
Keyword(s):  
Lie Groups ◽  
Point Of View ◽  
Partition Functions ◽  
Congruence Subgroups ◽  
Gauge Groups ◽  
Simple Lie Groups ◽  
Instanton Contribution ◽  
Elliptic Genera ◽  
Supersymmetric Gauge ◽  
The One

Abstract We study twisted circle compactification of 6d (2, 0) SCFTs to 5d $$ \mathcal{N} $$ N = 2 supersymmetric gauge theories with non-simply-laced gauge groups. We provide two complementary approaches towards the BPS partition functions, reflecting the 5d and 6d point of view respectively. The first is based on the blowup equations for the instanton partition function, from which in particular we determine explicitly the one-instanton contribution for all simple Lie groups. The second is based on the modular bootstrap program, and we propose a novel modular ansatz for the twisted elliptic genera that transform under the congruence subgroups Γ0(N) of SL(2, ℤ). We conjecture a vanishing bound for the refined Gopakumar-Vafa invariants of the genus one fibered Calabi-Yau threefolds, upon which one can determine the twisted elliptic genera recursively. We use our results to obtain the 6d Cardy formulas and find universal behaviour for all simple Lie groups. In addition, the Cardy formulas remain invariant under the twist once the normalization of the compact circle is taken into account.

scholarly journals Optimal control of affine connection control systems from the point of view of Lie algebroids

2014 ◽  
Vol 11 (09) ◽  
pp. 1450038 ◽  
Author(s):  
Lígia Abrunheiro ◽  
Margarida Camarinha
Keyword(s):  
Optimal Control ◽  
Vector Field ◽  
Control Systems ◽  
Lie Groups ◽  
Optimal Control Problems ◽  
Affine Connection ◽  
Point Of View ◽  
Lie Algebroids ◽  
Control Problems ◽  
Hamiltonian Vector Field

The purpose of this paper is to use the framework of Lie algebroids to study optimal control problems (OCPs) for affine connection control systems (ACCSs) on Lie groups. In this context, the equations for critical trajectories of the problem are geometrically characterized as a Hamiltonian vector field.

scholarly journals Contractions of Representations and Algebraic Families of Harish-Chandra Modules

2018 ◽  
Vol 2020 (11) ◽  
pp. 3494-3520 ◽  
Author(s):  
Joseph Bernstein ◽  
Nigel Higson ◽  
Eyal Subag
Keyword(s):  
Lie Groups ◽  
Unitary Group ◽  
Reductive Group ◽  
Mathematical Physics ◽  
Point Of View ◽  
Unitary Representations ◽  
Group Representations ◽  
Algebraic Point ◽  
Infinite Dimensional ◽  
Real Forms

Abstract We examine from an algebraic point of view some families of unitary group representations that arise in mathematical physics and are associated to contraction families of Lie groups. The contraction families of groups relate different real forms of a reductive group and are continuously parametrized, but the unitary representations are defined over a parameter subspace that includes both discrete and continuous parts. Both finite- and infinite-dimensional representations can occur, even within the same family. We shall study the simplest nontrivial examples and use the concepts of algebraic families of Harish-Chandra pairs and Harish-Chandra modules, introduced in a previous paper, together with the Jantzen filtration, to construct these families of unitary representations algebraically.

scholarly journals Controllability of affine right-invariant systems on solvable Lie groups

1997 ◽  
Vol Vol. 1 ◽  
Author(s):  
Yuri L. Sachkov
Keyword(s):  
Lie Groups ◽  
Lie Algebra ◽  
Lie Group ◽  
Point Of View ◽  
Theoretical Point ◽  
Solvable Lie Groups ◽  
International Audience ◽  
Systems On Lie Groups

International audience The aim of this paper is to present some recent results on controllability of right-invariant systems on Lie groups. From the Lie-theoretical point of view, we study conditions under which subsemigroups generated by half-planes in the Lie algebra of a Lie group coincide with the whole Lie group.

Discussion of “Geometric Algorithms for Robot Dynamics: A Tutorial Review” (F. C. Park, B. Kim, C. Jang, and J. Hong, 2018, ASME Appl. Mech. Rev., 70(1), p. 010803)

2018 ◽  
Vol 70 (1) ◽  
Author(s):  
Gregory S. Chirikjian
Keyword(s):  
Rigid Body ◽  
Lie Groups ◽  
Rigid Body Dynamics ◽  
Point Of View ◽  
Geometric Algorithms ◽  
Robot Kinematics ◽  
Robot Dynamics ◽  
Mathematical Tool ◽  
Theoretic Point ◽  
Physics Literature

Lie-theoretic methods provide an elegant way to formulate many problems in robotics, and the tutorial by Park et al. (2018, “Geometric Algorithms for Robot Dynamics: A Tutorial Review,” ASME Appl. Mech. Rev., 70(1), p. 010803) is simultaneously a complete and concise introduction to these methods as they pertain to robot dynamics. The central reason why Lie groups are a natural mathematical tool for robotics is that rigid-body motions and pose changes can be described as Lie groups, and allow phenomena including robot kinematics and dynamics to be formulated in elegant notation without introducing superfluous coordinates. The emphasis of the tutorial by Park et al. (2018, “Geometric Algorithms for Robot Dynamics: A Tutorial Review,” ASME Appl. Mech. Rev., 70(1), p. 010803) is robot dynamics from a Lie-theoretic point of view. Newton–Euler and Lagrangian formulation of robot dynamics algorithms with O(n) complexity were formulated more than 35 years ago using recurrence relations that use the serial structure of manipulator arms. This was done without using the knowledge of Lie theory. But issues such as why the ω× terms in rigid-body dynamics appear can be more easily understood in the context of this theory. The authors take great efforts to be understandable by nonexperts and present extensive references to the differential-geometric and Lie-group-centric formulations of manipulator dynamics. In the discussion presented here, connections are made to complementary methods that have been developed in other bodies of literature. This includes the multibody dynamics, geometric mechanics, spacecraft dynamics, and polymer physics literature, as well as robotics works that present non-Lie-theoretic formulations in the context of highly parallelizable algorithms.

scholarly journals On orbit spaces of representations of compact Lie groups

2014 ◽  
Vol 2014 (691) ◽  
Author(s):  
Claudio Gorodski ◽  
Alexander Lytchak
Keyword(s):  
Lie Groups ◽  
Metric Spaces ◽  
Point Of View ◽  
Irreducible Representations ◽  
Compact Lie Groups ◽  
Orbit Spaces ◽  
Quotient Spaces ◽  
Orthogonal Representations

Abstract.We investigate orthogonal representations of compact Lie groups from the point of view of their quotient spaces, considered as metric spaces. We study metric spaces which are simultaneously quotients of different representations and investigate properties of the corresponding representations. We obtain some structural results and apply them to study irreducible representations of small copolarity. As an important tool, we classify all irreducible representations of connected groups with cohomogeneity four or five.

Hom-Lie group and hom-Lie algebra from Lie group and Lie algebra perspective

2021 ◽  
pp. 2150068
Author(s):  
S. Merati ◽  
M. R. Farhangdoost
Keyword(s):  
Group Theory ◽  
Lie Groups ◽  
Lie Algebra ◽  
Lie Group ◽  
Group Structure ◽  
Point Of View ◽  
Lie Group Theory

A hom-Lie group structure is a smooth group-like multiplication on a manifold, where the structure is twisted by a isomorphism. The notion of hom-Lie group was introduced by Jiang et al. as integration of hom-Lie algebra. In this paper we want to study hom-Lie group and hom-Lie algebra from the Lie group’s point of view. We show that some of important hom-Lie group issues are equal to similar types in Lie groups and then many of these issues can be studied by Lie group theory.

scholarly journals Stable splittings, spaces of representations and almost commuting elements in Lie groups

2010 ◽  
Vol 149 (3) ◽  
pp. 455-490 ◽  
Author(s):  
ALEJANDRO ADEM ◽  
FREDERICK R. COHEN ◽  
JOSÉ MANUEL GÓMEZ
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Point Of View ◽  
Symmetric Products ◽  
Stable Splitting ◽  
Rank One ◽  
Finite Products

AbstractIn this paper the space of almost commuting elements in a Lie group is studied through a homotopical point of view. In particular a stable splitting after one suspension is derived for these spaces and their quotients under conjugation. A complete description for the stable factors appearing in this splitting is provided for compact connected Lie groups of rank one. By using symmetric products, the colimits Rep(ℤn, SU), Rep(ℤn, U) and Rep(ℤn, Sp) are explicitly described as finite products of Eilenberg–MacLane spaces.

The Geosciences Applied to Lunar Exploration

1962 ◽  
Vol 14 ◽  
pp. 169-257 ◽  
Author(s):  
J. Green
Keyword(s):  
Lunar Surface ◽  
Probable Mechanism ◽  
Point Of View ◽  
Lunar Exploration ◽  
Geological Model ◽  
Apply Geophysics ◽  
Surface Features ◽  
The Impact

The term geo-sciences has been used here to include the disciplines geology, geophysics and geochemistry. However, in order to apply geophysics and geochemistry effectively one must begin with a geological model. Therefore, the science of geology should be used as the basis for lunar exploration. From an astronomical point of view, a lunar terrain heavily impacted with meteors appears the more reasonable; although from a geological standpoint, volcanism seems the more probable mechanism. A surface liberally marked with volcanic features has been advocated by such geologists as Bülow, Dana, Suess, von Wolff, Shaler, Spurr, and Kuno. In this paper, both the impact and volcanic hypotheses are considered in the application of the geo-sciences to manned lunar exploration. However, more emphasis is placed on the volcanic, or more correctly the defluidization, hypothesis to account for lunar surface features.

Sign in / Sign up

Export Citation Format

Share Document